Singularly nonautonomous semilinear evolution equations with almost sectorial operators

Singularly nonautonomous semilinear evolution equations with almost sectorial operators (Ecuaciones de evoluci´ on semilineales singularmente no aut´ onomas con operadores casi sectoriales) Memoria escrita por Maykel Boldrin Belluzi para optar al grado de doctor del Programa Oficial de Doctorado “Matem´ aticas” en r´ egimen de cotutela con la Universidade Federal de S˜ ao Carlos - Brasil (UFSCar) VoBoLos Directores del Trabajo Tom´ as Caraballo Garrido Karina Schiabel (Universidad de Sevilla - Espa˜ na) (UFSCar - Brasil) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico Facultad de Matem´ aticas Universidad de Sevilla Agosto 2021 This work has been partially supported by Coordenac¸˜ ao de Aperfeic¸oamento de Pessoal de N´ ıvel Superior - Brasil (CAPES); by Conselho Nacional de Desenvolvimento Cient´ ıfico e Tecn´ ol´ ogico - Brasil (CNPq), grant N. 141512/2017-2; and by Fundac¸˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao Paulo - Brasil (FAPESP), grant N. 2017/09406-0 and N. 2017/17502-0. Acknowledgement The first ones I should express my gratitude towards are my parents: Marcos Antonio Belluzi and Renata Boldrin Belluzi. They have always put my education first and they were the ones that had to put up with my initial struggles with Mathematics and my reluctances in memorizing the multiplicative tables when I was a child. I wish I can somehow return all those incentives that both of you provided me. I also wish to thank my brother and favorite physicist, Marlon Boldrin Belluzi, for the support and admiration. Know that it is reciprocal and hopefully, in a non distant future, we can merge our works. To all the teachers that have collaborated with my educational journey, I want you to know that I will be eternally grateful. Especially to Cleide Maria, my first ever Mathematics teacher to encourage me to insist on Mathematics. To my advisor Karina Schiabel, thank you for giving me the opportunity to pursuit my career as a Mathematician. If it wasn’t for you, I would probably be in a different position and I would not be doing doing what I like the most. To my co-advisor Marcelo, thank you for the enlightening discussions and the time dedicated to this work. Hopefully there will be more collaborations in the future. To my advisor Tom´ as Caraballo, thank you for being so receptive during my stay in Sevilla. It was really a pleasure to meet you, work with you and learn from you. To my Ph.D fellows, thank you for the times we shared during this journey. It is such a pitty that the presential discussion had to be cut shorten, but I hope our paths cross again one more time in the future. A special gratitude towards Renata Oliveira Figueira. She is an incredible mathematician, an amazing person and an outstanding friend. I am also very grateful to each person that I met during my stay at Universidad de Sevilla. I enjoyed very much this period of my life and I will cherish it for a long time. Finally, I would like to express my gratitude to Coordenac¸˜ ao de Aperfeic¸oamento de Pessoal de N´ ıvel Superior (CAPES); to Conselho Nacional de Desenvolvimento Cient´ ıfico e Tecn´ ol´ ogico (CNPq), and to Fundac¸˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao Paulo (FAPESP) for the financial support (CAPES grant N. 141512/2017-2; FAPESP grant N. 2017/09406-0 and N. 2017/17502-0). i Abstract In this work we consider the singularly nonautonomous semilinear parabolic problem ut+A(t)u=F(u), t > τ, u(τ) = u0, in a Banach space X, where A(t), t ∈R, is a family of uniformly almost sectorial operators. The term singularly nonautonomous express the fact that the linear part of the equation, A(t) : D⊂X→X, is time-dependent and the almost sectoriality of the family A(t)comes from a deficiency in its resolvent estimate. For this semilinear problem in the abstract setting we study local well-posedness, regularity of the solution and the asymptotic dynamics of the problem. To illustrate the ideas developed for the abstract initial value problem, we consider a singularly nonautonomous reaction-diffusion equation in a domain with a handle. This type of domain consists in a subset of RN,Ω0= Ω ∪R0, where Ωis an open set of RNand R0is diffeomorphic to a subset (0,1) ⊂R. The “handle” refers to this line segment R0attached to Ω. In Ω0we consider the following reaction-diffusion equation              wt−div(a(t, x)∇w) + w=f(w), x ∈Ω, t > τ, ∂w ∂n = 0, x ∈∂Ω, vt−∂r(a(t, r)∂rv) + v=f(v), r ∈R0, t > τ, v(p0) = w(p0)and v(p1) = w(p1). This equation generates a singularly nonautonomous evolution equation with almost sectorial operator and local well-posedness, existence of strong solution and existence of pullback attractor are studied, in the lights of the abstract theory developed. In particular, in order to obtain existence of attractors, the system above will be decouple, originating two evolution equations: one with Neumann homogeneous boundary condition in Ωand another with nonhomogeneous and time-dependent Dirichlet boundary conditions in R0. The properties of those two decoupled equations are thoughtfully studied and from them, estimates on the pullback attractor are obtained. Key words: Singularly nonautonomous parabolic problems, almost sectorial operators, regularization, asymptotic dynamics, pullback attractor. iii Resumen En este trabajo consideramos el problema parab´ olico semilineal singularmente no aut´ onomo ut+A(t)u=F(u), t > τ, u(τ) = u0, en un espacio de Banach X, donde A(t), t ∈R, es una familia de operadores uniformemente casi sectoriales. El t´ ermino singularmente no aut´ onomo expresa el hecho de que la parte lineal de la ecuaci´ on, A(t) : D⊂X→X, es dependiente del tiempo y la casi sectorialidad de la familia A(t)proviene de una deficiencia en la estimaci´ on para la resolvente. Para este problema semilineal en el contexto abstracto, estudiamos el buen planteamento local de la ecuaci´ on, la regularidad de la soluci´ on y la din´ amica asint´ otica del problema. Para ilustrar las ideas desarrolladas para el problema con valor inicial abstracto, consideramos una ecuaci´ on de reacci´ on-difusi´ on singularmente no aut´ onoma en un dominio con una asa. Este dominio consiste en un subconjunto de RN,Ω0= Ω ∪R0, donde Ωes un conjunto abierto de RNyR0es difeom´ orfico a un subconjunto (0,1) ⊂R. La ”asa” se refiere a este segmento de l´ ınea R0unido a Ω. En Ω0consideramos la siguiente ecuaci´ on de reacci´ on-difusi´ on              wt−div(a(t, x)∇w) + w=f(w), x ∈Ω, t > τ, ∂w ∂n = 0, x ∈∂Ω, vt−∂r(a(t, r)∂rv) + v=f(v), r ∈R0, t > τ, v(p0) = w(p0)and v(p1) = w(p1). Esta ecuaci´ on genera una ecuaci´ on de evoluci´ on singularmente no aut´ onoma con operador casi sectorial y el buen plantemento local, la existencia de soluci´ on fuerte y existencia de atractores pullback son estudiados, a la luz de la teor´ ıa abstracta desarrollada. En particular, para obtener la existencia de atractores, el sistema anterior ser´ a desacoplado, originando dos ecuaciones de evoluci´ on: una con condici´ on de frontera de Neumann homog´ enea en Ωy otra con condiciones de frontera de Dirichlet no homog´ eneas y dependientes del tiempo en R0. Las propiedades de esas dos ecuaciones desacopladas son estudiadas y, a partir de ellas, se obtienen estimativas del atractor pullback. Key words: Problema parab´ olico semilineal singularmente no aut´ onomo, operadores casi sectoriales, regularizaci´ on, din´ amica asint´ otica, atractor pulback. v List of Figures 1Domain with a handle ..................................... 4 2 Dumbbelldomain...................................... 5 1.1 Sector Σϕand its contour Γ................................ 15 1.2 Diagramofembeddings................................... 26 2.1 Diagramofembeddings................................... 40 2.2 Maximal growth N= 3,δ=3 4............................... 43 2.3 Maximal growth N= 3,δ=1 4............................... 43 5.1 Graph of P(q)when N= 3 and δ=3 4........................... 88 5.2 Maximal growth N= 3,δ=3 4............................... 91 5.3 Maximal growth N= 3,δ=1 4............................... 91 6.1 Sectorial and positive operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Regularization and smoothing effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.1 Embeddings of Yθ q. ..................................... 104 7.2 Embeddings of Vθ q..................................... 121 xiii List of Tables 1.1 ConditionsofTheorem1.24 ................................ 31 2.1 Conditions of Proposition 2.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 ConditionsofTheorem4.27 ................................ 85 5.1 Lower bounds for q..................................... 90 xiv List of Symbols List of symbols used in the abstract theory Symbol Meaning Location αSectoriality of A(t)in X(P.2) δH¨ older exponent of A(t)(P.3) ρPolynomial growth of F(G) ωSectoriality of AY(t)(P.4) βCompatibility condition between A(t)and Y(P.4) Auxiliary constants Meaning Location ηH¨ older continuity of t7→ ϕ1(t, ·),t7→ Φ(t, ·)in L(X)Lemma 3.12 ωH¨ older continuity of t7→ ϕ1(t, ·),t7→ Φ(t, ·)in L(Y)Corollary 3.14 µH¨ older continuity in L(X)of h7→ T−A(τ)(t+h)Lemma 1.6 νH¨ older continuity in L(Y)of h7→ T−A(τ)(t+h)Lemma 3.4 xv List of symbols used for the application Symbol Meaning Location ΩOpen subset of RN R0Line segment Ω0Domain with a handle Ω0= Ω ∪R0 xVariable that takes values in Ω rVariable that takes values in R0 t, s, τ ∈RTime variables X=U0 pPhase space U0 p=Lp(Ω) ×Lp(0,1) Y=U0 qPhase space U0 q=Lq(Ω) ×Lq(0,1) α∈(0, α+) 0 < α < 1−N 2p:= α+(2.11) β∈(0, β+) 0 < β < 1−N 2q−1 21 q−1 p:= β+(2.14) ω∈(0, ω+) 0 < ω < 1−N 2q:= ω+(2.13) M,γdissipativeness condition on f(7.3) k0≥log2p k0positive integer such that L2k0,→Lp(7.12) xvi Introduction A great variety of phenomena that happens in nature and in complex systems relies on the theory of differential equations to be properly modeled and explained. Physics, chemistry, mechanics and several other applied sciences contribute to, and benefit from, the development of this branch of Mathematics. Inside this subject called differential equations, many subclassifications appear, depending on different features of the problem being studied and on which of those features one wishes to emphasize. In this work we focus on a specific class of partial differential equations named singularly nonautonomous semilinear evolution equations with almost sectorial operator. Each term that appears in this nomenclature refers to one characteristic of the problem, as we specify next. The abstract model for this equation is given by    ut+A(t)u=F(u), t > τ, u(τ) = u0∈X, (1) where Xis a Banach space, A(t),t∈R, is a family of closed linear operators defined on a fixed dense subspace Dof Xand Fis a nonlinearity defined in X. The term singularly nonautonomous is used to express the fact that the linear operator A(t)is time-dependent whereas the term semilinear designates the fact that Fdepends on the function u.Evolution equation refers to the fact that the equation models a state in the Banach space Xthat evolves as the time goes by, represented by utin the equation. Finally, almost sectorial operator express a certain property that the family A(t), t ∈R, possesses. As we discuss throughout this work, this almost sectoriality property allows us to define two families of linear operators in Xthat help the construction of solution (in an appropriate sense) for (1). Those families are called semigroup of growth 1−α, denoted by T−A(τ)(t), and linear process of growth 1−α, denoted by U(t, τ). Just as there are several types of differential equations, there are also many different features that we can choose to study whenever we are analyzing a problem like (1). To study its local well-posedness means to prove that the problem, for any initial time τand initial condition u0∈X, has a unique solution u=u(t)defined at least for a small interval of time and behaving continuously for small changes in the initial condition. In case there exists this local solution for the problem, we can study the regularity of the solution, which consists in analyzing the properties (concerning continuity, differentiability and integrability) that u=u(t)possesses. The problem (1) considered in this work has strong regularization properties for 1 2Introduction its solutions, even when the initial condition u0lacks regularity. It belongs to a class of problem called parabolic problems. It is interesting in many cases to study the global well-posedness for the equation, which is the existence of the solution at any time t > τ. This is specially interesting if we wish to understand the long-time behavior of the problem. The global well-posedness is usually hitched to finding estimates for the solutions as well as sets in the phase space Xthat has the property of attracting this solution to them. Those sets are called attractors and they are an important tool to study the asymptotic dynamics of the problem. This work is organized in a way to attend the topics mentioned above. In order to facilitate the comprehension of the topics presented and illustrate the ideas developed, we will intercalate abstract theory with an application of it to a singularly nonautonomous reaction-diffusion equation in domain with a handle. The presentation is structured in the following manner: Chapter P is dedicated to preliminaries results necessary to the development of the theory. We then divided the work in three parts and in each one of them we address a different aspect of the problem studied. Part I of this works deals with local well-posedness of singularly nonautonomous evolution equations with almost sectorial operators. It consists of two chapters. •Chapter 1: we present an abstract approach developed in [19] to obtain local well-posedness for (1). The treatment and notation we use in this part slightly diverge from the treatment provided in [19]. In there, the authors studied local solvability by understanding the relation that the nonlinearity F has with the domain of fractional powers of the operators A(t). We choose to pose the problem and the results in a different setting, which seemed more suitable to the application we had in mind. •Chapter 2: the reaction-diffusion equation in a domain with a handle is rigorously presented at this point. This application is a nonautonomous version of the autonomous problem presented in the series of articles [7, 8, 9]. The general properties of the problem are posed and the theory developed in Chapter 1 is applied to achieve local well-posedness for the problem. The definition of solution established in Part I lacks any property of differentiability so far. As a matter of fact, the local solution obtained there is what we call mild solution and might not be a solution in the “usual” sense (by usual we understand a function that is differentiable in time tand satisfies the equation u0(t) = −A(t)u+F(u),t > τ). Part II is dedicated to prove the differentiability property of the solution, as well as other results on improving the properties of u=u(t), which, as mentioned before, is called regularization. This part is organized in three chapters: •Chapter 3: is dedicated to obtain H¨ older continuity properties for several functions that feature in the problem studied. •Chapter 4: the results on regularization for abstract singularly nonautonomous evolution equations with almost sectorial operators are presented. The theory developed in this part complements the 3Introduction theory of regularization provided, for instance, by [25, 37] for the autonomous setting where Ais sectorial, by [31] for the autonomous case with almost sectorial operator and by [21, 55] for the nonautonomous case where the family A(t), t ∈R, is sectorial. To the best of our knowledge, there were no preview studies on regularization properties of local mild solution for the case considered in this work. •Chapter 5: this effect of regularization is studied at an example, a reaction-diffusion equation in a domain with a handle. The results in Part II were presented in two articles: [15] which deals with regularizing properties of semilinear autonomous equations with almost sectorial operators and [14] which concerns the singularly nonautonomous case. The content in the two first parts of this work prepares for the final one, Part III. This part deals with the global dynamics of the problem: •Chapter 6: the abstract theory on pullback attractors is briefly presented, following the approach adopted at [20] and some results on fractional powers of sectorial operators are also introduced. •Chapter 7: global well-posedness and existence of pullback attractor for the singularly nonautonomous reaction-diffusion equation are studied. To approach this problem, we will develop an iteration procedure (inspired in the ideas presented by Moser-Alikakos [2, 28, 29]) in order to obtain Lpand L∞estimates for the solution. Then, the smoothing effect of the equation (which is the property of improving the regularity of uand ut) allows us to extend those estimates to more regular spaces and finally obtain a compact set that pullback attracts. The new results established in Chapter 7 (alongside with local well-posedness established in Chapter 2) were presented in [12]. We also reserve a chapter to final considerations and further discussion. A particular topic discussed is how the theory developed in this work connects with the usual approach on parabolic problems (via fractional powers). There is also an Appendix Section (Appendix A) that deals with the smoothing effect that singularly nonautonomous differential parabolic equations has on the derivative utof the solution. We proved in this appendix that utalso belongs to regular spaces. The content of this appendix is crucial to the develop of Chapter 7 and also to treat the matter of classical solution and global existence for any other singularly nonautonomous parabolic equations. This content was presented in [13]. Before we advance to the theory, we would like to introduce, in a non rigorous exposition at first, the problem of reaction-diffusion equation in a domain with a handle that sets the tone of this work. Let Ω⊂RNbe a bounded smooth domain to which a line segment R0is attached at. We denote Ω0= Ω ∪R0and, in order to simplify the description of this set, we will assume that Ωis formed by two disjoint components, one at the left side of R0and one at the right side, while the line segment R0 4Introduction is given by R0={(r, 0) ∈R×RN−1;r∈(0,1)}. Furthermore, Ωand R0are connected by the points (0,0) ∈R×RN−1and (1,0) ∈R×RN−1, as illustrated in Figure 1. Ω Ω R0 Figure 1: Domain with a handle In this domain, we consider the coupled system of reaction-diffusion equation:              wt−div(a(t, x)∇w) + w=f(w), x ∈Ω, t > τ, ∂w ∂n = 0, x ∈∂Ω, vt−1 g∂r(g(r)a(t, r)∂rv) + v=f(v), r ∈R0, t > τ, v(p0) = w(p0)and v(p1) = w(p1), (2) where p0and p1are the end points of R0,a:R×Ω0→R+is a positive function that represents variations of the rate of diffusion at different points of Ω0and at different times t∈R. The function f denotes a nonlinear function defined in R. A brief glance at the equation allows us to point out some considerations: 1. First we note the presence of a function ginside the differential term in the second equation. This function gappears as a consequence of the way the problem is obtained. In physical terms, this problem could represent, for instance, two isolated tanks (the left and right side of the set Ω), at certain temperature distribution, to which we attach a cable connecting them. It might be of interest to determine how the heat flow occurs in this new domain, formed by to sets in R3(the two tanks) and the line segment, R0, represented by the cable. However, even though the cable has proportions much smaller than the tanks, it is actually a set in R3, not a line segment. The correct representation of the domain being studied should be given as in Figure 2 (and it is called dumbbell domain), where Rεrepresents the cable with diameter proportion of a small order ε. The limit case, where ε= 0 and the channel becomes the line R0, could be seen as an approximation of the problem. This limiting procedure gives rise to equation (2) and the function gappears as a consequence of the geometry of channel Rεand how it collapses to R0as ε→0+. 5Introduction Ω Ω Rε Figure 2: Dumbbell domain This type of analysis considering differential equations on thin domains, that is, domains in which the size in one direction is much smaller than the size in other directions, is an interesting topic and have been studied by several authors. Hale and Raugel [35, 36] made important contributions to the development of this field. The dumbbell domains belong to this class of thin domains and have also been studied by several authors, for instance Jimbo in a series of works [38, 39, 40, 41], Arrieta et. al. ([7, 8, 9]) in the articles that inspired this work and Carvalho et. al. [19] that considered a nonautonomous version of the problem earlier posed by Arrieta et. al. The studies on this type of constricted domain emerged in the literature as a counterpart of working with convex domains. One of the reasons to work with them is: if we consider an autonomous reaction-diffusion equation in a convex domain, stable equilibria for the equation are constant in the domain (see, for instance, [23, 45]). In order to obtain stable equilibria that are not spatially constant, we can not allow the domain to be convex. This is when the dumbbell domains appeared as prototype of nonconvex domains. Its constriction prevents the diffusion phenomena to take place properly, creating the possibility of existence of stable equilibrium that is not spatially constant. 2. The system (2) is a weakly coupled system, which means that the differential equations are coupled in only one direction. Indeed, the first equation given in terms of wis independent of the second equation (in terms of v), whereas the second one depends on the values assumed by wat the junction points. Returning to the example of the tanks and the cable, this would mean that the temperature distribution of the tanks is insensible to what happens in the cable. On the other hand, the temperature of the cable would be given in terms of the temperature of the tanks in the points where its extremes are attached. 3. The conditions at p0and p1only make sense if w∈ C(Ω). This implies restrictions on the space of function that we will pose the problem. Part I: Local well-posedness We consider the local well-posedness of the singularly nonautonomous evolution equation ut+A(t)u=F(u), t > 0, u(0) = u0∈X, (P.3) where the family of linear operators A(t), t ∈R, is time-dependent and uniformly almost sectorial, and Fis a nonlinearity defined in R×X. As we shall see during this part, almost sectorial operator fails to generate C0−semigroups. Nevertheless, it generates a special type of integrated semigroups, called semigroup of growth 1−α. The theory of semigroups of growth is presented in Chapter 1 as well as its application on solving autonomous evolution equations. To the family A(t), t ∈R, we also associate a two parameter family of linear operators called linear process of growth 1−α, that inherits several of the properties of the semigroup of growth 1−αand can be associated to the solution of a singularly nonautonomous evolution equation. The usual strategy to approach (P.3), adopted for instance in [19, 21, 55], consists in “breaking” the problem into less complex problems, in order to solve initially the simplest one and then use its solution to solve the more complex ones. Chapter 1 is structured accordingly to this partition and we fix the nomenclature that will appear throughout this entire work. 1. Autonomous linear evolution equation: in this case we fix τ∈Rand consider A(τ), one single operator of the family A(t), t ∈R. The operator A(τ)is almost sectorial and associated to it there exists a parabolic equation ut+A(τ)u= 0, t > 0; u(0) = u0∈X. (P.4) 2. Singularly nonautonomous and homogeneous linear evolution equation: we consider the equation with the nonautonomous linear operator ut+A(t)u= 0, t > τ;u(τ) = u0∈X. (P.5) 3. Singularly nonautonomous and nonhomogeneous linear evolution equation: this case differs from the previous one by a perturbation of the problem with a nonlinearity depending only on time t, that is, ut+A(t)u=G(t), τ < t < τ +T;u(τ) = u0∈X. (P.6) 12 13 Part I: Local well-posedness 4. Singularly nonautonomous semilinear evolution equation: this is the semilinear problem (1), where the nonlinearity depends on u: ut+A(t)u=F(u), τ < t < τ +T;u(τ) = u0∈X. (P.7) The local well-posedness for the semilinear abstract problem provided in Theorem 1.24 slightly differs from a version given in [19] and it is more suitable to the application we are dealing. This alternative result on local well-posedness was compiled in [12]. CHAPTER 1 Local well-posedness for singularly nonautonomous evolution equations Let Xbe a Banach space with norm given by k·kX. We denote by L(X)the set of all bounded linear operators S:X→X, which is also a Banach space with the norm kSkL(X)= supkxkX≤1kSxkX. If S(t)∈ L(X),t∈[0, T], is a one parameter family of linear operators, we say that the function [0, T]3t→S(t)is continuous in the uniform topology if the map [0, T]3t→S(t)∈ L(X)is continuous or, equivalently, S(·)∈ C([0, T ],L(X)). We say that [0, T ]3t→S(t)is strongly continuous if for each x∈X,[0, T]3t→S(t)x∈Xis continuous or, equivalently, S(·)x∈ C([0, T ], X). Continuity in the uniform topology implies strong continuity, but the reverse is not true. Also, uniform continuity and strong continuity could be stated for tin any interval in R(open or closed, finite or infinite) or any other metric space where the parameter lies. The same nomenclature also applies to functions in two parameters (t, s)7→ S(t, s)∈ L(X). To carry out the program proposed in Introduction, we assume that A(t), t ∈R, is a family of linear operators satisfying: (P.1) A(t) : D(A(t)) ⊂X→Xis closed, densely defined and D(A(t)) = D=X1, for all t∈R. (P.2) There exist constants ϕ∈π 2, π,C > 0and α∈(0,1), independent of t∈R, such that, if Σϕ represents the sector Σϕ:= {λ∈C;|argλ| ≤ ϕ},then Σϕ∪{0} ⊂ ρ(−A(t)) (see Figure 1.1) and  (λ+A(t))−1 L(X)≤C |λ|α,∀λ∈Σϕ.(1.1) We refer to this property as the family A(t), t ∈R, being uniformly almost sectorial or α−uniformly almost sectorial if we intend to emphasize the constant. We refer to αas the constant of almost sectoriality. 14 15 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations (P.3) There are constants C > 0and 0< δ ≤1such that, for any t, τ, s ∈R, k[A(t)−A(τ)]A−1(s)kL(X)≤C|t−τ|δ.(1.2) To express this fact we say that the function R3t7→ A(t)A−1(s)∈ L(X)is uniformly H¨ older continuous or δ−uniformly H¨ older continuous if we intend to emphasize the constant. We refer to δas the H¨ older exponent. Re Im Γ ϕ Σϕ⊂ρ(−A(t)) Figure 1.1: Sector Σϕand its contour Γ Some immediate consequences of the properties (P.1) - (P.3) are: if τ=sin (1.2), then  A(t)A(s)−1 L(X)≤1 + C|t−s|δ(1.3) and A(t)A(s)−1is a bounded linear operator in X. In future calculations, (t, s)will be located in a compact set K⊂R2and we will use (1.3) as kA(t)A(s)−1kL(X)≤C. From the fact that 0∈ρ(−A(t)) and from the continuity of the resolvent map ρ(−A(t)) 3λ7→ (λ+A(t))−1∈ L(X)in the uniform topology, the inequality (1.1) is equivalent to  (λ+A(t))−1 L(X)≤C 1 + |λ|α,∀λ∈Σϕ∪{0}. Still from (1.1) and the resolvent’s equality, we deduce  A(t)(λ+A(t))−1 L(X)≤1 + C|λ|1−α,∀λ∈Σϕ∪{0}.(1.4) Whenever we fix τ∈R, the linear operator A(τ)enjoys the properties (P.1) and (P.2) stated above. The constant α∈(0,1) that features in estimate (1.1) prevents us from concluding that −A(τ)generates aC0−semigroup, since Hille-Yosida’s necessary conditions are not fulfilled (see [52, Theorem 1.3.1]). However, this almost sectorial operator (we drop the “uniform” from the name when we fix one time τ∈R) generates a special type of semigroup, called semigroup of growth 1−α, which we introduce in the next section. 16 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations 1.1 Autonomous linear evolution equation Almost sectorial operators have a close connection with generation of semigroups of growth. These semigroups were first introduced by Da Prato in [26], where the growth considered was given by a positive integer n. Later on, this concept was generalized to semigroups of growth β, for any β > 0, and its properties were studied by several authors, like [50, 51, 56, 63]. The definition of this type of semigroup is presented in the sequel. Definition 1.1. [51, Definition 1.1] Let Xbe a Banach space and α∈(0,1). A family {T(t)∈ L(X) : t≥0}is a semigroup of growth 1−αif 1. T(0) = Iand T(t)T(s) = T(t+s), for all t, s > 0. 2. There exists M, γ > 0such that kt1−αT(t)kL(X)≤M, for all 0≤t≤γ. 3. If T(t)x= 0 for every t > 0, then x= 0. 4. X0=St>0T(t)[X]is dense X. The connection between almost sectorial operators and semigroups of growth 1−αwas then explored in several works. It was proved in [19] that, for a fixed τ∈R, the operator −A(τ)generates a family of linear operators T−A(τ)(t),t > 0, given by T−A(τ)(t) = 1 2πi ZΓ eλt(λ+A(τ))−1dλ, (1.5) where Γis the contour of Σϕ(see Figure 1.1), that is, Γ = {re−iϕ :r > 0}∪ {reiϕ :r > 0}and it is oriented with increasing imaginary part. This family satisfies the following properties. Proposition 1.2. If T−A(τ)(t), t > 0, is the family defined in (1.5), then: 1. Each operator T−A(τ)(t)is bounded and  T−A(τ)(t) L(X)≤Ctα−1,∀t > 0.(1.6) 2. T−A(τ)(t) : X→D,A(τ)T−A(τ)(t)is a bounded linear operator and  A(τ)T−A(τ)(t) L(X)≤Ctα−2,∀t > 0.(1.7) 3. There exists ξ > 0(independent of τ) such that T−A(τ)(t)has an exponential decay  T−A(τ)(t) L(X)≤Ctα−1e−ξt,∀t > 0.(1.8) 17 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Proof. We briefly mention the ideas used to obtain those estimates. For a detailed description of the proof, we recommend [8, Section 2]. Inequality (1.6) follows from the resolvent estimate (1.1). Indeed, parameterizing the part of Γwith positive imaginary part by λ=reiϕ,r∈[0,∞), where ϕis a fixed value in (π 2, π), and doing the analogous for the negative imaginary part, we have kT−A(τ)(t)kL(X)≤1 2π2Z∞ 0 ercos(ϕ)tC rαdr ≤tα−11 πZ∞ 0 ecos(ϕ)uC uαdu =Ctα−1. From (1.4), we obtain RΓeλtA(τ)(λ+A(τ))−1dλ converges and from the fact that A(τ)is closed, (1.7) follows. As for the exponencial decay in (1.8), since 0∈ρ(−A(·)) and the resolvent is an open set, we can slightly shift the sector Σϕto the left by a positive constant ξ > 0. In this case, ξI −A(τ)is also almost sectorial with kTξI−A(τ)(t)kL(X)≤Ctα−1, for a (possibly different) constant C. This implies that eξtkT−A(τ)(t)kL(X)=kTξI−A(τ)(t)kL(X)≤Ctα−1. Up to this point, we have been calling T−A(τ)(t), t > 0, just a family of bounded linear operators. The next result states that u(t) = T−A(τ)(t)u0is a classical solution of the autonomous problem ut+A(τ)u= 0, t > 0; u(0) = u0∈X, which allows us to conclude, from the uniqueness of the solution, that T−A(τ)(t)T−A(τ)(s) = T−A(τ)(t+s) for any t, s > 0. Since conditions 3and 4of Definition 1.1 are readily verified, we conclude that T−A(τ)(t) is a semigroup of growth 1−α. Lemma 1.3. ([8, Lemma 2.1 and Lemma 2.4]) Let T−A(τ)(t)be the linear operator defined in (1.5). The mapping T−A(τ)(t) : (0,∞)→ L(X)is differentiable and d dtT−A(τ)(t) = −A(τ)T−A(τ)(t) = 1 2πi ZΓ λeλt(λ+A(τ))−1dλ. That is, for u0∈X, d dtT−A(τ)(t)u0+A(τ)T−A(τ)(t)u0= 0,∀t > 0, and u(t) = T−A(τ)(t)u0is a classical solution of (P.4). As a consequence of the previous lemma, we obtain the following result. Proposition 1.4. For τ∈R, the family T−A(τ)(t), t > 0, defined in (1.5), satisfies: kA(τ)2T−A(τ)(t)kL(X)≤Ctα−3, t > 0.(1.9) 18 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Proof. It follows from Lemma 1.3 that A(τ)2T−A(τ)(t) = −A(τ)d dtT−A(τ)(t) = −1 2πiA(τ)ZΓ λeλt(λ+A(τ))−1dλ. If λ=reiϕ,r∈[0,∞), is the parametrization of the branch of Γwith positive imaginary part and λ=re−iϕ the parametrization of the negative branch, we obtain     1 2πi ZΓ λeλtA(τ)(λ+A(τ))−1dλ   L(X)≤CZ∞ 0 ercos(ϕ)trkA(τ)(λ+A(τ))−1kL(X)dr ≤CZ∞ 0 ercos(ϕ)tr2−αdr ≤CZ∞ 0 e−uu2−α t2−αcos2−α(ϕ) 1 cos(ϕ)tdu ≤Ctα−3Γ(3 −α) ≤Ctα−3, where we used the Gamma function defined in (P.1). Therefore, from the closedness of A(τ)added to the existence of the integral estimated above, A(τ)ZΓ λeλt(λ+A(τ))−1dλ =ZΓ λeλtA(τ)(λ+A(τ))−1dλ and (1.9) follows. The semigroups of growth 1−αare not necessarily continuous at t= 0 and their singular behavior distinguishes them from the usual C0−semigroups. However, for any initial condition in D, the continuity at t= 0 holds, as we will see next. Moreover, if x∈D2, then −A(τ)satisfies a property (item 3below) that resembles the definition of infinitesimal generator for C0−semigroups. Lemma 1.5. Let T−A(τ)(t),t > 0, be the semigroup of growth 1−αobtained by −A(τ). 1. If x∈D, then  T−A(τ)(t)x−x X→0when t→0+. 2. If x∈D, then A(τ)T−A(τ)(t)x=T−A(τ)(t)A(τ)x. 3. If x∈D2, then limt→0+ T−A(τ)(t)x−x t=−A(τ)x. 4. If x∈D2, then T−A(τ)(t)xis continuously differentiable in [0,∞)(including t= 0) and d dtT−A(τ)(t)x=   −A(τ)T−A(τ)(t)x, if t > 0, −A(τ)x, if t= 0. 5. Given any x∈Xand 0< s1< s2, T−A(τ)(s2)x−T−A(τ)(s1)x=−Zs2 s1 A(τ)T−A(τ)(s)xds. (1.10) If s1= 0, then equality holds only for x∈D2. 19 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Proof. First statement was proved in [8, Proposition 2.6]. For the second one, if x∈D, it follows from the closedness of A(τ)that A(τ)T−A(τ)(t)x=1 2πi ZΓ eλtA(τ)(λ+A(τ))−1xdλ =1 2πi ZΓ eλt(λ+A(τ))−1A(τ)xdλ =T−A(τ)(t)A(τ)x. The proof of the third statement is given in [8, Proposition 2.7]. We then use this information to prove the fourth statement. If x∈D2, d dtT−A(τ)(t)x=   −A(τ)T−A(τ)(t)x, t > 0, −A(τ)x, t = 0. The continuity for t > 0is already known. To prove the continuity at t= 0, we note that d dtT−A(τ)(t)x=−A(τ)T−A(τ)(t)x=−T−A(τ)(t)A(τ)x→ −A(τ)x, since x∈D2and A(τ)x∈D. Last statement follows from the fact that (0,∞)3t7→ d dt T−A(τ)(t)x=−A(τ)T−A(τ)(t)xis continuous. If x∈D2, then this map is continuous including at t= 0. The semigroup of growth 1−αalso presents a certain type of H¨ older continuity when we consider h7→ T−A(τ)(t+h), for t > 0. Lemma 1.6. Let T−A(τ)(t),t > 0, be the semigroup of growth 1−αobtained by −A(τ). Given any 0< µ < α2, for t, h > 0, we have kT−A(τ)(t+h)−T−A(τ)(t)kL(X)≤Chµtα−1−µ α,(1.11) and α−1−µ α∈(−1,0). Proof. Note that kT−A(τ)(t+h)x−T−A(τ)(t)xkX=kT−A(τ)(h)T−A(τ)(t)x−T−A(τ)(t)xkX =   Zh 0 d dξ T−A(τ)(ξ)T−A(τ)(t)xdξ   X≤Zh 0kT−A(τ)(ξ)kL(X)dξkA(τ)T−A(τ)(t)xkX ≤CZh 0 ξα−1dξtα−2kxkX=Chαtα−2kxkX. A positive exponent for happeared, but at the downside thas a power in the negative interval (−2,−1), which is not convenient when convergence of integrals is being considered. However, we already know that kT−A(τ)(t+h)−T−A(τ)(t)kL(X)≤(t+h)α−1+tα−1≤Ctα−1. 20 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations In order to improve the estimate, we interpolate the estimates already obtained kT−A(τ)(t+h)x−T−A(τ)(t)xkL(X)≤Chαtα−2 kT−A(τ)(t+h)x−T−A(τ)(t)xkL(X)≤Ctα−1, with coefficients µ αand (1 −µ α),0≤µ≤α, resulting in kT−A(τ)(t+h)−T−A(τ)(t)kL(X)≤Chµtα−1−µ α.(1.12) The exponent of twill be in the interval (−1,0) provided that 0< µ < α2. The results up to now explored the properties of the semigroup T−A(τ)(t)for τ∈Rfixed and we only required the uniformly sectoriality of the family A(t), t ∈R. This is enough to deal with the autonomous problem (P.4). However, to treat the singularly nonautonomous case (P.5), other estimates are necessary. For instance, we can transfer the H¨ older continuity of the family A(t),t∈R, to the semigroup generated by this family. Lemma 1.7. [19, Lemma 2.2] Let A(t), t ∈R,be a family satisfying (P.1),(P.2) and (P.3). Given t, s ∈R, we have kT−A(t)(τ)−T−A(s)(τ)kL(X)≤Cτ−2+2α(t−s)δ, τ > 0.(1.13) In other words, the function R3t7→ T−A(t)(·)is H¨ older continuous with exponent δ. This transference of the H¨ older continuity of the family A(t),t∈R, to the semigroup will play an essential role in Chapter 3. We will also see in this chapter that other families associated to A(t), t ∈R, inherit the H¨ older continuity of this family. In order to facilitate future calculations, we gather in the sequel properties of continuity for certain maps. The results on continuity in the uniform topology are proved in [19, Lemma 2.3 and Corollary 2.1], whereas the strong continuity of the functions below follows from the results and estimates already mentioned in this section. Proposition 1.8. [Continuity in the uniform topology] The following maps are continuous in the uniform topology: (0,∞)×R3(τ, s)7→ T−A(s)(τ)∈ L(X), (0,∞)×R×R3(τ, t, s)7→ A(t)T−A(s)(τ)∈ L(X), {(t, τ)∈R2;t > τ} 3 (t, τ)7→ T−A(τ)(t−τ)∈ L(X), {(t, τ)∈R2;t > τ} 3 (t, τ)7→ T−A(t)(t−τ)∈ L(X), {(t, τ)∈R2;t > τ} 3 (t, τ)7→ [A(τ)−A(t)]T−A(τ)(t−τ)∈ L(X), {(t, τ)∈R2;t > τ} 3 (t, τ)7→ [A(τ)−A(t)]T−A(t)(t−τ)∈ L(X). 21 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Note that all continuities pointed above avoid the initial time, as it is expected. We would only have continuity of the first map in time t= 0 at the uniform topology if A(t),t∈R, were a continuous family of bounded linear operator (and T−A(τ)(t)a uniformly continuous semigroup), for example. However, we can expect strong continuity at t= 0 provided that x∈D. Corollary 1.9. [Continuity in the strong topology] Let x∈D. The following maps are strongly continuous: [0,∞)×R3(τ, s)7→ T−A(s)(τ)x∈ L(X), [0,∞)×R×R3(τ, t, s)7→ A(t)T−A(s)(τ)x∈ L(X) {(t, τ)∈R2;t≥τ} 3 (t, τ)7→ T−A(τ)(t−τ)x∈ L(X), {(t, τ)∈R2;t≥τ} 3 (t, τ)7→ T−A(t)(t−τ)x∈ L(X), {(t, τ)∈R2;t≥τ} 3 (t, τ)7→ [A(τ)−A(t)]T−A(τ)(t−τ)x∈ L(X), {(t, τ)∈R2;t≥τ} 3 (t, τ)7→ [A(τ)−A(t)]T−A(t)(t−τ)x∈ L(X). Remark 1.10. For C0−semigroups, all continuities in Corollary 1.9 hold for any x∈X. 1.2 The nonautonomous linear associated problem Consider the singularly nonautonomous equation (P.5) and the associated family A(t), t ∈R. Unlike the previous case, in the present situation we search for a two parameter family of linear operator U(t, τ) that, in some sense, is connected to the solution of the evolution equation. Ideally, if u(t, τ, u0)is the local solution for (P.5), we would be searching for a family U(t, τ)such that U(t, τ)u0=u(t, τ, u0). This two parameter family of linear operators U(t, τ)replaces in the nonautonomous case the one parameter semigroup T−A(τ)(t)and it inspires the following definition: Definition 1.11. Let Xbe a Banach space and α∈(0,1). A family {U(t, s)∈ L(X); t>s}is a process of growth 1−αif 1. U(t, t) = Iand U(t, τ)U(τ, s) = U(t, s), for all s<τ<t. 2. There exists M > 0such that k(t−s)1−αU(t, s)kL(X)≤M, for all t>s. 3. (t, s, x)→U(t, s)xis continuous for t>sand for all x∈X. Kato in [42, 43, 44] was the first to prove the existence of this process {U(t, τ); t≥τ}associated to the family A(t), t ∈R. However, the family A(t), t ∈R,considered was a hyperbolic family of linear operators and U(t, τ)was obtained through an approximation procedure. The parabolic problem where each operator A(t)is sectorial (that is, the family A(t), t ∈Ris uniformly sectorial) was studied simultaneously by Sobolevskii in [55] and Tanabe in [59, 58, 60]. They proved existence of a family U(t, τ)associated to A(t)satisfying (P.1), (P.2) and (P.3). Both authors 28 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Proof. Item (1) follows from the resolvent estimates in L(Y, X)and L(Y), respectively. For the semigroups in item (2), all those estimates are obtained in the same way as they were in Proposition 1.2, when we considered k(λ+A(t))−1kL(X)≤C 1+|λ|α. For the linear family ϕ1(t, τ), it follows from its formulation in (1.15), that is, kϕ1(t, τ)kL(Y)≤ k[A(τ)−A(t)]A(τ)−1kL(Y)kT−A(τ)(t−τ)kL(Y)≤c(t−τ)δ+ω−2. For Φ(t, τ), note that condition ω+δ > 1is necessary and it replaces α+δ > 1in Theorem 1.13, as we see next: kΦ(t, τ)kL(Y)≤ kϕ1(t, τ)kL(Y)+Zt τkϕ1(t, s)kL(Y)kΦ(s, τ)kL(Y)ds ≤C(t−τ)ω+δ−2+Zt τ C(t−s)ω+δ−2kΦ(s, τ)kL(Y)ds, since ω+δ−2>−1, we can apply a generalized Gronwall inequality (see Lemma P.7) and obtain kΦ(t, τ)kL(Y)≤C(t−s)δ+ω−2. Finally, formulation for U(t, τ)given in (1.14), implies kU(t, τ)kL(Y,X)≤ kT−A(τ)(t−τ)kL(Y,X)+Zt τkT−A(s)(t−s)kL(Y,X)kΦ(s, τ)kL(Y)ds ≤C(t−τ)β−1+Zt τ C(t−s)β−1(s−τ)ω+δ−2ds ≤C(t−τ)β−1+C(t−τ)β−1+ω+δ−1B(β, ω +δ−1). By hypothesis ω+δ−1>0and, consequently, kU(t, τ)kL(Y,X)≤C(t−τ)β−1.The estimate for kU(t, τ)kL(Y)is identical to the estimate in kU(t, τ)kL(X)with αreplaced by ω. 1.3.2 Local mild solution The type of solution for the semilinear problem that we explore in this section is a generalized one, called mild solution, which we define next. Definition 1.22. A function u: (τ, τ +t0]→Xis a mild solution in the interval (τ, τ +t0]for (P.7) if 1. u(·)∈ C((τ, τ +t0], X)and limt→τ+ku(t)−U(t, τ)u0kX= 0. 2. u(·)is given by the variation of constants formula u(t) = U(t, τ)u0+Zt τ U(t, s)F(u(s))ds, ∀t∈(τ, τ +t0].(1.19) Remark 1.23. The condition limt→τ+ku(t)−U(t, τ)u0kX= 0 states that the mild solution replies the same type of discontinuity that the process U(t, τ), t > τ, might posses as t→τ+. 29 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Given τ∈Rand u0∈X, we will search for mild solutions in the following space K(t0, u0) = v∈C((τ, τ +t0], X); supt∈(τ,τ+t0]kv(t)−U(t, τ)u0kX≤µ, where µis a positive constant and t0will be suitably chosen later. K(t0, u0)is a Banach space with norm kξkK= supt∈(τ,τ+t0]kξ(t)−U(t, τ)u0kX.Note that (t−τ)1−αkv(t)kX≤(t−τ)1−αkv(t)−U(t, τ)u0kX+ (t−τ)1−αkU(t, τ)u0kX ≤t1−α 0µ+Cku0kX, and in case u0is in a bounded set B⊂X, we can obtain uniform estimates for (t−τ)1−αkv(t)kX, that is, (t−τ)1−αkv(t)kX≤k, where k:= t1−α 0µ+Csup u0∈Bku0kX.(1.20) The following theorem proves the existence of mild solution for the problem studied. It is similar to Theorem 3.1 in [19]. However, the authors there posed the problem in a different setting (using fractional power spaces, as mentioned in Remark 1.20). We rewrite the proof using the setting established in this section (with Xand Yspaces). This alternate version is presented in [12]. Theorem 1.24. Let X, Y be Banach spaces with X ,→Y. Suppose A(t), t ∈R, is α−uniformly sectorial and δ−uniformly H¨ older continuous. Additionally, we assume that A(t), t ∈Rsatisfies the condition in (P.4). If α+δ > 1,ω+δ > 1and F:X→Yis a nonlinearity satisfying (G) with 1≤ρ < β 1−α,(1.21) then, for every u0∈X, there exists t0>0, such that the initial value problem ut+A(t)u=F(u), t > τ;u(τ) = u0∈X, has a unique mild solution defined in (τ, τ +t0]. This t0depends on u0, but can be chosen uniformly for u0 in bounded sets of X. Furthermore, we can extend this mild solution to a maximal interval (τ, τM(u0)). Proof. The conditions α+δ > 1and ω+δ > 1guarantees the existence of the family U(t, τ)in L(X) and L(Y, X)(see Corollary 1.14 and Lemma 1.21). We can consider the operator (Tv)(t) := U(t, τ)u0+Zt τ U(t, s)F(u(s))ds, t ∈(τ, τ +t0], defined in K(t0, u0). From the continuity of {(t, τ)∈R2;t > τ} 3 (t, τ)7→ U(t, τ)∈ L(X)stated in Corollary 1.14, it follows readily that T:K(t0, u0)→C((τ, τ +t0], X). We prove that, for small values of t0,Tis a contraction in K(t0, u0). The Banach Fixed Point Theorem will ensure then the existence of a unique fixed point for Tin K(t0, u0). 30 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Given v∈K(t0, u0), in order for T v(·)to be in K(t0, u0), we must have kTv(t)−U(t, τ)u0kX≤µ, for all t∈(τ, τ +t0).Using the estimates available for the process, for the nonlinearity and for v∈ K(t0, u0)in (1.20), we have kTv(t)−U(t, τ)u0kX≤Zt τkU(t, s)F(v(s))kXds ≤Zt τkU(t, τ)kL(Y,X)kF(v(s))kYds ≤CZt τ (t−s)β−1(1 + kv(s)kρ X)ds ≤CZt τ (t−s)β−1(1 + kρ(s−τ)(α−1)ρ)ds ≤CZt τ (t−s)β−1ds +CkρZt τ (t−s)β−1(s−τ)(α−1)ρds ≤C(t−τ)β+C(t−τ)β−(1−α)ρB(β, 1−(1 −α)ρ). Condition (1.21) implies β−(1 −α)ρ > 0and also 1−(1 −α)ρ > 0.Therefore, kTv(t)−U(t, τ)u0kX≤C[tβ 0+tβ−(1−α)ρ 0]≤µ for t0small enough. Moreover, kTv(t)−U(t, τ)v0kX t→τ+ −−−→ 0. Also, kTv(t)−T w(t)kX≤Zt τkU(t, s)kL(Y,X)kF(v(s)) −F(w(s))kYds ≤Zt τ C(t−s)β−1kv(s)−w(s)kX(1 + kv(s)kρ−1 X+kw(s)kρ−1 X)ds ≤CZt τ (t−s)β−11+2kρ−1(s−τ)−(1−α)(ρ−1)ds kv−wkK ≤CZt τ (t−s)β−1+Zt τ (t−s)β−1(s−τ)−(1−α)(ρ−1)dskv−wkK ≤C(t−τ)β+ (t−τ)β−(1−α)(ρ−1)B(β, 1−(1 −α)(ρ−1))kv−wkK and from condition (1.21), we have ρ−1<β 1−α, which implies β−(1−α)(ρ−1) >0and, consequently, 1−(1 −α)(ρ−1) >0. Therefore, for t0sufficiently small and any t∈(τ, τ +t0), kTv(t)−T w(t)kX≤Chtβ 0+tβ−(1−α)(ρ−1) 0ikv−wkK≤1 2kv−wkK. If vis the fixed point of the contraction T, then it is a mild solution for the semilinear equation, since kv(t)−U(t, τ)v0kX t→τ+ −−−→ 0and v(t) = U(t, τ)u0+Zt τ U(t, s)F(v(s))ds. The uniformity of t0in bounded sets B⊂Xfollows from the fact that kin (1.20) can be chosen uniform in B. 31 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Note that condition (1.21) indicates that for any ρ < β 1−αthe problem can be solved, whereas for ρ=β 1−α, it is impossible to ensure convergence of the integrals that appears in the proof of the theorem. However, for the problem being studied, we have more than one condition that will impose restrictions on the growth of F. There are actually two sets of conditions required for the well-posedness to hold: 1. One related to the existence of the process U(t, τ)in L(X)and L(Y, X). Those are the conditions α+δ > 1and ω+δ > 1. 2. The other related to the blow-up at initial time t=τ:(t−τ)α−1.Fcannot have a growth ρso large that this blow-up becomes incontrollable. This is expressed by condition (1.21). We will distinguish them in order to emphasize where each condition is being required, as we see in Table 1.1. Local well-posedness I. Conditions on the existence of U(t, τ)II. Conditions to ensure existence of mild solution α+δ > 1(in L(X))1≤ρ < β 1−α ω+δ > 1(in L(Y, X)) Table 1.1: Conditions of Theorem 1.24 We prove next that the mild solution obtained depends continuously on initial data and satisfies an alternative blow-up. Theorem 1.25. Under the conditions of Theorem 1.24, for each u0∈X, the unique mild solution u: (τ, τM(u0)) →X, defined in its maximal interval, satisfies: 1. τM(u0)=+∞or lim supt→τM(u0)−ku(t)kX= +∞. 2. The solution is continuous with respect to the initial data in the following sense: given u0∈Xand δ > 0, there exists τ < t∗< τM(u0)sufficiently small, such that, for all v0∈BX[u0, δ], the mild solution v(t, τ, v0)associated to v0is defined in (τ, t∗]and ku(t, τ, u0)−v(t, τ, v0)kX≤C(t−τ)α−1ku0−v0kX, t ∈(τ, t∗]. Proof. To prove the first claim, suppose τM(u0)<∞. We show that lim supt→τM(u0)ku(t, τ, u0)kX= ∞.If that is not the case, then there exists M > 0such that lim supt→τM(u0)ku(t, τ, u0)kX≤M. From the continuity of uin (τ, τM(u0)), given any T∈(τ, τM(u0)),it follows that there exists K > 0such that ku(t, τ, u0)kX≤K, ∀t∈[T, τM(u0)) . 32 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations Let B⊂Xbe the closed ball of radius Kand centered in zero. From the uniformity in the time of existence of the mild solution obtained in Theorem 1.24, there exists t0>0such that, for every v0∈B, the problem vt+A(t)v=F(v), t > τ;v(τ) = v0, admits a unique mild solution in the interval (τ, τ +t0). ( τ [ T| ˜ t ) τM(u0) ) ˜ t+t0 | | θ Let ˜ t=tM(u0)−θ, with θ= min nt0 4,τM(u0)−T 4o. Then ˜ t∈[T, τM(u0)) and  u(˜ t) X≤K. By taking v0=u(˜ t), there exists v: (τ, τ +t0)→Xmild solution of vt+A(t)v=F(v)t > τ;v(τ) = u(˜ t). From the uniqueness of solution, u(t+˜ t, τ, u0) = v(τ+t),for all t∈(0, t0)and we can extend u to (τ, t0+˜ t), which is larger than (τ, τM(u0)), contradicting the maximality of the interval (τ, τM(u0)). Therefore, lim supt→τM(u0)ku(t, τ, u0)kX=∞. As for the second item, we prove that solutions u(t, τ, u0)and v(t, τ, v0)strating arbitrarily close has a behavior similar to U(t, τ)(u0−v0), that is, ku(t)−v(t)kXis similar to kU(t, τ)(u0−v0)k ≤ C(t−τ)α−1ku0−v0kX, at least for tclose to τ. Let (τ, τ +t0]be an interval in which both uand vare defined. From (1.20), there exists k > 0such that ku(t)kX≤k(t−τ)α−1and kv(t)kX≤k(t−τ)α−1, for all t∈(τ, τ +t0]. Let Ψ(t) = ku(t)−v(t)−U(t, τ)(u0−v0)kX then, for every t∈(τ, τ +t0], kΨ(t)kX=ku(t)−v(t)−U(t, τ)(u0−v0)kX ≤Zt τkU(t, s)kL(Y,X)kF(u(s)) −F(v(s))kYds ≤Zt τ C(t−s)β−1ku(s)−v(s)kX1 + ku(s)kρ−1 X+kv(s)kρ−1 Xds ≤Zt τ C(t−s)β−11 + ku(s)kρ−1 X+kv(s)kρ−1 Xku(s)−v(s)−U(s, τ)(u0−v0)kXds +Zt τ C(t−s)β−11 + ku(s)kρ−1 X+kv(s)kρ−1 XkU(s, τ)(u0−v0)kXds ≤CZt τ (t−s)β−11+2kρ−1(s−τ)−(1−α)(ρ−1)Ψ(s)ds +CZt τ (t−s)β−11+2kρ−1(s−τ)−(1−α)(ρ−1)(s−τ)−(1−α)dsku0−v0kX 33 Chapter 1. Local well-posedness for singularly nonautonomous evolution equations ≤CZt τ (t−s)β−1Ψ(s)ds +Zt τ (t−τ)β−1(s−τ)−(1−α)(ρ−1)Ψ(s)ds +C(t−τ)α+β−1B(β, α)+(t−τ)β−ρ(1−α)B(β, 1−ρ(1 −α))ku0−v0kX Since all the exponents of (t−τ)in the last line are positive, Ψ(t)≤Cku0−v0kX+CZt τ (t−s)β−1Ψ(s)ds +Zt τ (t−τ)β−1(s−τ)−(1−α)(ρ−1)Ψ(s)ds. The functions u(t)and v(t)are mild solution, therefore ku(t)−U(t, τ)u0kX t→τ+ −→ 0and kv(t)− U(t, τ)v0kX t→τ+ −→ 0. In this case, Ψ(t)is bounded in (τ, τ +t0]. For any t∈(τ, τ +t0],let Ψ∗(t) = sup s∈(τ,t]kΨ(s)kX. We have, Ψ(t)≤Cku0−v0kX+Cntβ 0+tβ−(1−α)(ρ−1) 0B(β, 1−(1 −α)(ρ−1)oΨ∗(τ+t0). Taking the supreme on the left side (for t∈(τ, τ +t0]) and choosing t0small enough so that Cntβ 0+tβ−(1−α)(ρ−1) 0B(β, 1−(1 −α)(ρ−1)o<1 2, we achieve 1 2Ψ∗(τ+t0]≤Cku0−v0kX which proves the continuous dependence. CHAPTER 2 Domains with a handle: Existence of mild solution In order to illustrate the ideas and results presented in Chapter 1, we consider the following example of a reaction-diffusion equation in a domain with a handle that will follow us through this entire work. Let Ω⊂RNbe a bounded smooth domain formed by two disjoint components: Ω=ΩL∪ΩR, ΩL∩ΩR=∅. Attached to this Ω, consider the line segment R0given by R0={(r, 0) ∈R×RN−1;r∈ (0,1)}. We assume that Ωand R0are connected by the points (0,0) ∈R×RN−1and (1,0) ∈R×RN−1 and that there exists a cylinder centered in the line segment R0that only intersects Ωin its bases, as illustrated in Figure 1. We denote Ω0= Ω ∪R0and in this domain we consider the following one-sided coupled reactiondiffusion equation:              wt−div(a(t, x)∇w) + w=f(w), x ∈Ω, t > τ, ∂w ∂n = 0, x ∈∂Ω, vt−∂r(a(t, r)∂rv) + v=f(v), r ∈R0, t > τ, v(p0) = w(p0)and v(p1) = w(p1), (2.1) where p0= (0,0, ..., 0) ∈RNand p1= (1,0, ..., 0) ∈RNare the junction points between the sets Ωand R0. We also assume that: (A.1). Ω⊂RNis a bounded domain with regular boundary (C2) formed by two smooth disjoint components: ΩLand ΩR, with p0∈∂ΩLand p1∈∂ΩR. (A.2). The function a:R×Ω0→R+is continuously differentiable, that is, a∈ C1(R×Ω0,R+)and has its image in a closed interval [a0, a1]⊂(0,∞). Furthermore, we denote by b(t, x) := ∇xa(t, x) 34 35 Chapter 2. Domains with a handle: Existence of mild solution the gradient function (in x) of a(t, x)and we assume that a0(t, x)and b(t, x)are both bounded, that is, a0(t, x), b(t, x)∈L∞(Ω0). (A.3). Both functions a(·,·)and b(·,·)are H¨ older continuous in the first variable with same H¨ older exponent δ∈(0,1]: |a(t, x)−a(s, x)| ≤ C|t−s|δ,|b(t, x)−b(s, x)| ≤ C|t−s|δ.(2.2) (A.4). The nonlinearity fis continuously differentiable, f∈ C1(R,R)and satisfies a polynomial growth condition, that is, |f0(ξ)| ≤ C(1 + |ξ|ρ−1),for some ρ≥1.(2.3) Remark 2.1. To avoid misunderstands, we will fix the variables used for each situation. We will save xfor the variable that takes values in Ω,rfor the variable that takes values in R0and t, s, τ ∈Rfor variables representing a given instant of time. Note that rhas the form (z, 0) ∈R×RN−1, with z∈[0,1]. As consequence, at some points, we will consider ras an element in the interval [0,1] and treat vr(t, r) as the derivative of vin the real variable r∈[0,1]. Remark 2.2. Equations (2) and (2.1) differ only by the presence of the function g. As mentioned at the Introduction, this function gcomes from the geometry of the sequence of channels Rεthat collapses to R0 as ε→0+. In order to simplify the calculations, we assume that g≡1. This would be the case if equation (2.1) was obtained as a limit case of reaction-diffusion equations ut−div(a(t, x)∇u) + u=f(u)in dumbbell domains Ω∪Rεwhere the channels Rεwere straight cylinders with diameter εand collapsed to R0as ε→0(see [7]). The phase space in which we consider this equation is given by U0 p=Lp(Ω) ×Lp(0,1) with norm k(w, v)kU0 p=kwkLp(Ω) +kvkLp(0,1). The space (U0 p,k · kU0 p)is Banach and (2.1) originates the following abstract singular semilinear evolution problem: (w, v)t+A0(t)(w, v) = F0(w, v), t > τ, (w, v)(τ) = (w0, v0)∈U0 p,(2.4) where A0(t) : D(A0(t)) ⊂U0 p→U0 pis the linear operator with time-independent domain D=D(A0(t)) given by D=(w, v)∈W2,p(Ω) ×W2,p(0,1) : ∂nw= 0 in ∂Ωand v(pi) = w(pi), i = 0,1,(2.5) A0(t)(w, v)=(−div(a(t, x)∇w) + w, −∂r(a(t, r)∂rv) + v),for (w, v)∈D, (2.6) and the nonlinearity F0is given by F0(w, v)(x) =    f(w(x)), x ∈Ω, f(v(r)), r ∈R0. (2.7) 36 Chapter 2. Domains with a handle: Existence of mild solution Remark 2.3. The condition imposed at p0and p1in (2.5) only makes sense if w∈C(Ω). Therefore, the restriction on p > N 2must be required at this point, which ensures that W2,p(Ω) ,→C(Ω) [1, Theorem 5.4]. Under those conditions, we have the following properties for the family A0(t), t ∈R. Lemma 2.4. Let A0(t) : D⊂U0 p→U0 pbe the family of linear operators defined in (2.6). If condition (2.2) holds, then R3t7→ A0(t)A0(τ)−1∈ L(U0 p)is H¨ older continuous with exponent δ, that is, k[A0(t)−A0(s)]A(τ)−1kL(U0 p)≤C|t−s|δ,for all τ, s, t ∈R. Proof. For (w, v)∈D, we have A0(t)(w, v)−A0(s)(w, v)=(−div ([a(t, x)−a(s, x)]∇w),−∂r([a(t, r)−a(t, s)]∂rv)) and ZΩ|div([a(t, x)−a(s, x)]∇w(x))|pdx =ZΩ|∇x([a(t, x)−a(s, x)])∇w(x)+[a(t, x)−a(s, x)]∆w|pdx ≤(t−s)δp ZΩ|∇xa(t, x)−∇xa(s, x)| |t−s|δp |∇w(x)|pdx + (t−s)δp ZΩ|a(t, x)−a(s, x)| |t−s|δp |∆w(x)|pdx ≤C(t−s)δp nk∇wkp Lp(Ω) +k∆wkp Lp(Ω)o ≤C(t−s)δpkwkp W2,p(Ω). The same calculation now on the line segment R0gives ZΩ|∂r([a(t, r)−a(s, r)]∂rv(r))|pdx ≤C(t−s)δpkvkp W2,p(0,1). Therefore, k[A0(t)−A0(s)](w, v)kp U0 p≤C|t−s|pδk(w, v)kp D,for all (w, v)∈D. Taking the p−th roots on both sides and replacing (w, v)by A0(τ)−1( ˜w, ˜v), we have k[A0(t)−A0(s)]A(τ)−1( ˜w, ˜v)kU0 p≤C|t−s|δk( ˜w, ˜v)kU0 p,∀( ˜w, ˜v)∈U0 p. In [19, Proposition 4.1] and [8, Proposition 3.1] several properties of the family A0(t), t ∈R, are presented, including its almost sectoriality. We enunciate it in the sequel and in the last statement we provide information about the spectrum of A0(t)that can be found in [8, Section 3.2]. Proposition 2.5. The family of linear operators A0(t), t ∈R,satisfies: 37 Chapter 2. Domains with a handle: Existence of mild solution 1. A0(t)is a closed linear operator and it has a fixed dense domain D. 2. A0(t)has compact resolvent and the semigroup T−A0(t)(s)is compact. 3. There exists ϕ∈π 2, πand C > 0(independent of t) such that Σϕ⊂ρ(−A0(t)), for all t∈R, and, for N 2< q ≤p,λ∈Σϕ∪{0}, we have k(λ+A0(t))−1kL(U0 q,U0 p)≤C |λ|β+ 1, for each 0< β < 1−N 2q−1 21 q−1 p. In particular, the case p=qyields k(λ+A0(t))−1kL(U0 p)≤C |λ|α+ 1, kA0(t)(λ+A0(t))−1kL(U0 p)≤C(1 + |λ|1−α), for 0< α < 1−N 2p<1. 4. The spectrum of A0(t)consists entirely of isolated eigenvalues, all of them positive and real. To be more precise, if {µi(t)}∞ i=1 are the eigenvalues of −div(a(t, x)∇w) + w=λw, x ∈Ω, ∂nw= 0, x ∈∂Ω, and {τi(t)}∞ i=1 the eigenvalues of −∂r(a(t, r)∂rv) + v=λv, s ∈(0,1), v(0) = 0 = v(1), then σ(A0(t)) = {µi(t)}∞ i=1 ∪{τi(t)}∞ i=1. They can be arranged in a way such that σ(A0(t)) = {λi(t) : 1 = λ1(t)≤λ2(t)≤... ≤λn(t)≤...}. Remark 2.6. The operator A0(t),t∈R, given in (2.6) differs from the operators considered in [8, 19]. In [8] the authors work with an autonomous version of the linear operator given by A0(w, v) = (−∆ + I, −d2 dr2+I), whereas [19] deals with a nonautonomous version A0(t)(w, v)=(−a(t, x)∆ + I, −a(t, r)d2 dr2+I). Despite the difference, the proof of each statement above is exactly the same as the one presented in [8], since it only depends on the sectoriality of the operator −∆ + Iin Ω, on the sectoriality of −d2 dr2+I (with Dirichlet boundary condition) in R0, and on Sobolev embeddings. Even though the linear operators on Ωand on R0are sectorial, the condition at p0and p1imposes restriction on the estimate of the resolvent that culminates with A0(w, v)=(−∆ + I, −d2 dr2+I)being almost sectorial. Part II: Regularization Up to now, we were able to established the existence of a two parameter family of linear operators U(t, τ)associated to the family A(t), t ∈R. Using this family, a generalized notion of solution for the semilinear problem was given in (1.19), called mild solution. However, the connection of the family U(t, τ)with equation (P.5) or the connection of the mild solution with the equation (P.7) is not yet established. We dedicate this part to prove that the solutions obtained so far recovers a classical idea of solution for a partial differential equation. The mainly inspiration for the topics studied here were the work of Sobolevskii [55] and the series of papers by Tanabe [58, 59, 60]. The classical work by Henry [37] was also an inspiration in many of the underlying ideas. The almost sectorial case and the discontinuity at the initial time t=τthat this case carries with it imply that some of the convergence arguments used in the works just mentioned fails to occur. In Chapter 4 we present a way to overcome this. The ideas and results proved in this part were organized in two articles: [15] which is a compiled of the results developed in Chapter 3 and 4 but focused on the existence of strong solution for the autonomous case with almost sectorial operators (which was an open problem) and [14] which concerns the singularly nonautonomous case and the differentiability properties associated to the linear process U(t, τ). 44 CHAPTER 3 H¨ older continuities The mild solution u(t, τ, u0):(τ, τM(u0)) →Xobtained in Theorem 1.24 satisfies the equation in the sense of Definition 1.22, that is, usatisfies the integral equation u(t) = U(t, τ)u0+Rt τU(t, s)F(u(s))ds. It is of interest to know whether or not this solution uis differentiable and satisfies the equation in the usual sense. In order to obtain this differentiability, the differential quotient must be evaluated, that is, u(t+h)−u(t) h=1 hnU(t+h, τ)u0−U(t, τ)u0+Rt+h τU(t+h, s)F(u(s))ds −Rt τU(t, s)F(u(s)dso. In this chapter we present properties of H¨ older continuity for several functions, with the purpose of using them in the next chapter to obtain the desired differentiability. We call the attention for one point: whereas the mild solution is placed in the phase space X, the problem ut+A(t)u=F(u), t > τ;u(τ) = u0, takes place in the less regular space Y, since F:X→Y. Therefore, we must analyze what happens to    u(t+h)−u(t) h  Yas h→0. This justify the results in this chapter, in which we present several of the estimates obtained in Chapter 1 for T−A(τ)(t),ϕ1(t, τ),Φ(t, τ)and U(t, τ)in L(X), now for the spaces L(Y, X)and L(Y). Some of those have already been introduced in Lemma 1.21. Remark 3.1. This type of analysis in which is necessary to obtain estimates for linear operators in L(Z1, Z2)is usual when we are dealing with scales of fractional power spaces. Usually, if A(t)is sectorial, positive and generates fractional powers spaces, one has to estimate ke−A(τ)(t)kL(Zθ,Zγ), as well as kU(t, τ)kL(Zθ,Zγ), in order to study the properties of the differential equation in which the family A(t)features (see for instance [19, 21]). Those fractional power spaces and the estimates L(Zθ, Zγ) would be the parallel to the spaces X,Yand the estimates L(X),L(Y)and L(Y, X)considered in this work (actually, this approach with Xand Yincorporates the case of fractional powers, as discussed in Chapter 8). 45 46 Chapter 3. H¨ older continuities 3.1 Estimates in L(Y, X)and L(Y) Since A(t), t ∈R, acting in Yis an almost sectorial operator, all the estimates for T−A(τ)(t),ϕ1(t, τ), Φ(t, τ)and U(t, τ)in the space L(Y)are the same as the estimates in L(X), with αreplaced by ω(and a possibly correction of the constant). However, in order to obtain estimates in L(Y, X)a careful analysis must be done. 3.1.1 H¨ older continuity of t7→ T−A(t)(·)in L(Y, X)and L(Y) We prove the result in Lemma 1.7 now for the spaces L(Y, X)and L(Y). Corollary 3.2. Let t, s ∈Rand τ > 0, then kT−A(t)(τ)−T−A(s)(τ)kL(Y,X)≤C(t−s)δτα+β−2,(3.1) kT−A(t)(τ)−T−A(s)(τ)kL(Y)≤C(t−s)δτ2ω−2.(3.2) Proof. From the resolvent equality (λ+A(t))−1−(λ+A(s))−1= (λ+A(t))−1(A(s)−A(t))(λ+A(s))−1 we obtain T−A(t)(τ)−T−A(s)(τ) = 1 2πi ZΓ eλτ [(λ+A(t))−1−(λ+A(s))−1]dλ =1 2πi ZΓ eλτ [(λ+A(t))−1(A(s)−A(t))(λ+A(s))−1]dλ. Parameterizing the branch of Γwith positive imaginary part by λ=reiϕ,r∈[0,∞), where ϕ∈ (π 2, π), and doing the analogous for the negative imaginary part, it follows from the estimates on item (1) of Lemma 1.21 that kT−A(t)(τ)−T−A(s)(τ)kL(Y,X) ≤CZΓ|eλτ |k(λ+A(t))−1kL(X)k[A(t)−A(s)]A(s)−1kL(X)kA(s)(λ+A(s))−1kL(Y,X)d|λ| ≤CZ∞ 0 erτ cos ϕ1 rα(t−s)δ(r1−β)dr ≤C(t−s)δτα+β−2Γ(2 −α−β) and in the last inequality we used the fact that cos ϕ∈(−1,0). Inequality (3.2) follows exactly as (1.13) with αreplaced by ω. Remark 3.3. The appearance of βand αat the same time in (3.1) is caused by the chaining of the two norms k[A(t)−A(s)]A(s)−1kL(X)and kA(s)(λ+A(s))−1kL(Y,X). 47 Chapter 3. H¨ older continuities 3.1.2 H¨ older continuity of h7→ T−A(τ)(t+h)in L(Y) In Lemma 1.6 it was established that, for τ∈R,t > 0and h > 0, given any 0< µ < α2, kT−A(τ)(t+h)−T−A(τ)(t)kL(X)≤Chµtα−1−µ α. By simply replacing αto ω, we can restate the result in L(Y). Lemma 3.4. Let T−A(τ)(t),t > 0, be the semigroup obtained by −A(τ). Given any 0< ν < ω2,we have kT−A(τ)(t+h)−T−A(τ)(t)kL(Y)≤Chνtω−1−ν ω, h > 0, and ω−1−ν ω∈(−1,0). Remark 3.5. In Lemmas 1.6 and 3.4, µand νare auxiliary constants that establish a range of possible estimates for the difference T−A(τ)(t+h)−T−A(τ)(t). They feature in several moments in this chapter and they play an essential role in Chapter 4, especially when studying the existence of regular solutions for the nonautonomous linear equation ut+A(t)u=G(t). There is a certain trade-off in choosing the value of those constants. For example, if µin Lemma 1.6 is close to α2, the exponent of H¨ older continuity in hµis close to its maximum, but that causes the exponent of tα−1−µ αto be more negative. On the other hand, if µis close to zero, the blow-up at t= 0 is more controlled, but we also have a smaller H¨ older exponent for h. The optimal choice for the value of µwill depend on other features of the problem being considered. 3.2 H¨ older continuity of t7→ F(u(t)) ∈Y Assume that all the conditions on Theorem 1.24 are satisfied, that is, A(t),t∈Ris α−uniformly almost sectorial, δ−uniformly H¨ older continuous, satisfies (P.4), α+δ > 1,ω+δ > 1and F:X→Y is a nonlinearity satisfying (G), with 1≤ρ < β 1−α. Then there exists a local mild solution u: (τ, τM(u0)) →Xfor the semilinear problem (P.7). Let (τ, τM(u0)) 3t7→ F(u(t)) ∈Y. We prove in this section that under certain conditions, this map is H¨ older continuous. Before we do that, we need the following technical result: Proposition 3.6. Let τ < t and h > 0. Then, for any 0≤η < ω +δ−1, we have kϕ1(t+h, τ +h)−ϕ1(t, τ)kL(Y)≤Chη(t−τ)ω+δ−2−η, kΦ(t+h, τ +h)−Φ(t, τ)kL(Y)≤Chη(t−τ)ω+δ−2−η. Proof. Note that ϕ1(t+h, τ +h)−ϕ1(t, τ) = [A(τ+h)−A(t+h)]T−A(τ+h)(t−τ)−[A(τ)−A(t)]T−A(τ)(t−τ) 48 Chapter 3. H¨ older continuities = [A(τ+h)−A(t+h)−A(τ) + A(t)]T−A(τ+h)(t−τ) + [A(τ)−A(t)][T−A(τ+h)(t−τ)−T−A(τ)(t−τ)] = [A(τ+h)−A(τ)]A(ξ)−1A(ξ)T−A(τ+h)(t−τ) + [A(t)−A(t+h)]A(ξ)−1A(ξ)T−A(τ+h)(t−τ) + [A(τ)−A(t)]A(ξ)−1A(ξ)[T−A(τ+h)(t−τ)−T−A(τ)(t−τ)]. The differences above are estimated in L(Y, X)by Chδ(t−τ)ω−2,Chδ(t−τ)ω−2and C(t−τ)δhδ(t− τ)2ω+δ−2, respectively, the last one follows from (3.2). Therefore, kϕ1(t+h, τ +h)−ϕ1(t, τ)kL(Y)≤Chδ(t−τ)ω−2. On the other hand, this difference can be estimated by kϕ1(t+h, τ +h)−ϕ1(t, τ)kL(Y)≤c(t−τ)ω+δ−2. Interpolating those two estimates with exponents η δand 1−η δ, for 0≤η < δ, we obtain kϕ1(t+h, τ +h)−ϕ1(t, τ)kL(Y)≤Chη(t−τ)ω+δ−2−η. The last assertion follows from kΦ(t+h, τ +h)−Φ(t, τ)kL(Y) ≤    ϕ1(t+h, τ +h)−ϕ1(t, τ)+Zt+h τ+h ϕ1(t+h, s)Φ(s, τ +h)ds −Zt τ ϕ1(t, s)Φ(s, τ)ds    L(Y) ≤Chη(t−τ)(δ−η)−1+   Zt τ [ϕ1(t+h, s +h)−ϕ(t, s)]Φ(s+h, τ +h)ds    L(Y) +   Zt τ ϕ1(t, s)[Φ(s+h, τ +h)−Φ(s, τ)]ds    L(Y) ≤Chη(t−τ)ω+δ−2−η+CZt τ hη(t−s)ω+δ−2−η(s−τ)ω+δ−2ds +CZt τ (t−s)ω+δ−2kΦ(s+h, τ +h)−Φ(s+h, τ +h)kL(Y)ds ≤Chη(t−τ)ω+δ−2−η+CZt τ (t−s)ω+δ−2kΦ(s+h, τ +h)−Φ(s, τ)kL(Y)ds. Applying Gronwall’s inequality (Lemma P.7), we obtain kΦ(t+h, τ +h)−Φ(t, τ)kL(X)≤Chη(t−τ)ω+δ−2−η. 49 Chapter 3. H¨ older continuities With those properties for the families ϕ1(t, τ)and Φ(t, τ)and the estimates obtained earlier, the H¨ older continuity of t7→ F(u(t)) follows, and this property is essential to derive the regularity of the solution. However the almost sectoriality, given by αand ωcauses several restrictions on the values of ρ in terms of α, β, ω. This is a consequence of the initial blow-up (t−τ)α−1that the solution has, which is amplified by the growth of F. Proposition 3.7. Assume that conditions of Theorem 1.24 are satisfied and let u: (τ, τM(u0)) →Xbe the local mild solution obtained for the semilinear problem given by u(t) = U(t, τ)u0+Rt τU(t, s)F(u(s)), t ∈(τ, τM(u0)). If 0<µ<min{α2, ω +δ−1}and 1≤ρ < min{β 1−α+ 1,δ 1−α,α−µ α(1−α)}, then kF(u(t+h)) −F(u(t))kY≤Chmin{µ,1−ρ(1−α)}(t−τ)min{−µ α,δ−1,β−α}(t−τ)−ρ(1−α), for any t>τand h > 0. Moreover, the exponents for (t−τ)given by min −µ α−ρ(1 −α), δ −1−ρ(1 −α), β −α−ρ(1 −α), belong to (−1,∞). Proof. From the growth condition on Fwe obtain kF(u(t+h)) −F(u(t))kY≤Cku(t+h)−u(t)kX1 + ku(t)kρ−1 X+ku(t+h)kρ−1 X. We already know from (1.20) that there exists a constant ksuch that, ku(t+h)kX,ku(t)kX≤k(t−τ)α−1. Taking this into account, we obtain kF(u(t+h)) −F(u(t))kY≤Cku(t+h)−u(t)kX(t−τ)−(ρ−1)(1−α) (t−τ)(ρ−1)(1−α)kF(u(t+h)) −F(u(t))kY≤Cku(t+h)−u(t)kX.(3.3) Let Ψ(t) = (t−τ)(ρ−1)(1−α)kF(u(t+h)) −F(u(t))kY. Inequality (3.3) is rewritten as Ψ(t)≤Cku(t+h)−u(t)kX.(3.4) We study in the sequel properties of the difference ku(t+h)−u(t)kXin order to obtain the desired result. Before we attend to it, let us point out that kF(u(t))kYcan be locally estimated from the local estimate we have for u(t): kF(u(t))kY≤C(1 + ku(t)kρ X)≤C(1 + (t−τ)−ρ(1−α))≤C(t−τ)−ρ(1−α). 50 Chapter 3. H¨ older continuities From the variation of constant formula, we obtain u(t+h)−u(t) = U(t+h, τ)u0−U(t, τ)u0+Zτ+h τ +Zt+h τ+hU(t+h, s)F(u(s))ds −Zt τ U(t, s)F(u(s))ds = [U(t+h, τ)−U(t, τ)]u0+Zτ+h τ U(t+h, s)F(u(s))ds +Zt τ [U(t+h, s +h)F(u(s+h)) −U(t, s)F(u(s))]ds =I1+I2+I3, (see Remark 3.9 below for a discussion over the way we grouped the terms above). We estimate each of the terms above in k·kX. From Proposition 1.17 and for any 0< µ < α2, we obtain kU(t+h, τ)u0−U(t, τ)u0kX≤Chµh(t−τ)α−1−µ α+ (t−τ)α+δ−2i. The second term follows the same idea,    Zτ+h τ U(t+h, s)F(u(s))ds   X≤Zτ+h τkU(t+h, s)kL(Y,X)kF(u(s))kYds ≤CZτ+h τ (t+h−s)β−1(s−τ)−ρ(1−α)ds ≤Ch1−ρ(1−α)(t−τ)β−1. The last term requires more reasoning, as we see in the sequel. To perform the necessary calculations, we use formulation (1.14) for the process: I3=Zt τ U(t+h, s +h)F(u(s+h)) −U(t, s)F(u(s))ds =Zt τT−A(s+h)(t−s)F(u(s+h)) −T−A(s)(t−s)F(u(s))ds +Zt τZt+h s+h T−A(ξ)(t+h−ξ)Φ(ξ, s +h)F(u(s+h))dξds −Zt τZt s T−A(ξ)(t−ξ)Φ(ξ, s)F(u(s))dξds. Adding and subtracting T−A(s)(t−s)F(u(s+h)) inside the integral on the first term and performing a simple change of variable in the second integral, we obtain I3=Zt τ[T−A(s+h)(t−s)−T−A(s)(t−s)]F(u(s+h))ds 51 Chapter 3. H¨ older continuities +Zt τT−A(s)(t−s)[F(u(s+h)) −F(u(s))]ds +Zt τZt s T−A(ξ+h)(t−ξ)Φ(ξ+h, s +h)F(u((s+h))dξds −Zt τZt s T−A(ξ)(t−ξ)Φ(ξ, s)F(u(s))dξds =Zt τ[T−A(s+h)(t−s)−T−A(s)(t−s)]F(u(s+h))ds +Zt τT−A(s)(t−s)[F(u(s+h)) −F(u(s))]ds +Zt τZt s [T−A(ξ+h)(t−ξ)−T−A(ξ)(t−ξ)]Φ(ξ+h, s +h)F(u(s+h))dξds +Zt τZt s T−A(ξ)(t−ξ)[Φ(ξ+h, s +h)−Φ(ξ, s)]F(u(s+h))dξds +Zt τZt s T−A(ξ)(t−ξ)Φ(ξ, s)[F(u(s)) −F(u(s+h))]dξds =S1+S2+S3+S4+S5. We estimate each item separately due to their particularity. Initially, recall that 1≤ρ < β 1−α, which implies α+β−1>0. From (3.1), we obtain kS1kX≤Zt τkT−A(s+h)(t−s)−T−A(s)(t−s)kL(Y,X)kF(u(s+h))kYds CZt τ hδ(t−s)α+β−2(s−τ)−ρ(1−α)ds ≤Chδ(t−τ)(α−1)+(β−ρ(1−α)). For S2, using item (2) of Lemma 1.21, we have kS2kX≤Zt τkT−A(s)(t−s)kL(Y,X)kF(u(s+h)) −F(u(s))kYds ≤CZt τ (t−s)β−1kF(u(s+h)) −F(u(s))kYds ≤CZt τ (t−s)β−1(s−τ)−(ρ−1)(1−α)Ψ(s)ds. Term S3also follows from (3.1) and condition α+β−1>0(as in S1), as well as the estimate for Φ(·,·)given in item (3) of Lemma 1.21 kS3kX≤Zt τZt sk[T−A(ξ+h)(t−ξ)−T−A(ξ)(t−ξ)kL(Y,X)kΦ(ξ+h, s +h)kL(Y)kF(u(s+h))kYdξds ≤CZt τZt s hδ(t−ξ)α+β−2(ξ−s)ω+δ−2dξ(s−τ)−ρ(1−α)ds 52 Chapter 3. H¨ older continuities ≤CZt τ hδ(t−s)(α+β−1)+(ω+δ−1)−1(s−τ)−ρ(1−α)ds ≤Chδ(t−τ)(α+β−1)+(ω+δ−1)−ρ(1−α). From Proposition 3.6, for any 0≤η < ω +δ−1, we obtain kS4kX≤Zt τZt skT−A(ξ)(t−ξ)kL(Y,X)k[Φ(ξ+h, s +h)−Φ(ξ, s)]kL(Y)kF(u(s+h))kYdξds ≤CZt τZt s (t−ξ)β−1hη(ξ−s)ω+δ−2−ηdξ(s−τ)−ρ(1−α)ds ≤ChηZt τ (t−s)β+(ω+δ−2−η)(s−τ)−ρ(1−α)ds ≤Chη(t−τ)(ω+δ−1−η)+(β−ρ(1−α)). For the last term, note that items (2) and (3) of Lemma 1.21 imply kS5kX≤Zt τZt skT−A(ξ)(t−ξ)kL(Y,X)kΦ(ξ, s)kL(Y)k[F(u(s)) −F(u(s+h))]kYdξds ≤CZt τZt s (t−ξ)β−1(ξ−s)ω+δ−2dξ(s−τ)−(ρ−1)(1−α)Ψ(s)ds ≤CZt τ (t−s)(β+ω+δ−1)−1(s−τ)−(ρ−1)(1−α)Ψ(s)ds. Using the above estimates and selecting the terms with smallest exponents for hand the most negative for (t−τ), we obtain kI3kX≤Chδ(t−τ)(α−1)+(β−ρ(1−α)) +Chδ(t−τ)(α−1)+(β−ρ(1−α))+(ω+δ−1) +Chη(t−τ)(ω+δ−1−η)+(β−ρ(1−α)) +CZt τ (t−s)β−1(s−τ)−(ρ−1)(1−α)Ψ(s)ds +CZt τ (t−s)β−1+(ω+δ−1)(s−τ)−(ρ−1)(1−α)Ψ(s)ds ≤Chη(t−τ)(α−1)+(β−ρ(1−α)) +CZt τ (t−s)β−1(s−τ)−(ρ−1)(1−α)Ψ(s)ds. From I1,I2and I3, we conclude that, for any 0< µ < α2and 0< η < ω +δ−1, ku(t+h)−u(t)kX≤Chµh(t−τ)α−1−µ α+ (t−τ)α+δ−2i +Ch1−ρ(1−α)(t−τ)β−1 +Chη(t−τ)(α−1)+(β−ρ(1−α)) +CZt τ (t−s)β−1(s−τ)−(ρ−1)(1−α)Ψ(s)ds 53 Chapter 3. H¨ older continuities ≤Chmin{µ,1−ρ(1−α),η}(t−τ)min{α−1−µ α,α+δ−2,β−1} +CZt τ (t−s)β−1(s−τ)−(ρ−1)(1−α)Ψ(s)ds. and we used above α+δ−2< α −1+(β−ρ(1 −α)) to group the terms. Indeed, α+δ−2 = α−1+(δ−1) < α −1+(β−ρ(1 −α)), since δ−1<0< β −ρ(1 −α). Therefore, replacing the estimate obtained in (3.4), we obtain Ψ(t)≤Chmin{µ,1−ρ(1−α),η}(t−τ)min{α−1−µ α,α+δ−2,β−1}+CZt τ (t−s)β−1(s−τ)−(ρ−1)(1−α)Ψ(s)ds. In order to apply the generalized version of Gronwall inequality given in Lemma P.8, the following inequalities must hold: α−µ α−(ρ−1)(1 −α)>0⇒ρ < α−µ α(1 −α), α+δ−1−(ρ−1)(1 −α)>0⇒ρ < δ 1−α, β−(ρ−1)(1 −α)>0⇒ρ < β 1−α+ 1. In this case, Ψ(t)≤Chmin{µ,1−ρ(1−α),η}(t−τ)min{α−1−µ α,α+δ−2,β−1}, kF(u(t+h)) −F(u(t))kY≤Chmin{µ,1−ρ(1−α),η}(t−τ)min{α−1−µ α,α+δ−2,β−1}(t−τ)−(ρ−1)(1−α), kF(u(t+h)) −F(u(t))kY≤Chmin{µ,1−ρ(1−α),η}(t−τ)min{−µ α,δ−1,β−α}(t−τ)−ρ(1−α), and we can group the exponents µand ηtogether, requiring that 0<µ<min{α2, ω +δ−1}. Remark 3.8. If A(t)were sectorial, then α=ω= 1, all the conditions above would be trivially satisfied and t7→ F(u(t)) would be µ−H¨ older continuous for any 0≤µ < min{12,1 + δ−1}=δ. This agrees with the results on [55, (2.84)] Remark 3.9. The way we separate the integral Rt+h τ=Rτ+h τ+Rt+h τ+hdiffers from the way we did in the previous results, as Rt+h τ=Rt τ+Rt+h t. This is done in order to force the difference F(u(t+h))−F(u(t)) to appear, which is useful later at the moment we apply Gronwall’s inequality. The same procedure is used for the nonsingular and sectorial case (A(t) = A), but in that case the analysis is much simpler once U(t+h, s +h) = T−A(t+h−s−h) = T−A(t−s). The existence of strong solution for the autonomous semilinear equations with almost sectorial is compiled and presented it in [15]. 60 Chapter 4. Regularization: Abstract theory This definition of strong Y−solution acknowledges that we might have discontinuity at the initial time t=τand to prove that the problem has a strong Y−solution, we have to understand the connection of U(t, τ)to the equation ut+A(t)u= 0. This is how this chapter is structured: 1. In the first section we prove that the family U(t, s)associated to A(t), t ∈R, is strongly differentiable in L(X)and satisfies ∂tU(t, s) = −A(t)U(t, s). In this case, U(t, τ)recovers the solution of the homogeneous equation if we define u(t) = U(t, τ)u0and ut=−A(t)u(t). 2. We then study in the second section the nonhomogeneous (but still linear) evolution equation ut= A(t)u+G(t), τ < t < τ +Tin the space X, whose solution is given by u(t) = U(t, s)u0+Zt τ U(t, s)G(s)ds, t ∈(τ, τ +T]. We prove that this function u(·)is actually a strong solution for the problem, under certain conditions on the perturbation G(t)and on the constant αof almost sectoriality and exponent δof H¨ older continuity of the family A(t),t∈R. 3. Since our goal is to solve the semilinear equation in Y, we translate the results obtained in these two preceding sections for the equation in Y. 4. In the last section we prove that the mild solution for the semilinear case (P.7) is a strong Y−solution for the problem. As we will see during this chapter, the estimates obtained in Chapter 3 are of extreme importance. The auxiliary constant ηintroduced in Section 3.3 plays an essential role in the differentiability of U(t, τ) in L(X)(in the same sense, ϑplays an essential role in the differentiability of U(t, τ)in the space L(Y)), whereas the auxiliary constant µintroduced in Section 3.1.2 controls the magnitude of the discontinuity at the initial time t=τand is connected to the H¨ older continuity properties of t7→ F(u(t)). This causes in the problem the same type of behavior that we observed in Chapter 1 and Chapter 2: there is a set of conditions accountable of ensuring differentiability of the process, and on the other hand, a set of conditions that ensures the discontinuity at t=τis under control. Remark 4.3. To obtain the differentiability results in this chapter we will only evaluate the right-side derivative of the functions being studied. If the right-side derivative is continuous (which will be the case) Lemma P.9 allow us to conclude the continuous differentiability. This results in this chapter were presented in [14]. 4.1 Differentiability of U(t, τ)in L(X) We start this section enunciating the main theorem of this chapter, which states the differentiability of the process. We call the attention for the fact that U(t, τ)being differentiable does not depend on the 61 Chapter 4. Regularization: Abstract theory nonlinearity F:X→Y. Therefore, only the constants αand δfeature in the conditions established to ensure differentiability of U(t, τ)in L(X). In the same sense, only ω(constant of almost sectoriality in Y) and δshows up in the conditions to ensure differentiability of U(t, τ)in L(Y). We present all the results and prove them in space L(X). During the proofs, note that they can all be translated to L(Y), by simply switching αto ωand Dto DY, since both of operators A(t) : D⊂X→X and AY(t) : DY⊂Y→Yare almost sectorial. Theorem 4.4. Let A(t), t ∈R, be a family of linear operators in Xsatisfying (P.1) - (P.3), α∈(0,1) the constant of almost sectoriality and δ∈(0,1] the exponent of H¨ older continuity. 1. If α+δ > 1, then there exists a unique linear process of growth 1−α,U(t, τ), associated to the family A(t), t ∈R. This process satisfies kU(t, τ)kL(X)≤C(t−τ)α−1. 2. In addition, if α2+αδ −1>0∗, then (a) U(t, τ) : X→D, for any τ < t. (b) {(t, τ)∈R2;τ < t} 3 (t, τ)7→ U(t, τ)∈ L(X)is strongly differentiable, that is, for each x∈X,{(t, τ)∈R2;τ < t} 3 (t, τ)7→ U(t, τ)x∈Xis differentiable. (c) The derivative ∂tU(t, τ)is a bounded linear operator, strongly continuous in {(t, τ)∈R2;τ < t}and satisfies: ∂tU(t, τ) + A(t)U(t, τ) = 0,∀t > τ, (4.1) k∂tU(t, τ)kL(X)=kA(t)U(t, τ)kL(X)≤C(t−τ)α−2,∀t > τ, (4.2)  A(t)U(t, τ)A(τ)−1 L(X)≤C(t−τ)α−1,∀t > τ. (4.3) Remark 4.5. For the sectorial case (α= 1), any δ > 0ensures that α2+αδ −1>0and the differentiability of U(t, τ)holds. This agrees with the classical results on the sectorial case presented in [55, Section 1.5]. Remark 4.6. Sometimes the inequality α+δ 2−1>0will be necessary, but this follows readily from α2+αδ −1>0. Indeed α > √δ2+4 2−δ 2>1−δ 2. The first statement has already been proved in Corollary 1.14. The second one allows us to conclude that u(t) = U(t, τ)u0is a strong solution for the evolution equation ut+A(t)u= 0,t > 0,u(τ) = u0 and U(t, τ)is indeed a linear process of growth 1−α(see Remark 1.15). To demonstrate the second statement, we use the following strategy: Given any γ > 0and t0> τ +γ, it is enough to prove the strong differentiability of U(t, τ)for t∈[τ+γ, t0]. From the arbitrariness of γ and t0, the result will follow. ∗This restriction appears as a consequence of the conditions required on Lemma 3.15 and Lemma 4.9. 62 Chapter 4. Regularization: Abstract theory Therefore, given u0∈X, consider U(t, τ)u0=T−A(τ)(t−τ)u0+Zt τ T−A(s)(t−s)Φ(s, τ)u0, t ∈[τ+γ, t0]. If we tried to evaluate the derivative of U(t, τ)u0directly from the expression above, we would face a problem of convergence in the integral, since the expected value for the derivative inside the integral would be −A(s)T−A(s)(t−s)Φ(s, τ)u0and kA(s)T−A(s)(t−s)Φ(s, τ)kL(X)≤C(t−s)α−2(s−τ)α+δ−1. To overcome this problem, we consider the auxiliary family of bounded linear operators {Uρ(t, τ); t∈ [τ+γ, t0]}given by Uρ(t, τ) = T−A(τ)(t−τ) + Zt−ρ τ T−A(s)(t−s)Φ(s, τ), t ∈[τ+γ, t0], where ρ > 0is small enough so that t−ρ>τ+γ. This slightly retreat in the domain of integration implies that the integrand is continuously differentiable in (τ, t −ρ]and from Lemma P.6 we have [τ+γ, t0]3 t7→ Uρ(t, τ)u0∈Xis continuously differentiable, with derivative given by d dtUρ(t, τ)u0=−A(τ)T−A(τ)(t−τ)v0+T−A(t−ρ)(ρ)Φ(t−ρ, τ)v0 +Zt−ρ τ−A(s)T−A(s)(t−s)Φ(s, τ)v0ds. (4.4) We prove in the sequel the following: (1) Uρ(·, τ)u0converges as ρ→0to U(·, τ)u0in C([τ+γ, t0], X). (2) d dt Uρ(·, τ)u0converges as ρ→0to −A(·)U(·, τ)u0in C([τ+γ, t0], X). Then, differentiability of t7→ U(t, τ)u0for t∈[τ+γ, t0]follows from the fact that C1([τ+γ, t0], X) is a complete metric space. Moreover, d dt U(·, τ)u0=−A(·)U(·, τ)u0. Item (1) is easily obtained: for each t∈[τ+γ, t0]we have kUρ(t, τ)−U(t, τ)kL(X)=   Zt t−ρ T−A(s)(t−s)Φ(s, τ)ds   L(X)≤Zt t−ρ C(t−s)α−1(s−τ)α+δ−2ds ≤C(t−ρ−τ)α+δ−2Zt t−ρ (t−s)α−1ds ≤C(γ−ρ)α+δ−2ραρ→0 −→ 0. Item (2), on the other hand, is a more delicate matter. Ideally, we would like to rearrange the expression (4.4) for ∂tUρ(t, τ)in a way that becomes visible its convergence to −A(t)U(t, τ)u0=−A(t)T−A(τ)(t−τ)u0−A(t)Zt τ T−A(s)(t−s)Φ(s, τ)u0ds =−A(t)T−A(τ)(t−τ)u0−A(t)Zt τ T−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds −A(t)Zt τ T−A(s)(t−s)Φ(t, τ)u0ds. (4.5) 63 Chapter 4. Regularization: Abstract theory However, the expression above might not make sense, since it is not proved yet that U(t, τ) : X→D or that the integrals on the right side belong to D. Nonetheless, we will use it as a target of what we wish to achieve when we make ρ→0in the expression of ∂tUρ(t, τ). From first to second line in (4.5) we added and subtracted A(t)Rt τT−A(s)(t−s)Φ(t, τ)ds in order to obtain the difference [Φ(s, τ)−Φ(t, τ)], where we can use its H¨ older continuity in the first variable (Lemma 3.12) to study its convergence. We will rearrange (4.4) in a form that it approximates the most from the expression on the right side of our idealized equality (4.5). Lemma 4.7. The function [τ+γ, t0]3t7→ ∂tUρ(t, τ)can also be given as ∂tUρ(t, τ)u0=−A(t)T−A(τ)(t−τ)u0−Zt−ρ τ A(t)T−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds −Zt−ρ τ A(t)T−A(s)(t−s)Φ(t, τ)u0ds +Zt t−ρ ϕ1(t, s)Φ(s, τ)u0ds + [T−A(t−ρ)(ρ)−I]Φ(t, τ)u0 +T−A(t−ρ)(ρ)[Φ(t−ρ, τ)−Φ(t, τ)]u0. (4.6) Proof. Rearranging (4.4) and taking into account the expressions (1.15) and (1.16) for ϕ1(t, τ)and Φ(t, τ), respectively, we have: ∂tUρ(t, τ) = −A(τ)T−A(τ)(t−τ)u0+Zt−ρ τ−A(s)T−A(s)(t−s)Φ(s, τ)u0ds +T−A(t−ρ)(ρ)Φ(t−ρ, τ)u0 =−A(t)T−A(τ)(t−τ)u0+ [A(t)−A(τ)]T−A(τ)(t−τ)u0 +Zt−ρ τ [A(t)−A(s)]T−A(s)(t−s)Φ(s, τ)u0ds −Zt−ρ τ A(t)T−A(s)(t−s)Φ(s, τ)u0ds +T−A(t−ρ)(ρ)Φ(t−ρ, τ)u0 =−A(t)T−A(τ)(t−τ)u0−ϕ1(t, τ)u0−Zt−ρ τ ϕ1(t, s)Φ(s, τ)u0ds −Zt−ρ τ A(t)T−A(s)T−A(s)(t−s)Φ(s, τ)u0ds +T−A(t−ρ)(ρ)Φ(t−ρ, τ)u0 =−A(t)T−A(τ)(t−τ)u0−Zt−ρ τ A(t)T−A(s)(t−s)Φ(s, τ)u0ds −ϕ1(t, τ)u0 −Zt τ ϕ1(t, s)Φ(s, τ)u0ds +Zt t−ρ ϕ1(t, s)Φ(s, τ)u0ds +T−A(t−ρ)(ρ)Φ(t−ρ, τ)u0 =−A(t)T−A(τ)(t−τ)u0−Zt−ρ τ A(t)T−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds −Zt−ρ τ A(t)T−A(s)(t−s)Φ(t, τ)u0ds +Zt t−ρ ϕ1(t, s)Φ(s, τ)u0ds +T−A(t−ρ)(ρ)[Φ(t−ρ, τ)−Φ(t, τ)]u0 64 Chapter 4. Regularization: Abstract theory + [T−A(t−ρ)(ρ)−I]Φ(t, τ)u0. Remark 4.8. For the sectorial case, the terms in the third and fourth line of equality (4.6) vanish as ρ→0(see [52, 55]). The same can not be said when A(t),t∈R, is almost sectorial. The first line of (4.6) is already suited to our purpose and converges to the first line of the right side in equality (4.5) as we can see in the next lemma. Lemma 4.9. Assume that the constants αand δsatisfy the inequality α2+αδ −1>0. In this case, the integral Rt τT−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds belongs to Dand A(t)Rt−ρ τT−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds ρ→0 −→ A(t)Rt τT−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds, uniformly for t∈[τ+γ, t0]in the norm of X. Proof. If we prove that Rt τA(t)T−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]ds converges, then the result follows from Corollary P.4. From Lemma 3.15, there exists 0< η < α(α+δ−1) such that    Zt τ A(t)T−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds   L(X) ≤CZt τ (t−s)α−2(t−s)η(s−τ)α+δ−2−η αds =C(t−τ)(α+η−1)+(α+δ−1−η α)−1B(α+η−1, α +η−1−η α) and the entries on the function Bare positive, provided that 1−α < η < α2+αδ −α. The existence of a suitable ηrelies on the constants αand δto satisfy α2+αδ −α > 1−α, that is, α2+αδ −1>0. Remark 4.10. Until Lemma 4.9 we only had upper bounds for η(see Lemma 3.15). Now we must have 1−α < η < α(α+δ−1),and the existence of such ηhappens only if α2+αδ −1>0. Note that this is the condition that features in Theorem 4.4 and it represents a restriction on the possible values that αcan assume. For instance, if δ=3 4, this condition only holds for α > 1 8−1 + √73≈ 0.69. If δ=1 4, then α > 1 8−3 + √73≈0.88. The lesser δis, the harder it is to obtain differentiability for the process. For the remaining terms in (4.6), we will adopt a different strategy. Rather than evaluating what happens to them as ρ→0, we first study the existence of A(t)Zt τ T−A(s)(t−s)xds 65 Chapter 4. Regularization: Abstract theory for an arbitrary x∈X, and then we relate the outcome of this analysis to the remaining terms of (4.6). If T(t)is a C0−semigroup with inifnitesimal generator A, an important feature of T(t)is the fact that given any x∈X,Rt 0T(s)xds ∈D(A)and AZt 0 T(s)xds=T(t)x−x. The next results prove that Rt τT−A(s)(t−s)xds ∈D, for any x∈X, when A(t),t∈R, is almost sectorial, and a characterization of A(t)Rt τT−A(s)(t−s)xdsthat extends the one we have for C0−semigroups is obtained. Lemma 4.11. Let α2+αδ −1>0and consider the linear operator H(t, τ) : D2→X,t>τ, given by H(t, τ)w=A(t)Rt τT−A(s)(t−s)wds. Then H(t, τ)is a well defined operator, it is bounded in D2, satisfies kH(t, τ)wkX≤C(t−τ)α−1kwkX,∀w∈D2, and admits a bounded extension to X. Proof. The fact that H(t, τ)is well defined in D2follows from Corollary P.4 and the estimate    Zt τ A(t)T−A(s)(t−s)wds   X =   Zt τ T−A(s)(t−s)A(t)wds   X ≤CZt τ (t−s)α−1ds kA(t)wkX<∞. We prove in the sequel that there exists a constant C > 0such that, for all w∈D2,kH(t, τ)kX≤ C(t−τ)α−1kwkX.In Lemma 1.5 we proved that for any y∈D2, the function t7→ T−A(τ)(t)yis continuously differentiable in [0,∞)and A(t)Zt τ T−A(t)(t−s)yds =Zt τ A(t)T−A(t)(t−s)yds =Zt τ d ds T−A(t)(t−s)yds =y−T−A(t)(t−τ)y. (4.7) Also, the function t7→ T−A(t)(t−s−u)T−A(s)(u)wis continuously differentiable in [0, t −s]and d du T−A(t)(t−s−u)T−A(s)(u)w=T−A(t)(t−s−u)[A(t)−A(s)]T−A(s)(u)w. (4.8) Therefore, (4.8), a change of variable and Fubini’s Theorem ([34, Theorem 2.39]) imply H(t, τ)w=A(t)Zt τ T−A(s)(t−s)wds 66 Chapter 4. Regularization: Abstract theory =A(t)Zt τ T−A(t)(t−s)wds +A(t)Zt τ [T−A(s)(t−s)−T−A(t)(t−s)]wds (4.8) =A(t)Zt τ T−A(t)(t−s)wds +A(t)Zt τZt−s 0 T−A(t)(t−s−u)[A(t)−A(s)]T−A(s)(u)wduds =A(t)Zt τ T−A(t)(t−s)wds +A(t)Zt τZt s T−A(t)(t−ξ)[A(t)−A(s)]T−A(s)(ξ−s)wdξds =A(t)Zt τ T−A(t)(t−s)wds +A(t)Zt τZξ τ T−A(t)(t−ξ)[A(t)−A(s)]T−A(s)(ξ−s)wdsdξ =A(t)Zt τ T−A(t)(t−s)wds +A(t)Zt τ T−A(t)(t−ξ)Zξ τ [A(t)−A(s)]T−A(s)(ξ−s)wdsdξ =A(t)Zt τ T−A(t)(t−s)wds −A(t)Zt τ T−A(t)(t−ξ)Zt τ [A(s)−A(t)]T−A(s)(t−s)wdsdξ +A(t)Zt τ T−A(t)(t−ξ)Zt τ [A(s)−A(t)]T−A(s)(t−s)wdsdξ +A(t)Zt τ T−A(t)(t−ξ)Zξ τ [A(t)−A(s)]T−A(s)(ξ−s)wdsdξ. Note that in the last equality for H(t, τ)w, the first two terms are in the form A(t)Rt τT−A(t)(t−s)yds, where y∈D2. We know how to handle these expressions using (4.7). Returning to the expression H(t, τ)w, H(t, τ)w=w−T−A(t)(t−τ)w−Zt τ ϕ1(t, s)wds +T−A(t)(t−τ)Zt τ ϕ1(t, s)wds +A(t)Zt τ T−A(t)(t−ξ)Zt τ [A(s)−A(t)]T−A(s)(t−s)wdsdξ +A(t)Zt τ T−A(t)(t−ξ)Zξ τ [A(t)−A(ξ)]T−A(s)(ξ−s)wdsdξ +A(t)Zt τ T−A(t)(t−ξ)Zξ τ [A(ξ)−A(s)]T−A(s)(ξ−s)wdsdξ =w−T−A(t)(t−τ)w−Zt τ ϕ1(t, s)wds +T−A(t)(t−τ)Zt τ ϕ1(t, s)wds +A(t)Zt τ T−A(t)(t−ξ)Zt τ ϕ1(t, s)wdsdξ +A(t)Zt τ T−A(t)(t−ξ)[A(t)−A(ξ)]A(ξ)−1A(ξ)Zξ τ T−A(s)(ξ−s)wdsdξ +A(t)Zt τ T−A(t)(t−ξ)Zξ τ−ϕ(ξ, s)wdsdξ = [I−T−A(t)(t−τ)] w−Zt τ ϕ1(t, s)wds +A(t)Zt τ T−A(t)(t−ξ)Zξ τ ϕ1(t, s)wdsdξ 67 Chapter 4. Regularization: Abstract theory +A(t)Zt τ T−A(t)(t−ξ)Zt ξ ϕ1(t, s)wdsdξ −A(t)Zt τ T−A(t)(t−ξ)Zξ τ ϕ1(ξ, s)wdsdξ +A(t)Zt τ T−A(t)(t−ξ)[A(t)−A(ξ)](A(ξ))−1H(ξ, τ)wdξ = [I−T−A(t)(t−τ)] w−Zt τ ϕ1(t, s)wds +A(t)Zt τ T−A(t)(t−ξ)Zt ξ ϕ1(t, s)wds +Zξ τ [ϕ1(t, s)w−ϕ1(ξ, s)w]dsdξ +A(t)Zt τ T−A(t)(t−ξ)[A(t)−A(ξ)](A(ξ))−1H(ξ, τ)wdξ. Using the estimates (1.6) for the semigroup, (1.2) for the H¨ older continuity of A(t), (1.15) for the operators ϕ1(·,·)and (3.5) for the H¨ older continuity of t7→ ϕ1(t, ·), we have kH(t, τ)wkX≤C1+(t−τ)α−11 + Zt τ (t−s)α+δ−2dskwkX +CZt τ (t−ξ)α−2Zt ξ (t−s)α+δ−2ds +Zξ τ (t−ξ)η(ξ−s)α+δ−2−η αdsdξkwkX +Zt τ (t−ξ)α+δ−2kH(ξ, τ)wkXds ≤C1+(t−τ)α−11+(t−τ)α+δ−1kwkX +CZt τ (t−ξ)(α−2)+(α+δ−1)dξ+(t−τ)(α+η−1)+(α+δ−1−η α)B(α+η−1, α +δ−η α)kwkX +Zt τ (t−ξ)α+δ−2kH(ξ, τ)wkXds ≤C(t−τ)α−1kwkX +Ch(t−τ)2α+δ−2+ (t−τ)(α+η−1)+(α+δ−1−η α)B(α+η−1, α +δ−η α)ikwkX +Zt τ (t−ξ)α+δ−2kH(ξ, τ)wkXds ≤C(t−τ)α−1kwkX+Zt τ (t−ξ)α+δ−2kH(ξ, τ)wkXds. The arguments used in the above estimates only hold provided that 2α+δ−2>0and 1−α < η < α2+αδ −α, that is, α > 1−δ 2and α2+αδ > 1. The second one is more restrictive (see Remark 4.6). Finally, applying the generalized version of Gronwall inequality (see Lemma P.7) we have, for w∈D2, kH(t, τ)wkX≤C(t−τ)α−1kwkX. 68 Chapter 4. Regularization: Abstract theory Therefore, H(t, τ)can be extended to a bounded linear operator in X, which we denote the same. The fact that H(t, τ)is bounded allows us to prove the following result. Lemma 4.12. Let α2+αδ −1>0and w∈X. Then Rt τT−A(s)(t−s)wds belongs to Dand we can obtain an expression for A(t)Rt τT−A(s)(t−s)wds: for any 0< ρ < t −τ, A(t)Zt τ T−A(s)(t−s)wds =w−T−A(t−ρ)(ρ)w−Zt t−ρ ϕ1(t, s)wds+A(t)Zt−ρ τ T−A(s)(t−s)wds. (4.9) Furthermore, A(t)Rt τT−A(s)(t−s)ds is a bounded linear operator satisfying     A(t)Zt τ T−A(s)(t−s)ds   L(X)≤C(t−τ)α−1.(4.10) Proof. Let (wn)be a sequence in D2such that wn→w. Since Rt τT−A(s)(t−s)ds is a bounded linear operator in X, it follows that Rt τT−A(s)(t−s)wnds →Rt τT−A(s)(t−s)wds. The extension H(t, τ)is also a bounded linear operator and A(t)Zt τ T−A(s)(t−s)wnds =H(t, τ)wn→ H(t, τ)w. From the closedness of A(t), we derive that Rt τT−A(s)(t−s)wds ∈D(A(·)) and A(t)Zt τ T−A(s)(t−s)wds = lim n→∞A(t)Zt τ T−A(s)(t−s)wnds = lim n→∞A(t)Zt−ρ τ T−A(s)(t−s)wnds +A(t)Zt t−ρ T−A(s)(t−s)wnds =A(t)Zt−ρ τ T−A(s)(t−s)wds + lim n→∞Zt t−ρ A(s)T−A(s)(t−s)wnds +Zt t−ρ [A(t)−A(s)]T−A(s)(t−s)wnds =A(t)Zt−ρ τ T−A(s)(t−s)wds + lim n→∞wn−T−A(t−ρ)(ρ)wn−Zt t−ρ ϕ1(t, s)wnds =w−T−A(t−ρ)(ρ)w−Zt t−ρ ϕ1(t, s)wds +A(t)Zt−ρ τ T−A(s)(t−s)wds, and in the fourth line we used (1.10). The estimate in (4.10) follows immediately from the one obtained for H(t, τ)and the fact the A(t)Rt τT−A(s)(t−s)ds is the extension of this operator. Remark 4.13. Note that, even though Rt τT−A(s)(t−s)wds ∈Dfor any w∈X, it does not mean that Rt τA(t)T−A(s)(t−s)wds is defined. The second integral might not exist. We can only prove that A(t)Rt τT−A(s)(t−s)wds, with the operator outside the integral, exists. 69 Chapter 4. Regularization: Abstract theory From the results above, we can obtain all the properties enumerated in Theorem 4.4, as we see next. But prior to those conclusions, it is worth comparing such result with the existent theory for singularly nonautonomous problems with sectorial operator. At that case, to conclude the differentiability of the process, we prove that A(t)Zt−ρ τ T−A(τ)(t)xds ρ→0 −→ A(t)Zt τ T−A(τ)(t)xds, and this comes as consequence of kA(t)Rt−ρ τT−A(τ)(t)dsk ≤ C. For the almost sectorial case, such convergence does not necessarily occur. As we can see from (4.9), it will only happen if T−A(t−ρ)(ρ)wρ→0 −→ w, which we know is not necessarily true. Moreover, the order from which A(t)Rt−ρ τT−A(τ)(t)ds diverges from A(t)Rt τT−A(τ)(t)ds is the same of the semigroup of growth 1−α,T−A(τ)(t), at the initial instant t= 0. This is reinforced by the fact kA(t)Rt−ρ τT−A(τ)(t)dsk ≤ C(t−τ)α−1.We gather those considerations in the following corollary: Corollary 4.14. Let α2+αδ −1>0and w∈X. Then A(t)Zt t−ρ T−A(s)(t−s)wds =w−T−A(t−ρ)(ρ)w−Zt t−ρ ϕ1(t, s)wds and A(t)Rt t−ρT−A(s)(t−s)wds does not necessarily vanishes as ρ→0+. In particular, the expression A(t)Rt−ρ τT−A(s)(t−s)wds does not necessarily converges to A(t)Rt τT−A(s)(t−s)wds, as ρ→0. We are finally in conditions to return to the derivative ∂tUρ(t, τ)u0which last characterization was given in (4.6). Note that the second and third line are exactly the right side of (4.9) for w= Φ(t, τ)u0 (with a negative sign) and we obtain ∂tUρ(t, τ)u0=−A(t)T−A(τ)(t−τ)u0−Zt−ρ τ A(t)T−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds −A(t)Zt τ T−A(s)(t−s)Φ(t, τ)u0ds +T−A(t−ρ)(ρ)[Φ(t−ρ, τ)−Φ(t, τ)]u0. Lemma 4.7 already proved the uniform (for t∈[τ+γ, t0]) convergence of the second term to Rt τA(t)T−A(s)(t−s)[Φ(s, τ)−Φ(t, τ)]u0ds. The fourth term, the last remaining, converges uniformly to zero, since  T−A(t−ρ)(ρ)[Φ(t−ρ, τ)−Φ(t, τ)] L(X)≤Cρα+η−1(t−ρ−τ)α+δ−2−η αρ→0 −→ 0, provided that α+η−1>0(which is satisfied when 1−α < η < α2+αδ −α). This allows us to conclude the uniform convergence of ∂tUρ(t, τ)u0to −A(t)[T−A(τ)(t−τ)v0+ Rt τT−A(s)(t−s)Φ(s, τ)u0ds] = −A(t)U(t, τ). Hence, sup t∈[τ+γ,t0]{kUρ(t, τ)u0−U(t, τ)u0k+k∂tUρ(t, τ)u0+A(t)U(t, τ)u0k} ρ→0 −−→ 0 and items (a)−(b)in Theorem 4.4 are verified, as well as (4.1). The other estimates in item (c)we prove in the sequel. 76 Chapter 4. Regularization: Abstract theory 4.2 Regular solution for ut+A(t)u=G(t)in X In the previous section we proved that the linear process U(t, τ),t>τ, associated to the family A(t), t ∈R, recovers strong solutions for the evolution problem ut+A(t)u= 0,u(τ) = u0. In this section we study the nonhomogeneous linear problem ut+A(t)u=G(t), τ < t < τ +T;u(τ) = u0∈X. (4.19) It will be used in the sequel to study the semilinear equation (P.7). If G∈L1((τ, τ +T], X), then the function u: (τ, τ +T]→X, given by u(t) = U(t, τ)u0+Zt τ U(t, s)G(s)ds (4.20) is well defined and it is called mild solution of (4.19). If we impose further conditions on G, we can prove that this mild solution is actually a strong solution for the equation. As we did in the preceding section, we enunciate the main result and prove it throughout the section. The content in this section is compiled in [14]. Theorem 4.20. Let A(t), t ∈R, be a family of linear operators in the Banach space Xsatisfying (P.1) - (P.3), α∈(0,1) the constant of almost sectoriality and δ∈(0,1] the exponent of H¨ older continuity. Suppose that α2+αδ −1>0and let U(t, τ)be the strongly differentiable process of growth 1−α associated to A(t),t∈R. Also, assume G: (τ, τ +T]→Xis a continuous function that satisfies kG(t)−G(s)kX≤C(t−s)θ(s−τ)−ψ,for any τ < s < t, (4.21) kG(t)kX≤C(t−τ)−ψ,for any τ < t, (4.22) where θand ψare positive constants satisfying θ > 1−α,0< ψ < 1. Then, for each u0∈X, the mild solution (4.20) is a strong solution for (4.19), that is, 1. u(·)∈ C1((τ, τ +T], X),u(τ) = u0and u(t)∈D, for all τ<t<τ+T. 2. The equation d dt u(t) = −A(t)u(t) + G(t),τ<t<τ+T, is satisfied in the usual sense (in X) and the following expression for the derivative of u(·)holds ut(t) = −A(t)U(t, τ)u0−A(t)Zt τ U(t, s)[G(s)−G(t)]ds −A(t)Zt τ U(t, s)G(t)ds +G(t). Moreover, if u0∈D, then u(·)is continuous at t=τ. We prove this theorem in the sequel following a strategy similar to the one we adopted in Section 4.1 to treat differentiability of U(t, τ). If we tried to evaluate the derivative directly in the expression 77 Chapter 4. Regularization: Abstract theory (4.20), the first term would not pose any problem, that is, ∂tU(t, τ)u0=−A(t)U(t, τ)u0. However, the expression given by the integral would be troublesome, since the expected value inside the integral is −A(t)U(t, s)G(s)and we cannot prove convergence of the integral with such integrand (recall that kA(t)U(t, τ)kL(X)≤(t−τ)α−2). We denote this term as v(t), that is, v(t) = Zt τ U(t, s)G(s)ds. To overcome the problem mentioned above, we consider, for small ρ > 0, the approximations [τ+γ, t0]3t7→ vρ(t) = Zt−ρ τ U(t, s)G(s)ds, where γ > 0is arbitrary, t0∈(τ+γ, τ +T]and ρis small enough such that t−ρ > τ +γ. With this slight retreat in the domain of integration, we can prove the following result for this function. Lemma 4.21. The function vρ: [τ+γ, t0]→Xis continuously differentiable in Xand v0 ρ(t) = U(t, t −ρ)G(t−ρ)−A(t)Zt−ρ τ U(t, s)G(s)ds. (4.23) Proof. This follows readily from the fact that the integrand is continuously differentiable in (τ, t−ρ]and from an application of Lemma P.6. Once we know vρis differentiable, we prove: (1) vρ(·)converges as ρ→0to v(·)in C([τ+γ, t0], X). (2) v0 ρ(·)converges as ρ→0to −A(·)v(·) + G(·)in C([τ+γ, t0], X). Then, the differentiability of t7→ v(t)for t∈[τ+γ, t0]follows and v0(t) = −A(t)v(t)+G(t). From the arbitrariness of γ > 0and t0, we have the differentiability in (τ, τ +T). After these two steps, Theorem 4.20 will be proved, since u0(t) = −A(t)U(t, τ)u0+d dt Zt τ U(t, s)G(s)ds =−A(t)U(t, τ)u0+v0(t) =−A(t)U(t, τ)u0−A(t)v(t) + G(t) = −A(t)u(t) + G(t). Item (1) is easily obtained: for each t∈[τ+γ, t0]we have kvρ(t)−v(t)kX=   Zt t−ρ U(t, s)G(s)ds   X ≤Zt t−ρ C(t−s)α−1(s−τ)−ψds ≤C(t−ρ−τ)−ψραρ→0 −→ 0. Item (2), on the other hand, demands more attention. We first prove that v(t)∈D. 78 Chapter 4. Regularization: Abstract theory Lemma 4.22. For any t∈[τ+γ, t0],v(t)∈Dand −A(t)v(t) = −A(t)Zt τ U(t, s)G(s)ds =−A(t)Zt τ U(t, s)[G(s)−G(t)]ds −A(t)Zt τ U(t, s)G(t)ds. Proof. It follows from Lemma 4.18 that Rt τU(t, s)G(t)ds ∈D. Furthermore, from (4.21) with θ > 1−α, we conclude that Rt τA(t)U(t, s)[G(s)−G(t)]ds converges. Indeed,    Zt τ A(t)U(t, s)[G(s)−G(t)]ds   X≤Zt τ (t−s)α−2(t−s)θ(s−τ)−ψds ≤C(t−τ)(α+θ−1)−ψ<∞. (4.24) Therefore, Rt τU(t, s)[G(s)−G(t)]ds ∈Dand A(t)Zt τ U(t, s)[G(s)−G(t)]ds =Zt τ A(t)U(t, s)[G(s)−G(t)]ds. To prove item (2), we must check that v0 ρ(·)given by (4.23) converges to −A(·)v(·) + G(·)which is also given by: −A(t)v(t) + G(t) = G(t)−A(t)Zt τ U(t, s)G(s)ds =G(t)−A(t)Zt τ U(t, s)[G(s)−G(t)]ds −A(t)Zt τ U(t, s)G(t)ds. (4.25) We rearrange (4.23) in a way that it approaches the most the expression (4.25) above, that is, v0 ρ(t) = U(t, t −ρ)G(t−ρ)−A(t)Zt−ρ τ U(t, s)[G(s)−G(t)]ds −A(t)Zt−ρ τ U(t, s)G(t)ds. (4.26) The second term of (4.26) converges and it satisfies: Lemma 4.23. If G: (τ, τ +T]→Xsatisfies (4.21) with θ > 1−α, then A(t)Zt−ρ τ U(t, s)[G(s)−G(t)]ds ρ→0 −→ A(t)Zt τ U(t, s)[G(s)−G(t)]ds. Proof. This follows from the existence of Rt τA(t)U(t, s)[G(s)−G(t)]ds proved in Lemma 4.22, equation (4.24). Note that θ > 1−αwas necessary to ensure such existence. 79 Chapter 4. Regularization: Abstract theory For the other terms in (4.26), note that the discontinuity of the process at the initial time allow situations in which U(t, t −ρ)G(t−ρ)9G(t)and A(t)Zt−ρ τ U(t, s)G(t)ds 9A(t)Zt τ U(t, s)G(t)ds, as ρ→0. Therefore, we cannot work them separately and, in order to obtain the desired convergence, we have to find an alternative to overcome this situation. We will provide a way to write A(t)Rt τU(t, s)xds in terms of A(t)Rt−ρ τU(t, s)xds, for a given ρ > 0and x∈X. This is done in next lemma. Lemma 4.24. Let α2+αδ −1>0. Given any 0< ρ < t −τand x∈X, the following equality holds: A(t)Zt τ U(t, s)xds =A(t)Zt−ρ τ U(t, s)xds +x−T−A(t−ρ)(ρ)x−Zt t−ρ ϕ1(t, s)xds +A(t)Zt t−ρ T−A(ξ)(t−ξ)Zξ t−ρ [Φ(ξ, s)−Φ(t, s)]xdsdξ (4.27) +A(t)Zt t−ρ T−A(ξ)(t−ξ)Zt t−ρ Φ(t, s)xdsdξ (4.28) −A(t)Zt t−ρ T−A(ξ)(t−ξ)Zt ξ Φ(t, s)xdsdξ. (4.29) Moreover, the terms (4.27),(4.28) and (4.29) vanish as ρ→0+. Proof. Note first that, since Rt τU(t, s)xds ∈Dfor any τ, t ∈Rand x∈X(Lemma 4.18), we can separate the integrals as follows A(t)Zt τ U(t, s)xds =A(t)Zt−ρ τ U(t, s)xds +A(t)Zt t−ρ U(t, s)xds. The expression available for the process U(t, τ)(Corollary 1.14) and the result on Corollary 4.14 implies that the second term in the right-side of equality above satisfies: A(t)Zt t−ρ U(t, s)xds =A(t)Zt t−ρ T−A(s)(t−s)xds +A(t)Zt t−ρZt s T−A(ξ)(t−ξ)Φ(ξ, s)xdξds =x−T−A(t−ρ)(ρ)x−Zt t−ρ ϕ1(t, s)xds+A(t)Zt t−ρZt s T−A(ξ)(t−ξ)Φ(ξ, s)xdξds and we already obtain the first line of the desired equality: A(t)Zt τ U(t, s)xds =A(t)Zt−ρ τ U(t, s)xds +x−T−A(t−ρ)(ρ)x−Zt t−ρ ϕ1(t, s)xds +A(t)Zt t−ρZt s T−A(ξ)(t−ξ)Φ(ξ, s)xdξds. (4.30) 80 Chapter 4. Regularization: Abstract theory An application of Fubini’s Theorem and some algebraic manipulation on (4.30) yield A(t)Zt t−ρZt s T−A(ξ)(t−ξ)Φ(ξ, s)xdξds =A(t)Zt t−ρ T−A(ξ)(t−ξ)Zξ t−ρ Φ(ξ, s)xdsdξ =A(t)Zt t−ρ T−A(ξ)(t−ξ)Zξ t−ρ [Φ(ξ, s)−Φ(t, s)]xdsdξ +A(t)Zt t−ρ T−A(ξ)(t−ξ)Zt t−ρ Φ(t, s)xdsdξ −A(t)Zt t−ρ T−A(ξ)(t−ξ)Zt ξ Φ(t, s)xdsdξ =I1(ρ) + I2(ρ) + I3(ρ). The first statement of the lemma is already proved, it only remains to prove that I1(ρ),I2(ρ)and I3(ρ)vanish as ρ→0+. From Lemma 3.12, we obtain kI1(ρ)kX≤CZt t−ρ (t−ξ)α−2Zξ t−ρ (t−ξ)η(ξ−s)α+δ−2−η αkxkXdsdξ ≤CZt t−ρ (t−ξ)α+η−2(ξ−(t−ρ))α+δ−1−η αkxkXdξ ≤Cρ(α+η−1)+(α+δ−1−η α)kxkX ρ→0 →0. For I2(ρ), if wρ=Rt t−ρΦ(t, s)xds, then kwρkX ρ→0+ −→ 0and we have I2(ρ) = A(t)Zt t−ρ T−A(ξ)(t−ξ)wρdξ =A(t)Zt τ T−A(ξ)(t−ξ)wρdξ −A(t)A(t−ρ)−1A(t−ρ)AZt−ρ τ T−A(ξ)(t−ξ)wρdξ =H(t, τ)wρ−A(t)A(t−ρ)−1H(t−ρ, τ)wρ. Since H(·,·)is a bounded linear operator (Lemma 4.11), it follows that kI2(ρ)kX≤C(t−τ)α−1kwρkX+C(t−ρ−τ)α−1kwρkX ρ→0 →0. For the third term, we have kI3(ρ)kX≤CZt t−ρ (t−ξ)α−2Zt ξ (t−s)α+δ−2dsdξ ≤CZt t−ρ (t−ξ)α−2(t−ξ)α+δ−1dξ ≤Cρ2α+δ−2ρ→0 →0, since α+δ 2−1>0(as a consequence of α2+αδ −1>0). 81 Chapter 4. Regularization: Abstract theory Equality provided in Lemma 4.24 suits well our purpose. We use the result of this lemma to rewrite equation (4.26) for v0 ρ. If I1(ρ),I2(ρ)and I3(ρ)represent the terms (4.27), (4.28) and (4.29) with x=G(t)(all of them vanishing as ρ→0), we obtain v0 ρ(t) = U(t, t −ρ)G(t−ρ)−A(t)Zt−ρ τ U(t, s)[G(s)−G(t)]ds −A(t)Zt−ρ τ U(t, s)G(t)ds =T−A(t−ρ)(ρ)G(t−ρ)+Zt t−ρ T−A(s)(t−s)Φ(s, t−ρ)G(t−ρ)ds −A(t) Zt−ρ τ U(t, s)[G(s)−G(t)]ds −A(t)Zt τ U(t, s)G(t)ds +G(t)−T−A(t−ρ)(ρ)G(t)−Zt t−ρ ϕ1(t, s)G(t)ds +I1(ρ) + I2(ρ) + I3(ρ) =G(t)−A(t)Zt−ρ τ U(t, s)[G(s)−G(t)]ds −A(t)Zt τ U(t, s)G(t)ds (4.31) +T−A(t−ρ)(ρ)[G(t−ρ)−G(t)]+ Zt t−ρ T−A(s)(t−s)Φ(s, t −ρ)G(t−ρ)ds −Zt t−ρ ϕ1(t, s)G(t)ds +I1(ρ) + I2(ρ) + I3(ρ). First line in the last equality converges to G(t)−A(t)v(t), as needed (and uniformly for t∈[τ+γ, t0]), due to Lemma 4.23. We prove in the sequel that the remaining terms vanish as ρ→0. Note that the θ−H¨ older continuity of G(·)given in (4.21) is extremely important in the convergence analysis below, as well its controled discontinuity at initial time (given by the exponent ψ∈(0,1)) and the fact that 1−α < η < α(α+δ−1).We have kT−A(t−ρ)(ρ)[G(t−ρ)−G(t)]kX≤Cρα−1ρθ=Cρα+θ−1ρ→0 →0,    Zt t−ρ T−A(s)(t−s)Φ(s, t −ρ)G(t−ρ)ds   X≤CZt t−ρ (t−s)α−1(s−t−ρ)α+δ−2(t−ρ)−ψds ≤C(t−ρ)−ψZt t−ρ (t−s)α−1(s−t−ρ)α+δ−2ds ≤C(t−ρ)−ψρα+α+δ−2B(α, α +δ−2) ≤C(t−ρ)−ψρ2α+δ−2ρ→0 →0 and    Zt t−ρ ϕ1(t, s)G(t)ds   X≤CZt t−ρ (t−s)α+δ−2(t−τ)−ψds ≤C(t−τ)−ψρα+δ−1ρ→0 →0. Consequently, in the expression obtained for v0 ρ(·)we have (4.31) converging to G(t)−A(t)v(t) whereas the remaining terms converge to zero, which allow us to conclude sup t∈[τ+γ,T ]    v0 ρ(t)−G(t)−A(t)Zt τ U(t, s)G(s)ds   X ρ→0+ −−−→ 0 and Theorem 4.20 is proved. 82 Chapter 4. Regularization: Abstract theory 4.3 Regularity results stated in L(Y)and Y We restate the two main theorems presented in the previous sections now in the context of the Banach space Y. The proof is exactly the same, since AY(t) : DY⊂Y→Yis also an almost sectorial operator, but with constant of sectoriality ω. We first present the differentiability of U(t, τ)in Y. Theorem 4.25. Let A(t), t ∈R, be a family of linear operators in Ysatisfying (P.1) - (P.3), ω∈(0,1) the constant of almost sectoriality and δ∈(0,1] the exponent of H¨ older continuity. 1. If ω+δ > 1, then there exists a unique linear process of growth 1−ωin L(Y),U(t, τ), associated to the family A(t), t ∈R. This process satisfies kU(t, τ)kL(Y)≤C(t−τ)ω−1. 2. In addition, if ω2+ωδ −1>0, then (a) U(t, τ) : Y→DY, for any τ < t. (b) {(t, τ)∈R2;τ < t} 3 (t, τ)7→ U(t, τ)∈ L(Y)is strongly differentiable, that is, for each x∈Y,{(t, τ)∈R2;τ < t} 3 (t, τ)7→ U(t, τ)x∈Yis differentiable. (c) The derivative ∂tU(t, τ)is a bounded linear operator, strongly continuous in {(t, τ)∈R2;τ < t}and satisfies: ∂tU(t, τ) + A(t)U(t, τ)=0,∀t > τ, k∂tU(t, τ)kL(Y)=kA(t)U(t, τ)kL(Y)≤C(t−τ)ω−2,∀t > τ,  A(t)U(t, τ)A(τ)−1 L(Y)≤C(t−τ)ω−1,∀t > τ. Consider the nonhomogeneous problem ut+A(t)u=G(t), τ < t < τ +T;u(τ) = u0∈Y(4.32) now in the Banach space Y, and G∈L1((τ, τ +T), Y ).The function u: [τ, τ +T]→Y, given by u(t) = U(t, τ)u0+Zt τ U(t, s)G(s)ds (4.33) is the mild solution of (4.32). Theorem 4.26. Let A(t), t ∈R, be a family of linear operators in the Banach space Ysatisfying (P.1) - (P.3), ω∈(0,1) the constant of almost sectoriality and δ∈(0,1] the exponent of H¨ older continuity. Suppose that ω2+ωδ −1>0and let U(t, τ)be the strongly differentiable process of growth 1−ω associated to A(t),t∈R. Also, assume G: (τ, τ +T]→Yis a continuous function that satisfies kG(t)−G(s)kX≤C(t−s)θ(s−τ)−ψ,for any τ < s < t, 83 Chapter 4. Regularization: Abstract theory kG(t)kX≤C(t−τ)−ψ,for any τ < t, where θ,ψare positive constants satisfying θ > 1−ωand ψ∈(0,1). Then, for each u0∈Y, the mild solution (4.33) is a strong solution for (4.32), that is, 1. u(·)∈ C1((τ, τ +T], Y ),u(τ) = u0and u(t)∈DY, for all τ<t<τ+T. 2. The equation d dt u(t) = −A(t)u(t) + G(t)in Y,τ < t < τ +T, is satisfied in the usual sense. and the following expression for the derivative of u(·)holds ut(t) = −A(t)U(t, τ)u0−A(t)Zt τ U(t, s)[G(s)−G(t)]ds −A(t)Zt τ U(t, s)G(t)ds +G(t). Furthermore, if u0∈DY, then u(·)is continuous at t=τ, that is, u(·)∈ C([τ, ∞), Y )∩C1((τ, ∞), Y ). 4.4 Strong Y−solution for the semilinear equation We consider the semilinear evolution problem ut+A(t)u=F(u), t > τ;u(τ) = u0∈X, with mild solution u: (τ, τM(u0)) →X. Since the existence of this mild solution is already known, we look at the function (τ, τM(u0)) 3t7→ F(u(t)) ∈Y. If G(t) = F(u(t)), then Proposition 3.7 states that for any τ<s<t, given 0<µ<min{α2, ω+δ−1}, there exists C > 0, such that G(·)∈ Cθ ψ((τ, T ), Y ), that is, kG(t)−G(s)kY≤C(t−s)θ(s−τ)−ψ, where 0< θ < min{µ, 1−ρ(1 −α)}, for any 0<µ<min{α2, ω +δ−1}and −ψ= min −µ α−ρ(1 −α), δ −1−ρ(1 −α), β −α−ρ(1 −α). In order to obtain the results claimed in Theorem 4.26, we must have (i). µ > 1−ωand 1−ρ(1 −α)>1−ω. (ii). −ψ > −1. Since 0< µ < min{α2, ω +δ−1}, the interval (1 −ω, min{α2, ω +δ−1})only makes sense if α2+ω−1>0 ω+δ 2−1>0. 84 Chapter 4. Regularization: Abstract theory It follows from ω2+δω −1>that ω+δ 2−1>0is already satisfied. The other inequality in item (i) is satisfied if ρ < ω 1−α. In order for the conditions in (ii) to hold, we have already established in Proposition 3.7 that they must satisfy 1≤ρ≤min β 1−α+ 1,δ 1−α,α−µ α(1 −α). The minimum value allowed for µsuch that condition (i) holds is 1−ωand if we replace it in the expression above, we have 1≤ρ≤min β 1−α+ 1,δ 1−α,α+ω−1 α(1 −α). If all the previous conditions are satisfied, then Theorem 4.26 states that the mild solution u(t) = U(t, τ)u0+Rt τU(t, s)G(s)ds, t ∈(τ, τM(u0)), for the nonautonomous linear problem ut+A(t)u=G(t), t ∈(τ, τM(u0)); u(τ) = u0, is a strong solution u(·)∈ C1((τ, τM(u0)), Y ). But since u(·) : (τ, τM(u0)) →Xis a mild solution of ut+A(t)u=F(u), t ∈(τ, τM(u0)); u(τ) = u0, and G(t) = F(u(t)), it follows that u(·)is a strong Y−solution for the semilinear problem and the following theorem is proved. Theorem 4.27. Let X, Y be Banach spaces with X ,→Y. Suppose A(t), t ∈R, is α−uniformly sectorial in X,ω−uniformly sectorial in Yand δ−uniformly H¨ older continuous. Additionally, we assume that A(t), t ∈Rsatisfies the condition in (P.4) with constant β∈(0,1) and F:X→Yis a nonlinearity satisfying (G) with growth ρ≥1. We have: 1. If α+δ > 1and ω+δ > 1, then the linear process U(t, τ)exists in L(X),L(Y, X)and L(Y), respectively. 2. If ω2+ωδ −1>0, then U(t, τ)is strongly differentiable in L(Y). 3. Furthermore, if α2+ω−1>0 and 1≤ρ < min β 1−α,α+ω−1 α(1 −α),δ 1−α,ω 1−α 85 Chapter 4. Regularization: Abstract theory Then, for every u0∈X, the initial value problem ut+A(t)u=F(u), t > τ, u(τ) = u0∈X, has a unique strong Y−solution defined in (τ, τM(u0)). The list of necessary conditions to ensure existence of strong solutions is quite extensive. We sum them up in the next table, identifying where they are required. We also refer to each block of conditions (I, II, III or IV) as the conditions on the quadrant I, II, III or IV. Local well-posedness I. Conditions on the existence of U(t, τ)II. Conditions on the existence of mild solution α+δ > 1(in L(X))1≤ρ < β 1−α ω+δ > 1(in L(Y, X)) Regular solutions III. Strong differentiability U(t, τ)in L(Y)IV. Differentiability of the mild solution in Y ω2+ωδ −1>0α2+ω−1>0 1≤ρ < min nα+ω−1 α(1−α),δ 1−α,ω 1−αo Table 4.1: Conditions of Theorem 4.27 In Chapter 2 we started the discussion of local well-posedness for the equation in a domain with a handle and we saw how the conditions on quadrants I and II imply lower bounds for the values of q (and, consequently, upper bounds for the growth of F0:U0 p→U0 q). In the next chapter we discuss the conditions on III and IV. There will be several restriction taking place at once, but we will see that ω2+ωδ −1>0is the most restrictive one for the example considered and this single condition will determine the maximal growth of F0that ensures existence of regular solution. It is worth mentioning the duality in the origin of those conditions. Some of them come from the fact that we need to ensure good properties for the linear process U(t, τ). Those are the inequalities located at the left column of the table (quadrants I and III). On the other hand, the conditions on the right column (II and IV) are those responsible to ensure that the discontinuity at initial time t=τdoes not exceed a limit value that would disrupt the differentiability or existence of solution. In the usual case where A(t), t ∈R, is a family of sectorial operators (studied for instance in [21, 37, 52, 58, 59, 60]), all those conditions are trivially satisfied and we have existence of strong solution for the problem provided Fis locally Lipschitz and A(t)uniformly sectorial. Indeed, the right side of the table would not pose any problem (there is no more discontinuity at the initial time t=τ) and conditions on the left side are trivially satisfied (since α=ω= 1). This recovers the content of Theorem 7 of [55]. Part III: Long-time behavior In the last part of this work we focus our attention at the long-time behavior of the solution rather than its local properties. From Theorem 1.25, the singularly nonautonomous problem satisfies a blow-up alternative, that is, for each u0∈X, the solution u: (τ, τM(u0)) →X, starting at u0at the initial time τ and defined in its maximal interval, satisfies: τM(u0)=+∞or lim supt→τM(u0)−ku(t)kX= +∞. Therefore, estimates for u(t)in the phase space Xare useful in order to prove its global existence (also called global well-posedness of the problem). We dedicate this part to present an iterative procedure that allow us to obtain estimates for the solutions in L2k, for any k∈N, as well as identifying in the phase spaces sets with the properties of attracting the solutions to them as the evolution of the dynamics takes place. Those sets are called pullback attracting sets and they play an important role in describing the asymptotic dynamics of the system. The iterative procedure and the consequently existence of pullback attractor for the equation in the domain with a handle are presented in [12]. 92 CHAPTER 6 Global well-posedness and asymptotic dynamics In previous chapters we mentioned that the regularity properties of the solution for the semilinear problem could be helpful if one wishes to obtain estimates of the solution directly from the differential equation. Since we know u(·)∈ C1((τ, τM(u0)), Y )and u(t)∈DYfor t∈(τ, τM(u0)), we could perform some operations on the equation ut+A(t)u=F(u), t > τ, (6.1) to extract informations about kukY,kutkYand kA(t)ukY. To illustrate the ideas, consider the simple case presented in Example 1.19, where A=−∆is the Laplacian operator with Neumann boundary condition in a smooth bounded domain Ω, acting in L2(Ω). If u(·)is a solution of ut= ∆u+F(u)with regularity u(·)∈ C1((τ, τM(u0)), L2(Ω)) and u(t)∈H2 N for t∈(τ, τM(u0)), then we could take the inner product in L2(Ω) of the equation with u: ZΩ utudx =ZΩ (∆u)udx +ZΩ F(u)udx. The regularity that uhas allows us to integrate by parts and obtain 1 2 d dtkuk2 L2=−k∇uk2 L2+ZΩ F(u)udx. This equality, added to a suitable dissipativeness assumption on F, allows us to obtain estimates of kukL2. Those ideas will be discussed in Chapter 7 to the reaction-diffusion equation in the domain with a handle. Moreover, if usatisfies the equation (6.1), then any estimative for two of the three terms that features in this equation, implies an estimative for the third one. For instance, if we prove that F(u)and utare bounded in Y, then A(t)u=F(u)−utis also bounded in Y, implying ubounded in the stronger space DY. This chapter presents some techniques that allows us to obtain estimates for kukand kutk. 93 94 Chapter 6. Global well-posedness and asymptotic dynamics 6.1 Estimates for utin Y When it comes to obtain estimates for ut, one could try in the example above to take the inner product of the equation by utand to perform some calculation over it. However, it would be necessary to integrate by parts and a term ∇(ut)would appear. With the information we have so far on the regularity of ut (ut∈ C((τ, τM(u0)), Y )), this integration by parts would not be justified. There are techniques in the literature that overcome this problem by working with regular approximations of ut(for example, in [29, 30] the use of Steklov average to overcome the lack of regularity in the derivative, or the use of Galerkin approximation in [48, 54]). However, the time-dependence of the family A(t)makes it difficult to apply those techniques. On the other hand, the properties obtained so far for the linear process U(t, τ)associated to the family A(t),t∈R, allow us to obtain estimates for utby working with the mild formulation of u, (1.19). For the nonsingular (A(t) = A) and sectorial case, this type of result can be found in the Appendix Section of [18]. In order to reply it to the singular and almost sectorial case, we suppose that the following scenario for the problem holds: (S). Let A(t), t ∈R, be a family of almost sectorial operators and F:X→Ya nonlinearity as in Theorem 4.27. For u0∈X, let u=u(·, τ, u0)denotes the strong Y−solution of the semilinear problem. Assume that given a bounded set B⊂Xand T > τ, there exists a constant C=C(T, B) depending on Tand B, such that sup t∈(τ,T ] sup u0∈Bku(t, τ, u0)kX≤C(T, B)(t−τ)α−1,(6.2) where αis the constant of almost sectoriality of A(t),α∈(0,1). The estimate (6.2) on bounded sets for uis transferred through the variation of constants formula to the derivative ut, as we prove in the sequel. Theorem 6.1. Let X, Y be Banach spaces and assume that (S) is satisfied. Given any T > τ and B bounded set in X, there exists a constant Ddepending only on Tand B, such that sup t∈(τ,T ] sup u0∈Bkut(t, τ, u0)kY< D(T, B)(t−τ)ω−2, where ωis the almost sectoriality constant of A(t)in Y. Proof. From the conditions required in (S), the solution u(t) = U(t, τ)u0+Rt τU(t, s)F(u(s))ds is differentiable in tand, by taking the derivative over its mild formulation, we obtain ut(t) = −A(t)U(t, τ)u0−A(t)Zt τ U(t, s)F(u(s))ds +F(u(t)) 95 Chapter 6. Global well-posedness and asymptotic dynamics =−A(t)U(t, τ)u0−A(t)Zt τ U(t, s)[F(u(s))−F(u(t))]ds −A(t)Zt τ U(t, s)F(u(t))ds +F(u(t)). Since ku(t)kX≤C(T, B)(t−τ)α−1, the growth condition on Fimplies that kF(u(t))kY≤C(1 + ku(t)kρ X)≤D(T, B)(t−τ)−ρ(1−α).(6.3) We use this estimates and the results obtained in Chapter 4 to evaluate each term in the above equality for ut. Since X ,→Y, it follows from Theorem 4.25 that kA(t)U(t, τ)u0kY≤C(t−τ)ω−2ku0kY≤C(t−τ)ω−2ku0kX. From Lemma 4.18, A(t)Rt τU(t, s)ds is a bounded linear operator satisfying   A(t)Rt τU(t, s)ds  L(Y)≤C(t−τ)ω−1. Therefore, from the estimate on kF(u(t))kY, we obtain     A(t)Zt τ U(t, s)F(u(t))ds   Y≤D(T, B)(t−τ)ω−1−ρ(1−α) ≤D(T, B)(t−τ)ω−2, since ρ(1 −α)< β < 1. For the remaining term, recall from Proposition 3.7 (and Remark 3.10) that there exist θ > 0and ψ∈(0,1) such that kF(u(t)) −F(u(s))kY≤C(t−s)θ(s−τ)−ψ. Therefore,     A(t)Zt τ U(t, s)[F(u(s)) −F(u(t))]ds   Y≤C(T, B)Zt τ (t−s)ω−2(t−s)θ(s−τ)−ψds ≤C(T, B)(t−τ)ω+θ−1−ψB(ω+θ−1,1−ψ) ≤D(T, B)(t−τ)ω−2. The conclusion now follows from the above estimates. Rewriting the semilinear problem as A(t)u=F(u)−ut, we note that if the right side is bounded in (τ, T ](uniformly in B⊂X), then kA(t)ukYalso is, or equivalently, ku(t)kDY. Corollary 6.2. Under the conditions of Theorem 6.1, sup t∈(τ,T ] sup u0∈BkukDY< D(T, B)(t−τ)ω−2. 96 Chapter 6. Global well-posedness and asymptotic dynamics The same result holds if A(t)is sectorial (α=ω= 1). The estimate in this case is slightly improved, since A(t)Rt τU(t, s)ds is a bounded linear operator with norm no larger than C(t−τ)−1(it is enough to replay the almost sectorial case with constant of sectoriality equals 1). Recall that for the sectorial case, all conditions on Table 4.1 are trivially satisfied. Moreover, in the sectorial case there is no discontinuity at the initial time t=τand the estimate can be taken in [τ, T]. This justifies the following corollary that complements the theory for nonsingular problems for the sectorial case. Corollary 6.3. Suppose that A(t),t∈R, is a uniformly sectorial family of linear operators, T > τ and B⊂Xis a bounded set. If F:X→Yis a nonlinearity with growth condition (G) and the compatibility condition (P.4) between A(t)holds, then, whenever a solution u(t)of the semilinear problem ut+A(t)u=F(u), t > τ;u(τ) = u0∈X, is bounded in [τ, T ], sup t∈[τ,T ] sup u0∈Bkut(t, τ, u0)kX< C(T, B), the derivative utis also bounded in the same interval sup t∈[τ,T ] sup u0∈Bkut(t, τ, u0)kY< D(T, B)(t−τ)−1, where C(T, B)and D(T, B)are positive constants that depend only on T > τ and B⊂X. 6.2 Fractional powers of sectorial operators and smoothing effect of the differential equation Theorem 6.1 on its on is not enough to deal with the application we are investigating in this work: the reaction-diffusion equation in Ω0. As we will see in the next chapter, it will be necessary to estimate the solution (w, v)in more regular spaces than U0 pin order to obtain the existence of compact attracting sets. To tackle this problem we will use one of its features: the fact that the system is weakly coupled. By decoupling the problem, we will obtain two equations, one in Ωand other in R0, and each one of them, separately, will be associated to a family of sectorial operators (α=ω= 1). In order to take advantage of this fact, we introduce in this section the definition of fractional powers of sectorial operators and we study the smoothing effect that the differential equation with sectorial operators has. Therefore, unlike the rest of this work, this section is focused exclusively in the case where the family A(t), t ∈R, is sectorial. Remark 6.4. The fractional powers of almost sectorial operators can also be defined (as it is done, for instance, in [19]). However, due to the resolvent deficiency, some restrictions on the values of the powers 97 Chapter 6. Global well-posedness and asymptotic dynamics available for the almost sectorial operator Aemerges. For instance, Aθ, for a real number θ, is only defined if 1−α < θ < 1, where αis the constant of almost sectoriality. The restrictions increases if we wish to use a momentum type inequality. For this case, 1−α < θ < α(see [19, (8)]). Those conditions difficult the analysis of smoothing effect in which features almost sectorial operators. We assume that A:D(A)⊂X→Xis a sectorial operator in a Banach space Xsuch that Re(σ(A)) >0. We refer to it as being sectorial and positive. σ(A) Γ Re Im Figure 6.1: Sectorial and positive operator For an operator Awith those properties, if Γis a curve as above, orientated from ∞e−iψ to ∞eiψ (ψ∈(0,π 2)) in a way that σ(A)is contained inside the region determined by it, then, for θ > 0,A−θ given by A−θ=1 2πi ZΓ (−λ)−θ(λ+A)−1dλ. (6.4) For a deeper discussion on fractional powers of unbounded operators, we recommend [3, 49]. There are expressions equivalent to (6.4) that also defines the fractional power of A, as one can see in [11, 46, 47], even for more general operators A. We focus on the case Ais sectorial and positive and we gather some properties of this operator in the sequel. Proposition 6.5. [52, Section 2.6] Let A:D(A)⊂X→be a sectorial and positive operator. Then, for any θ > 0, the operator A−θgiven in (6.4) is well defined in Xand it is a bounded linear operator. Moreover, 1. A−θ:X→Xis one-to-one. 2. For ξ, θ > 0,A−θA−ξ=A−θ−ξ. 98 Chapter 6. Global well-posedness and asymptotic dynamics The fact that A−θis one-to-one allows the following definition: Definition 6.6. For every θ > 0,Aθ:= (A−θ)−1. We denote by Xθthe domain of Aθequipped with the norm kxkXθ=kAθxkX,∀x∈D(Aθ), and we refer to Xθas a fractional power space. For θ= 0,X0=X. Those fractional power spaces {Xθ}θ≥0are Banach and they establish a scale of spaces, as we see in the next proposition. Proposition 6.7. [25, Section 1.3.3] For each θ > 0,Xθis a Banach space. Moreover, if 0≤θ < ξ, then Xξis continuously embedded in Xθ, which we denote by Xξ,→Xθ. If Ahas compact resolvent, then Xξis compactly embedded in Xθ, denoted by Xξc ,→Xθ. We return to the family A(t),t∈R, of uniformly sectorial (α= 1) and uniformly δ−H¨ older continuous operators. For each t∈R,A(t)has its fractional powers well-defined. As a consequence of the H¨ older continuity (P.3), given any arbitrarily large compact set in R2, there exists a constant C > 0such that kA(t)A−1(τ)kL(X)≤C, for all (t, τ)in this compact set. In this case, for t, τ ∈[−M, M], the norms k · kD(A(t)) =kA(t)· kXand k · kD(A(τ)) =kA(τ)· kXdefined by the operators A(t)and A(τ), respectively, are equivalent. In the same way, k·kD(A(t)θ)=kA(t)θ· kXand k·kD(A(τ)θ)=kA(τ)θ·kXare equivalent. We will fix one operator A0=A(t0)as a reference and we shall refer to this norm as k · kX1. We notate by Xθthe domain of Aθ 0=A(t0)θendowed with the norm k·kXθ=kAθ 0·kX. From the equivalence obtained above, we can refer to Xθas domain of any operator A(t)θ. The scale of fractional powers associated to A(t),t∈R, allows us to work with semilinear problems in the following setting: ut+A(t)u=F(t, u), t > τ; u(τ) = u0∈Xθ,(6.5) where Fis a nonlinearity such that F:R×Xθ→X,0≤θ < 1. In [55, Theorem 7]) local wellposedness of (6.5) was proved, which we state next: Theorem 6.8. [55, Theorem 7] Let A(t),t∈R, be a family of uniformly sectorial operators (α= 1) and uniformly δ−H¨ older continuous, and F:R×Xθ→Xa locally H¨ older continuous function in the first variable and locally Lipschitz in the second variable, 0≤θ < 1. Then, given any u0∈Xθ, u(t) = U(t, τ)u0+Zt τ U(t, s)F(s, u(s))ds is a strong solution for (6.5), that is, 1. u(·)∈ C([τ, T), X)∩C1((τ, T ), X)and u(t)∈D, for τ < t < T; 99 Chapter 6. Global well-posedness and asymptotic dynamics 2. usatisfies the equation in the usual sense d dt u(t, x) = −A(t)u(t, x)+F(u(t, x)), for all t∈(τ, T ). Moreover, if ku(t)kXθis bounded in any bounded set [τ, t∗], then the solution is globally defined in time. Note that this type of result is similar to Theorem 4.27 proved in Chapter 4. The parabolic structure of the problem allows us to obtain regularity for the solution u(t), that is u(t)∈Xξfor t∈(τ, T)and any 0≤ξ≤1. As before, we call this property regularization. Another important feature of parabolic equation is the one called smoothing effect which refers to the increase of regularity of ut(t). From Theorem 6.8 above, ut(t)∈X, but we can prove that the derivative belongs to better spaces Xξ, with 0≤ξ < δ, where δis the H¨ older exponent of the family A(t). The theorem stated in the sequel extends Theorem 3.5.2 in [37], where the author proved the smoothing effect to the nonsingular (A(t) = A)and sectorial case. For the singular case with sectorial operators, an indication of such result is presented in [55]. We provide a proof of such fact in Appendix A, based on the theory developed in the previous chapter. This result was presented in [13] alongside its consequence in the study of the asymptotic dynamics of singularly nonautonomous problems. Theorem 6.9. Let A(t),t∈R, be a family of uniformly sectorial operators (α= 1) and uniformly δ−H¨ older continuous, and F:Xθ→Xa locally Lipschitz function, 0≤θ < 1. If u: [τ, T)→Xis the solution of ut(t) + A(t)u=F(u), t ∈(τ, T ); u(τ) = u0∈Xθ, then, for any 0≤ξ < δ,ut(t)∈Xξand satisfies the estimate kut(t)kXξ≤C(t−τ)−1−ξ+θku0kXθ. The illustration below describes the regularization effect (blue) and the smoothing effect (red) of the parabolic equation. In the vertical axis we represent the scale of fractional powers Xξ, starting at X0 and increasing upwards. In the horizontal axis, we have two points representing the solution uand its derivative ut. The vertical lines denote the spaces u/utbelongs. u(t)ut(t) X1 Xδ X Figure 6.2: Regularization and smoothing effect 100 Chapter 6. Global well-posedness and asymptotic dynamics Remark 6.10. The smoothing effect will play an essential role in the application when we transfer the information of the dynamics in Ωto the channel R0. Estimates of wtin stronger norms will be required at this point. To help us obtain estimate of the solutions in more regular spaces, we have the next results that provide estimates of the linear operators T−A(t)(s)and U(t, τ)in the space L(X, Xθ). Proposition 6.11. [37, Theorem 1.4.3] If A(τ)is sectorial and Re(σ(A(τ))) >0, for any θ, γ ≥0such that γ < θ, there exists a constant C(θ, γ)>0such that kT−A(τ)(t)kL(Xγ,Xθ)≤Ct−β+γ,for all τ > 0. Proposition 6.12. [21, Theorem 2.2] Let τ < t and 0≤γ≤θ < 1 + δ. Then kA(t)βU(t, τ)A(τ)−γkL(X)≤C(θ, β)(t−τ)γ−β. 6.3 Pullback attractor In this last section we provide a brief review in the theory of pullback attractors. For more details, we recommend [17, 20, 24]. Let Xbe a Banach space and {S(t, τ) : X→X;t≥τ}a family of operators satisfying: 1. S(t, t) = IX, for all t∈R. 2. S(t, s) = S(t, τ)S(τ, s), for all t≥τ≥s,s∈R. 3. (s, ∞)3t7→ S(t, s)xis continuous for all x∈X. Such family is called a process in Xand we also denote it by S(·,·). We will usually call it nonlinear process to distinguish from the family U(t, τ)obtained in Definition 1.14. Given any x∈X, there are two distinct manners of studying the asymptotic dynamics of such evolution process: One called the pullback dynamics that basically fixes the final time tand evaluate what happens to S(t, s)xwhen s→ −∞and the other called forward dynamics, which consider S(t, s)x when sis fixed and t→ ∞. The pullback dynamics can be described by an object in the phase space called pullback attractor. We recall in the sequel some basic concepts and results of the theory of pullback attractor. To compare the distance between two sets in the phase space X, we use the Hausdorff semidistance: given A, B ⊂X, the Hausdorff semidistance between Aand Bis dist(A, B) = sup a∈A inf b∈Bd(a, b). This semidistance measures how much the set Ais located outside the set B. If Aand Bare closed, then dist(A, B) = 0 ⇒A⊂B. 101 Chapter 6. Global well-posedness and asymptotic dynamics Definition 6.13. Let S(·,·)be a process. A family A(·) = {A(t)⊂X;t∈R}pullback attracts B⊂X if, for each t∈R, dist(S(t, s)B, A(t)) s→−∞ −−−−→ 0. Definition 6.14. The pullback attractor of S(·,·)is a family A(·) = {A(t)⊂X;t∈R}that satisfies: 1. A(t)is compact for all t∈R. 2. A(·)is invariant by S(·,·), that is, S(t, s)A(s) = A(t), for all t≥s,s∈R. 3. A(·)pullback attracts bounded sets of X. 4. A(·)is the minimal closed family that satisfies (3). The existence of such object in the phase space is guaranteed whenever we find a family of compact pullback attracting sets. Theorem 6.15. [20, Theorem 2.12] Let S(·,·)be a process. The statements below are equivalent: 1. S(·,·)has a pullback attractor A(·). 2. There exists a family of compact sets K(·)that pullback attracts bounded sets of X. Corollary 6.16. If there exists a fixed compact set K⊂Xsuch that, for any bounded set B⊂X dist(S(t, s)B, K)→0when s→ −∞, then S(·,·)has a pullback attractor A(·)such that ∪t∈RA(t)⊂K. The description of the pullback attractor can be given in terms of global bounded solutions, which we define in the sequel. Definition 6.17. A continuous function ξ:R→Xis a global solution of S(·,·)if for any t, s ∈R,t≥s, S(t, s)ξ(s) = ξ(t). Moreover, we say that a global solution ξ(·) : R→Xof S(·,·)is bounded in the past if there exists τ∈Rsuch that {ξ(t) : t≤τ}is bounded in X. From the pullback attraction property and the invariance of the global solution, we readily obtain: Proposition 6.18. Let S(·,·)be a process with pullback attractor A(·). If ξ(·)is a global solution bounded in the past, then ξ(t)∈ A(t),∀t∈R. 108 Chapter 7. Domains with a handle: Attractors 2γe2γ(t−τ)kwk2 L2(Ω) +e2γ(t−τ)d dtkwk2 L2(Ω) ≤e2γ(t−τ)2M|Ω| d dt he2γ(t−τ)kwk2 L2(Ω)i≤e2γ(t−τ)2M|Ω|.(7.6) Integrating from τto twe derive e2γ(t−τ)kw(t)kL2(Ω) −kw0kL2(Ω) ≤[e2γ(t−τ)−1]M γ|Ω| kw(t)k2 L2(Ω) ≤e−2γ(t−τ)kw0k2 L2(Ω) +M γ|Ω|. Taking the square roots on both sides and using the inequality |a+b|r≤2r(|a|r+|b|r)for any r > 0, we obtain the desired inequality. L2k−estimate for w The iteration technique consists in obtaining L2k−estimates of wby using the estimate in the L2k−1−norm. In other words, it is an inductive procedure. Therefore, from the L2−norm obtained above, we derive L4−estimate, than L8and so on. This is the procedure applied at the following lemma, but before we enunciate it, we discuss a convention that we adopt in the next results. Remark 7.7. Given a bounded set B⊂Lp(Ω) such that kw0kLp(Ω) ≤Lfor w0∈B, after any evolution in time, the solutions starting with initial conditions in Bbecome bounded in stronger norms. Indeed, from the continuity of the solution, for any τ < t∗, with t∗arbitrarily close to τ,kw(t, τ, w0)kLp(Ω) ≤CL, for all τ≤t≤t∗, and from the variation of constants formula, for θ∈[0,1), kw(t∗, τ, w0)kYθ q≤ kPΩ(t∗, τ)w0kYθ q+Zt∗ τkPΩ(t∗, s)F(w(s, τ, w0))kYθ qds ≤ kPΩ(t∗, τ)kL(Lq(Ω),Y θ q)kw0kLq(Ω)+Zt∗ τkPΩ(t∗, s)kL(Lq(Ω),Y θ q)kF(w(s, τ, w0))kLq(Ω)ds ≤C(θ)(t∗−τ)−θL+Zt∗ τ C(t∗−s)−θ(1 + kw(s, τ, w0)kρ Lp(Ω))ds ≤C(θ, ρ, kw0kLp(Ω)), where we used Theorem 6.12 to estimate kPΩ(t∗, τ)kL(Lq(Ω),Y θ q)and C(θ, ρ, kw0kLp(Ω))denotes a constant that depends on θ, ρ, the embedding Lp(Ω) ,→Lq(Ω) and kw0kLp(Ω). In particular, for θ > N 2q(see Figure 7.1) we obtain kw(t∗, τ, w0)kL∞(Ω) ≤C(ρ, kw0kLp(Ω)). In conclusion, given any bounded set Bin Lp(Ω), after an arbitrarily small evolution takes place, this set Bbecomes bounded in L∞(Ω). Since we are interested in the asymptotic dynamics of the problem, whenever we wish to estimate the solution, we will assume that given any bounded set of initial condition Bin Lp(Ω), this set will also be bounded in L∞(Ω). If that is not the case, we evolve the system any arbitrary time and restart the evolution from this point. For this reason, we de not lose generality in the next results by assuming that w0∈L∞(Ω) (consequently, w0∈L2k(Ω)). 109 Chapter 7. Domains with a handle: Attractors Lemma 7.8. Let w(·, τ, w0)be the solution of (7.1) and assume that w0∈L∞(Ω). Given any k∈N, there exists constant c > 0independent of ksuch that, for t>τ, kw(t)k2k L2k(Ω) ≤e−2k(t−τ)kw0k2k L2k(Ω) +c(2k)N 2+1e−2k(t−τ)Zt τ e2k(s−τ)kw(s)k2k L2k−1(Ω)ds+hM γ|Ω|i,(7.7) as long as the solution exists. Proof. Multiplying the equation in (7.1) by w2k−1and integrating in Ω, we obtain ZΩ wtw2k−1dx =ZΩ div(a(t, x)∇w)w2k−1dx −ZΩ w2kdx +ZΩ f(w)ww2k−2dx. The term on the left side can be replaced by 1 2k d dt ZΩ w2kdx =ZΩ wtw2k−1dx, whereas from the dissipativeness condition, we obtain ZΩ f(w)ww2k−2dx ≤ZΩ (1 −γ)w2kdx +MZΩ w2k−2dx ≤ZΩ (1 −γ)w2kdx +MZΩ w2k+ 1dx. In the last inequality we used the fact that a2k−2< a2k+ 1 for any positive a. This inequality holds since a2k−2<1, if a < 1, and a2k−2=a2k a2≤a2k, if a > 1. Thus, 1 2k d dt ZΩ w2kdx ≤ZΩ div(a(t, x)∇w)w2k−1dx + [M−γ]ZΩ w2kdx +M|Ω|. Integration by parts leads to −ZΩ [a(t, x)∇w](2k−1)w2k−2∇wdx =−(2k−1) ZΩ a(t, x)(∇w)2w2k−2dx. Note that ∇w2k−1= 2k−1w2(k−1)−1∇wand h∇w2(k−1) i2= 22(k−1)w2k−2(∇w)2, so the term w2k−2(∇w)2can be replaced by 1 22(k−1) h∇w2(k−1) i2 . Therefore, −(2k−1) ZΩ a(t, x)(∇w)2w2k−2dx ≤ −a0(2k−1)(22−2k)ZΩh∇w2(k−1) i2 dx and the inequality studied becomes (after multiplying both sides by 2k), d dt ZΩ w2kdx ≤ −a0(2k−1)(22−k)ZΩh∇w2(k−1) i2 dx + 2k[M−γ]ZΩ w2kdx + 2kM|Ω|.(7.8) If for a certain u∈W1,2(Ω) ∩L1(Ω) we apply Nirenberg-Gagliardo’s inequality (Lemma P.12) with j= 0,p= 2,m= 1,r= 2,q= 1 and θ=N N+2, we obtain kukL2(Ω) ≤C(N, Ω)k∇uk N N+2 L2(Ω)kuk 2 N+2 L1(Ω). 110 Chapter 7. Domains with a handle: Attractors If we also use the Young generalized inequality (Lemma P.13) with conjugated exponents ξ=1 θ= N+2 Nand ξ0=N+2 2, we obtain kukL2(Ω) ≤εk∇ukL2(Ω) +1 εN 2kukL1(Ω). Taking the square power on both sides (and rearranging ε2for ε), (1 −ε)kuk2 L2(Ω) ≤ kuk2 L2(Ω) ≤εk∇uk2 L2(Ω) +1 εN 2kuk2 L1(Ω). We apply the above inequality for u=w2(k−1) kw2(k−1) k2 L2(Ω) =ZΩ w2(2(k−1))dx =ZΩ w2kdx,   ∇hw2(k−1) i   2 L2(Ω) =ZΩ∇hw2(k−1) i 2 dx, kw2(k−1) k2 L1(Ω) =ZΩ w2(k−1) dx2 , and it becomes −ZΩ∇hw2(k−1) i 2 dx ≤1 εN 2+1 ZΩ w2(k−1) dx2 −(1 −ε) εZΩ w2kdx. We use this inequality with a proper choice of εand apply it at (7.8) in order to obtain a negative term multiplying RΩw2kdx. d dt ZΩ w2kdx ≤−a0 2k−1 (2k−2) (1 −ε) εZΩ w2kdx +a0 2k−1 (2k−2) 1 εN 2+1 ZΩ w2(k−1) dx2 + 2k[M−γ]ZΩ w2kdx + 2kM|Ω|. Note that 2≤2k−1 (2k−2)≤4, and we can readjust the preceding inequality to obtain d dt ZΩ w2kdx ≤−2a0 (1 −ε) ε+ 2k[M−γ]ZΩ w2kdx + 4a0 1 εN 2+1 ZΩ w2(k−1) dx2 + 2kM|Ω|. Choosing ε=c2−k, with csmall enough to ensure that 2a0 (1 −ε) ε= 2a0 1−c2−k c2−k>2a0 c2k>2k([M−γ] + 1), (for example, c=a0 ([M−γ]+1) ) we obtain d dt ZΩ w2kdx ≤−2kZΩ w2kdx + 4a0 1 cN 2+1 (2−k)N 2+1 ZΩ w2(k−1) dx2 + 2kM|Ω| 111 Chapter 7. Domains with a handle: Attractors =−2kZΩ w2kdx +c(2k)N 2+1 ZΩ w2(k−1) dx2 + 2kM|Ω|. We have then achieved the desired differential inequality 2kZΩ w2kdx +d dt ZΩ w2kdx ≤c(2k)N 2+1 ZΩ w2(k−1) dx2 + 2kM|Ω|.(7.9) Note that ZΩ w2(k−1) dx2 ="ZΩ w2(k−1) dx1 2(k−1) #2k =hkwkL2(k−1) (Ω)i2k . Inequality (7.9) becomes 2kkw(t)k2k L2k(Ω) +d dtkw(t)k2k L2k(Ω) ≤c(2k)N 2+1kw(t)k2k L2k−1(Ω) + 2kM|Ω| and then d dt he2k(t−τ)kw(t)k2k L2k(Ω)i≤e2k(t−τ)c(2k)N 2+1kw(t)k2k L2k−1(Ω) +e2k(t−τ)2kM|Ω|.(7.10) Integrating from τto t, we obtain e2k(t−τ)kwk2k L2k(Ω) ≤ kw0k2k L2k(Ω) +c(2k)N 2+1 Zt τ e2k(s−τ)kw(s)k2k L2k−1(Ω)ds + [e2k(t−τ)−1]M|Ω| kwk2k L2k(Ω) ≤e−2k(t−τ)kw0k2k L2k(Ω) +c(2k)N 2+1e−2k(t−τ)Zt τ e2k(s−τ)kw(s)k2k L2k−1(Ω)ds +M|Ω| and the statement of the lemma follows by noting that we can consider γ∈(0,1) and M|Ω| ≤ hM γ|Ω|i (this adjustment of constant is just to facilitate future calculus and notations). From the recurrence formula obtained in Lemma 7.8, we derive the next proposition: Proposition 7.9. Let M,γbe the constants obtained from the dissipativeness condition and assume w0∈L∞(Ω). Given any k∈N, there exists a constant D=D(N, k, γ)that depends on k,Nand γ, such that, for t > τ, kw(t)kL2k(Ω) ≤D(N, k, γ)e−γ(t−τ)kw0kL2k(Ω) +hM γ|Ω|i1 2, as long as the solution exists. Proof. We prove for k= 2 and k= 3 to see the pattern. The result follows from induction. We will use the inequality |a+b|r≤2r(|a|r+|b|r), for any r > 0, whenever we need to estimate the power of a sum of two terms. 112 Chapter 7. Domains with a handle: Attractors Let k= 2. Rather then replacing the value of kin the inequalities, we will keep it to help us obtain a generalization for any k∈N. We first estimate the integral that appears in (7.7) using the L2−bound obtained for the solution in Proposition 7.6: Zt τ e2k(s−τ)kw(s)k2k L2(Ω)ds ≤Zt τ e2k(s−τ)(21 2)2ke−γ(s−τ)kw0kL2(Ω) +hM γ|Ω|i1 22k ds ≤Zt τ e2k(s−τ)(21 2)2k(2)2k(e−2kγ(s−τ)kw0k2k L2(Ω) +hM γ|Ω|i2k 2)ds ≤(21 2)2k(2)2kZt τ e2k(1−γ)(s−τ)kw0k2k L2(Ω)ds + (21 2)2k(2)2khM γ|Ω|i2k 21 2ke2k(t−τ) ≤(21 2)2k(2)2ke2k(1−γ)(t−τ) 2k(1 −γ)kw0k2k L2(Ω) + (21 2)2k(2)2k1 2ke2k(t−τ)hM γ|Ω|i2k 2. Therefore, replacing it in (7.7), we have kw(t)k2k L2k(Ω) ≤e−2k(t−τ)kw0k2k L2k(Ω) +c(2k)N 2+1(21 2)2k(2)2k1 2k(1 −γ)e−2kγ(t−τ)kw0k2k L2(Ω) +c(2k)N 2+1(21 2)2k(2)2k1 2khM γ|Ω|i2k 2+hM γ|Ω|i ≤e−2k(t−τ)kw0k2k L2k(Ω) +c(2k)N 2+1(21 2)2k(2)2k1 2k(1 −γ)e−2kγ(t−τ)kw0k2k L2(Ω) +c(2k)N 2+1(21 2)2k(2)2k1 2hM γ|Ω|i2k 2+c(2k)N 2+1(21 2)2k(2)2k1 2hM γ|Ω|i2k 2 ≤e−2k(t−τ)kw0k2k L2k(Ω) +c(2k)N 2+1(21 2)2k(2)2k1 1−γe−2kγ(t−τ)kw0k2k L2(Ω) +c(2k)N 2+1(21 2)2k(2)2khM γ|Ω|i2k 2 and we assumed that c(2k)N 2+1 and M γ|Ω|are larger or equal than 1(we can increase c, M if that is not the case). Moreover, let dk−1,k denotes the embedding constant of L2k(Ω) ,→L2k−1(Ω). Then kw0kL2≤dk−1,kkw0kL2kand inequality above can be written as kw(t)k2k L2k(Ω) ≤e−2k(t−τ)kw0k2k L2k(Ω) +c(2k)N 2+1 d2k k−1,k 1−γ(21 2)2k(2)2ke−2kγ(t−τ)kw0k2k L2k(Ω) +c(2k)N 2+1(21 2)2k(2)2khM γ|Ω|i2k 2 ≤c(2k)N 2+1 1 1−γ(1 + d2k k−1,k)(21 2)2k(2)2k(e−2kγ(t−τ)kw0k2k L2k(Ω) +hM γ|Ω|i2k 2). Extracting the 2kroot and denoting C(k, N, γ) = nc(2k)N 2+1 1 1−γ(1 + d2k k−1,k)o1 2k, we obtain kw(t)kL2k(Ω) ≤C(k, N, γ)2(21 2)2 1 2ke−γ(t−τ)kw0kL2k(Ω) +hM γ|Ω|i1 2 113 Chapter 7. Domains with a handle: Attractors =D(2, N, γ)e−γ(t−τ)kw0kL2k(Ω) +hM γ|Ω|i1 2, where D(k, N, γ) = 2k−1Qk i=1 21 2iQk i=2 C(i, N, γ). For k= 3 the calculation is analogous. The previous information about kw(t)kL4(Ω) allows us to obtain now estimates on kw(t)kL23(Ω). First, the integral on (7.7) satisfies Zt τ e2k(s−τ)kw(s)k2k L4(Ω) ≤D(2, N, γ)2k(2)2kZt τ e2k(s−τ)(e−2kγ(s−τ)kw0k2k L4(Ω)+hM γ|Ω|i2k 2)ds ≤D(2, N, γ)2k(2)2k(1 2k(1 −γ)e2k(1−γ)(t−τ)kw0k2k L4(Ω) +1 2ke2k(t−τ)hM γ|Ω|i2k 2). Replacing this expression in (7.7), we obtain kw(t)k2k L2k(Ω) ≤e−2k(t−τ)kw0k2k L2k(Ω) +D(2, N, γ)2k(2)2kc(2k)N 2+1 1 2k 1 1−γe−2kγ(t−τ)kw0k2k L4(Ω) +D(2, N, γ)2k(2)2kc(2k)N 2+1 1 2khM γ|Ω|i2k 2+hM γ|Ω|i ≤e−2k(t−τ)kw0k2k L2k(Ω) +D(2, N, γ)2k(2)2kc(2k)N 2+1 1 1−γe−2kγ(t−τ)kw0k2k L4(Ω) +D(2, N, γ)2k(2)2kc(2k)N 2+1hM γ|Ω|i2k 2 ≤e−2k(t−τ)kw0k2k L2k(Ω) +D(2, N, γ)2k(2)2kc(2k)N 2+1 1 1−γe−2kγ(t−τ)d2k k−1,kkw0k2k L2k(Ω) +D(2, N, γ)2k(2)2kc(2k)N 2+1hM γ|Ω|i2k 2 ≤D(2, N, γ)2k(2)2kc(2k)N 2+1 1 1−γ(1 + d2k k−1,k)(e−2kγ(t−τ)kw0k2k L2k(Ω) +hM γ|Ω|i2k 2). Extracting the 2k−root and using nc(2k)N 2+1 1 1−γ(1 + d2k k−1,k)o1 2k=C(k, N, γ), we obtain kw(t)kL2k(Ω) ≤D(2, N, γ)C(3, N, γ)(2)(2 1 2k)e−γ(t−τ)kw0kL2k(Ω) +hM γ|Ω|i1 2 =D(3, N, γ)e−γ(t−τ)kw0kL2k(Ω) +hM γ|Ω|i1 2. In general, kw(t)kL2k(Ω) ≤D(k, N, γ)e−γ(t−τ)kw0kL2k(Ω) +hM γ|Ω|i1 2, where C(i, N, γ) = nc(2i)N 2+1 1 1−γ(1 + d2i i−1,i)o1 2iand D(k, N, γ)=2k−1Qk i=1 21 2iQk i=2 C(i, N, γ).(7.11) 114 Chapter 7. Domains with a handle: Attractors Yθ qestimates for w The estimates of the solution obtained so far allow us to derive estimates in better norms. To be precise, we can obtain estimates in each Yθ qspace for 0≤θ < 1. In (7.2) we proved that kF(w(t))kLq(Ω) ≤C(1 + kw(t)kρ Lp(Ω)). For p>Nfixed, let k0∈Nsuch that k0≥log2p. (7.12) In this case 2k0≥pand from classical Lr−embedding, we can rewrite inequality above as kF(w(t))kLq(Ω) ≤C(1 + kw(t)kρ L2k0(Ω)), adjusting the constant C. This fact and the results obtained earlier imply: Proposition 7.10. Let 0≤θ < 1,w0∈L∞(Ω) and k0∈Nsuch that k0≥log2p. There exist constants E1and E2depending on Ω,θ,ρ,N,k0,Mand γ, such that, for τ < t −1< t, kw(t, τ, w0)kYθ q≤E1e−γ(t−τ)kw0kL∞(Ω) +kw0kρ L∞(Ω)+E2, as long as the solution exists. Proof. From the variation of constants formula, Proposition 7.9, the embeddings Lr1(Ω) ,→Lr2(Ω) whenever r2≤r1(used for q≤p≤2k0) and estimate for PΩ(·,·)derived in Theorem 6.9, we obtain (adjusting constants when needed) kw(t, τ, w0)kYθ q=kw(t, t −1, w(t−1, τ, w0))kYθ q ≤ kPΩ(t, t −1)w(t−1, τ, w0)kYθ q+Zt t−1kPΩ(t, s)F(w(s, τ, w0))kYθ qds ≤C(θ)(1)−θkw(t−1, τ, w0)kLq(Ω) +Zt t−1 C(θ)(t−s)−θkF(w(s, τ, w0))kLq(Ω)ds ≤C(θ)|Ω|1 q−1 2k0kw(t−1, τ, w0)kL2k0(Ω) +C(θ)Zt t−1 (t−s)−θ1 + kw(t)kρ L2k0(Ω)ds ≤C(θ)|Ω|1 q−1 2k0D(k0, N, γ)e−γ(t−1−τ)kw0kL2k0(Ω) +hM γ|Ω|i1 2 +C(θ)Zt t−1 (t−s)−θ1 +D(k0, N, γ)ρe−γ(s−τ)kw0kL2k0(Ω) +hM γ|Ω|i1 2ρds ≤C(θ)|Ω|1 q−1 2k0D(k0, N, γ)e−γ(t−τ)kw0kL2k0(Ω) +hM γ|Ω|i1 2 +C(θ)Zt t−1 (t−s)−θ1 +D(k0, N, γ)ρ(2ρ)e−ργ(s−τ)kw0kρ L2k0(Ω) +hM γ|Ω|iρ 2ds 115 Chapter 7. Domains with a handle: Attractors ≤C(θ)|Ω|1 q−1 2k0D(k0, N, γ)e−γ(t−τ)kw0kL∞(Ω)|Ω|1 2k0+C(θ)|Ω|1 q−1 2k0D(k0, N, γ)hM γ|Ω|i1 2 +C(θ) + C(θ)D(k0, N, γ)ρ(2ρ)e−ργ(t−1−τ)kw0kρ L2k0(Ω) +C(θ)D(k0, N, γ)ρ(2ρ)hM γ|Ω|iρ 2 ≤C(θ)|Ω|1 qD(k0, N, γ)e−γ(t−τ)kw0kL∞(Ω) +C(θ)D(k0, N, γ)ρ(2ρ)e−ργ(t−τ)kw0kρ L∞(Ω)|Ω|ρ 2k0 +C(θ) + C(θ)D(k0, N, γ)|Ω|1 q−1 2k0hM γ|Ω|i1 2+C(θ)D(k0, N, γ)ρ(2ρ)hM γ|Ω|iρ 2 ≤E1e−γ(t−τ)kw0kL∞(Ω) +kw0kρ L∞(Ω)+E2, where D(k0, N, γ)is the constant given in (7.11) and we assumed that D(k0, N),hM γ|Ω|i>1to group the terms above as E1=C(θ)D(k0, N, γ)ρ(2ρ) max{|Ω|1 q,|Ω|ρ 2k0}, E2= 3C(θ)D(k0, N, γ)ρ(2ρ) max{1,|Ω|1 q−1 2k0}hM γ|Ω|iρ 2. (7.13) 7.1.2 Global well-posedness and Pullback attractor The Yθ q−estimate obtained in Proposition 7.10 for N 2qρ−1 ρ< θ < 1(see Figure 7.1) implies, in particular, that kw(t, τ, w0)kLp(Ω) is finite in each bounded interval [τ, T ]. Then, w(t, τ, w0)is globally defined in time (Theorem 6.8) and we can obtain a nonlinear process SΩ(t, τ) : Lp(Ω) →Lp(Ω) given by the solution w(t, τ, w0), that is, SΩ(t, τ)w0=w(t, τ, u0). If PΩ(t, τ)is the linear process associated to B(t), t ∈R, then SΩ(t, τ)w0=PΩ(t, τ)w0+Zt τPΩ(t, s)F(w(s))ds. We prove this result and the existence of pullback attractor in next two theorems. Theorem 7.11. Let w(·, τ, w0)be the solution of (7.1) and M, γ the constants obtained from the dissipativeness condition (7.3). Then w(·, τ, w0)is globally defined and associated to it there is a nonlinear process SΩ(t, τ)in Lp(Ω) given by SΩ(t, τ)w0=w(t, τ, w0), for all t≥τ. Moreover, the closed ball in Yθ qcentered in zero and with radius E2,BYθ q[0, E2], is a pullback attracting set for the process SΩ(t, τ)in the topology of Yθ q, where E2is given in (7.13) and depends on θ, ρ, N, k0, M and γ. Proof. Let w0∈Lp(Ω). It follows from Remark 7.7 that after any arbitrarily small evolution in time, w∗=w(t∗, τ, w0)∈L∞(Ω). Therefore, if we start the evolution at instant t∗and at the point w∗, Proposition 7.10 implies that kw(t, t∗, w∗)kYθ qis finite in any bounded interval [t∗, T]. For θ > N 2qρ−1 ρ, this boundedness implies global existence of the solution. 116 Chapter 7. Domains with a handle: Attractors Moreover, let B⊂Lp(Ω) be a bounded set such that kw0kLp(Ω) ≤Lfor any w0∈B. It also follows from Remark 7.7 that after an arbitrarily small evolution in time, t∗> τ, the elements w∗=w(t∗, τ, w0) are bounded in L∞(Ω), that is kw∗kL∞(Ω) ≤˜ L. Therefore, kw(t, t∗, w∗)kYθ q≤E1e−γ(t−t∗)(˜ L+˜ Lρ) + E2, and dist(SΩ(t, t∗)SΩ(t∗, τ)w0, BYθ q[0, E2]) = dist(w(t, t∗, w∗), BYθ q[0, E2]) t−t∗→∞ −−−−−→ 0, uniformly for w0∈B. The existence of pullback attractor for SΩ(t, τ)is now a consequence of the previous result. Theorem 7.12. Assume that p>Nand max nN 2δ,p(2N+1) 2p+1 o< q ≤p,X=U0 p,Y=U0 q,a:R×Ω0→ R+satisfies (A.2) and (A.3) and f:R→Rsatisfies (A.4) and (D). The solution w(t)for the equation (7.1) in Ωdefines a nonlinear process SΩ(t, τ)in Lp(Ω) which has a pullback attractor AΩ(t)in Lp(Ω). Moreover, ∪t∈RAΩ(t)⊂ C1,η(Ω), for some η > 0, and pullback attracts bounded sets of Lp(Ω) in the topology of C1,η(Ω). Proof. The conditions required ensure existence of local mild solution for the problem in Ω0= Ω ∪R0 (see Proposition 2.11). In particular, they ensure the existence of solution in Ω. Moreover, in Theorem 7.11 we proved the existence of a pullback attracting bounded set in Yθ qfor any 0≤θ < 1. Since Yθ qis compactly embedded in Lp(Ω) for θ > N 2q(ρ−1 ρ)(as seen in Lemma 7.1), we conclude that BYθ q[0, E2]is a compact pullback attracting set for the process SΩ(t, τ)in Lp(Ω). Therefore, from Corollary 6.16, there exists a pullback attractor AΩ(t)⊂BYθ q[0, E2]c ,→Lp(Ω),∀t∈R. that attracts bounded sets of Lp(Ω) in the topology of Yθ q. Moreover, if θ > 1 2+N 2q, then Yθ q,→ C1,η(Ω) and the last statement follows. Remark 7.13. Note we only required the conditions posed in Proposition 2.11 rather then the more restrictive conditions of Proposition 5.2. We do that because once the problem is decoupled, the fact that B(t)is sectorial ensures differentiability of the mild solution w. 7.1.3 Properties of wt Before we proceed to the analysis in the channel R0, some properties of wt, the derivative of win Ω, are required. As we will see in the next section, the nonhomogeneous condition at the junction points (given by w(t, p0)and w(t, p1)) will be incorporated at the equation in the channel via an appropriate change of variable. This will cause the appearance of a term depending on wtin the equation and the properties presented in this section will be necessary. 117 Chapter 7. Domains with a handle: Attractors Let us first recall that one of the conditions required to obtain local solution for the equation in Ω0= Ω ∪R0is that q > N 2δ(see Table 2.1), that is δ > N 2q. This condition, as we see in the next proposition, ensures that wt(t, x)belongs to Yθ q,→Cν(Ω), for θ∈(N 2δ, δ), and estimate for the derivative in the Cν(Ω)−norm is available (see Theorem 6.9). Moreover, after a certain time, those derivatives are enclosed in a compact set of the phase space and the variation of the solution in the long-time is somehow controlled. We say in the next proposition that a given set Kpullback absorbs wt(t, τ, w0)in the sense that there exists T∗>0such that t−τ > T∗implies that wt(t, τ, w0)∈K. Proposition 7.14. Let 0≤θ < δ and w0∈L∞(Ω). Then there exists constants F1and F2depending on Ω,θ,ρ,k0,N,Mand γ, such that, for τ < t −1< t, kwt(t, τ, w0)kYβ q≤F1e−γ(t−τ)kw0kL∞(Ω) +kw0kρ L∞(Ω)+F2, and, for any ε > 0,BYθ q[0, F2+ε]is a pullback absorbing bounded set for wt(t, τ, w0)in the topology of Yθ q. Moreover, for N 2q< θ < δ, there exists ν > 0such that BCν(Ω) [0, F2+ε]c ,→Lp(Ω) is a compact pullback absorbing set for wt(t, τ, w0)in the topology of Cν(Ω). Proof. From Theorem 6.9, wt(t)∈Yθ qfor any 0≤θ < δ and t>τ. Together with Proposition 7.10, we obtain kwt(t, τ, w0)kYβ q≤C(θ)(t−(t−1))−1−θkw(t−1, τ, w0)kLq(Ω) ≤C(θ)E1e−γ(t−τ)kw0kL∞(Ω) +kw0kρ L∞(Ω)+C(θ)E2. Taking Fi=C(θ)Ei,i= 1,2, we derive the desired inequality. The other statements follows from Lemma 7.1. From the proposition above we conclude that t7→ |wt(t, x)| ∈ Ris bounded for each x∈Ω. In this case, the solution t7→ |w(t, x)| ∈ Rcan increase/decrease in the long-time dynamics, but those variations are somehow controlled and limited. 7.2 The equation on the line segment R0 Now that we have estimates for both wand wtin Ω, we turn our attention to the reaction-diffusion equation that takes place at the line segment R0. Unlike the dynamics in Ωthat is indifferent to what happens at the line segment, the evolution in R0 is subordinated to the values wassume at the points, p0and p1, at the time t. This works as a boundary condition for the equation in R0which can be seen as a heat equation in [0,1] with nonhomogeneous and 124 Chapter 7. Domains with a handle: Attractors and we conclude that zψ(t, z)≤(1 −γ)z2+H1e−γ(t−τ)kw0kL∞(Ω) +kw0kρ L∞(Ω) +kw0kρ2 L∞(Ω)+H2, where H1=r12C2ρ3ρGρ 1, H2=r1(C2ρ(1 + rρ 1)+2C2ρ3ρGρ 2)(7.24) and G1, G2are constants given in (7.21). To simplify the notation in next sections, we denote by m2(w0) = H1kw0kL∞(Ω) +kw0kρ L∞(Ω) +kw0kρ2 L∞(Ω), M2(t, τ, w0) = e−γ(t−τ)m2(w0). (7.25) Remark 7.22. Note that (1) the function M2(t, τ, w0)is decreasing in tand M2(t, τ, w0)t−τ→∞ −−−−→ 0,uniformly for w0in bounded sets of L∞(Ω). (2) for any t > t −1> τ, M2(t, τ, w0) = e−γ(t−τ)m2(w0) = e−γe−γ(t−1−τ)m2(w0) = e−γM2(t−1, τ, w0). (3) the dissipativeness condition is restated as sψ(t, s; (τ, w0)) ≤(1 −γ)s2+M2(t, τ, w0) + H2. 7.2.3 Estimates in the channel R0 The ideas for this section are essentially the same of the ones used to obtain estimates for the function win Ω. However, rather than dealing with v, we will estimate the function z, solution of the associated problem with homogeneous boundary condition: zt−L(t)z=ψ(t, z), t > τ;z(τ, r) = z0(r) = v0(r)−ξ(0, r)∈Lp(0,1), where L(t)is the family of linear operators given in Lemma 7.17. Once we obtain estimates for z, those are transferred to v, since we know that ξis bounded (in Cν(Ω)) and z=v−ξ. The outline to obtain the estimates is the following: 1. We first evaluate kz(t)kL2(0,1) by performing formal calculus over the differential equation. 2. Then, via the iterative procedure we evaluate kz(t)kL2k(0,1). 125 Chapter 7. Domains with a handle: Attractors 3. From the previous items we can obtain estimates for z(t)in the spaces Vθ q. Moreover, we will denote by PR0(t, τ) : Lq(0,1) →Lq(0,1) the linear process associated to the family L(t), t ∈R, and, from the variation of constants formula, zcan be given as z(t, τ, z0) = PR0(t, τ)z0+Zt τPR0(t, s)ψ(t, z(s))ds. (7.26) Due to the similarity of the calculations involved in determining those estimates and the ones done in Section 7.1, we simply point out the differences in the proofs. Those differences come from the dissipativeness condition (7.22) that slightly differs from the one used for the problem in Ω. Note that in this case, the dissipation depends on (τ, w0). Remark 7.23. The parabolic structure of the associated problem (7.20) implies in similar regularization properties as the ones discussed in Remark 7.7, this time applied to the initial condition z0=v0− ξ(τ, w0). Therefore, if z0∈Lp(0,1), after any arbitrarily small evolution τ < t∗in time, z(t∗, τ, z0)will be an element of L∞(0,1) (and consequently, v(t∗) = z(t∗)−ξ(t∗)∈L∞(0,1)). Moreover, bounded sets of Lp(0,1) are taken in bounded sets of L∞(0,1) through the variation of constants formula (7.26). For this reason, we do not lose generality in the next results by assuming that (w0, v0)∈L∞(Ω) × L∞(0,1) and, consequently, z0∈L∞(0,1). L2−estimate for z Proposition 7.24. Let (w0, v0)∈L∞(Ω) ×L∞(0,1) and z0=v0−ξ(τ, w0)∈L∞(0,1). If H1, H2are the constants obtained in Proposition 7.21 depending only on Ω,ρ,k0,N, and γ(but not on t, τ or w0) and M2(t, τ, w0)is given in (7.25), then the solution z(·, τ, z0)to (7.20) satisfies, as long as it exists, kz(t, τ, z0)kL2(0,1) ≤21 2e−γ(t−τ)kz0kL2(0,1) +h2 γM2(t, τ, w0) + 2 γH2i1 2. Proof. We take the inner product in L2(0,1) of the equation (7.20) with z, 1 2 d dtkzk2 L2(0,1) =−Z1 0 a(t, r)|∂rz|2dr −kzk2 L2(0,1) +Z1 0 ψ(t, z)zdr and item (3) of Remark 7.22 leads to inequality kzk2 L2(0,1) +1 2 d dtkzk2 L2(0,1) ≤(1 −γ)kzk2 L2(0,1) +M2(t, τ, w0) + H2. Adjusting the terms above and using M(t, τ, w0) = e−γ(t−τ)m2(w0), we obtain 2γkzk2 L2(0,1) +d dtkzk2 L2(0,1) ≤2 [M2(t, τ, w0) + H2] = e−γ(t−τ)2m2(w0)+2eγ(t−τ)H2. 126 Chapter 7. Domains with a handle: Attractors The steps here differ from Proposition 7.6 at (7.6) due to the time dependence of the right side of inequality. This time dependence is important and must not be overlooked since it provides the exponential decay for the terms in which kw0kL2(Ω) features. After multiplying by e2γ(t−τ)we obtain d dt he2γ(t−τ)kzk2 L2(0,1)i≤eγ(t−τ)2m2(w0) + eγ(t−τ)2H2. Integrating from τto t, and using that m2(w0) + eγ(t−τ)H2is increasing in t, we obtain e2γ(t−τ)kz(t)k2 L2(0,1) −kz0k2 L2(0,1) ≤2m2(w0) + eγ(t−τ)2H2Zt τ eγ(s−τ)ds ≤eγ(t−τ)1 γ2m2(w0) + eγ(t−τ)2H2. Therefore, kz(t)k2 L2(0,1) ≤e−2γ(t−τ)kz0k2 L2(0,1) +2 γM2(t, τ, w0) + 2 γH2. Taking the square roots on both sides and using the inequality |a+b|r≤2r(|a|r+|b|r), for r > 0, we obtain the desired inequality. L2k−estimate for z With the same iterative procedure used for the solution in Ω, we estimate L2k−norm of the solution in R0. Lemma 7.25. Let (w0, v0)∈L∞(Ω) ×L∞(0,1),z0=v0−ξ(τ, w0)∈L∞(0,1) and z(·, τ, z0)the solution of (7.20). Given any k∈N, there exists constant c > 0independent of ksuch that, for t>τ, kz(t)k2k L2k(0,1) ≤e−2k(t−τ)kz0k2k L2k(0,1) +c(2k)3 2e−2k(t−τ)Zt τ e2k(s−τ)kz(s)k2k L2k−1(0,1)ds +h2 γM2(t, τ, w0) + 2 γH2i, (7.27) as long as the solution exists. The constant H2is given in (7.24) and the function M2(t, τ, w0)in (7.25). Proof. The proof follow the exactly same steps as the proof of Lemma 7.8 up the inequality (7.10). From this point forward, some differences appear due to the time dependence of the term M2(t, τ, w0). Notice that, in this case, we replace |Ω|by the measure of (0,1), which is 1, and the dimension of the space is N= 1. Therefore, we have d dt he2k(t−τ)kz(t)k2k L2k(0,1)i≤e2k(t−τ)c(2k)3 2kz(t)k2k L2k−1(0,1) +e2k(t−τ)2k[M2(t, τ, w0) + H2].(7.28) We will integrate the above inequality from τto t, but in order to understand the behavior of the term in which M2(t, τ, w0)appears, we evaluate it separately: Zt τ e2k(s−τ)2k[M2(s, τ, w0) + H2]ds =Zt τ e2k(s−τ)2ke−γ(s−τ)[m2(w0) + eγ(s−τ)H2]ds 127 Chapter 7. Domains with a handle: Attractors ≤2k[m2(w0) + eγ(t−τ)H2]Zt τ e(2k−γ)(s−τ)ds ≤2k 2k−γe(2k−γ)(t−τ)[m2(w0) + eγ(t−τ)H2] ≤2k 2k−γe2k(t−τ)[M2(t, τ, w0) + H2]. Therefore, integrating (7.28) from τto tand some manipulation imply kz(t)k2k L2k(0,1) ≤e−2k(t−τ)kz0k2k L2k(0,1) +c(2k)3 2e−2k(t−τ)Zt τ e2k(s−τ)kz(s)k2k L2k−1(0,1)ds +2k−1 2k−1h2 γM2(t, τ, w0) + 2 γH2i. The statement of the lemma follows by noting that γ∈(0,1) and 2k−1 2k−1<1(these adjustment of constant is just to facilitate future calculus and notations). From the recurrence formula obtained in Lemma 7.25, we derive the next proposition: Proposition 7.26. Let (w0, v0)∈L∞(Ω) ×L∞(0,1),z0=v0−ξ(τ, w0)∈L∞(0,1) and z(·, τ, z0)the solution of (7.20). Given any k∈N, there exists a constant ˜ D(k, γ)that depends on kand γsuch that, for t > τ, kz(t)kL2k(0,1) ≤˜ D(k, γ)e−γ(t−τ)kz0kL2k(0,1) +h2 γM2(t, τ, w0) + 2 γH2i1 2, as long as the solution exists. The constant H2is given in (7.24) and the function M2(t, τ, w0)in (7.25). Proof. We verify for k= 2 to see the pattern. The result follows from induction. We will use the inequality |a+b|r≤2r(|a|r+|b|r), for r > 0, whenever we need to estimate the power of a sum of two terms. Let k= 2. Rather then replacing the value of kin the inequalities, we keep it to help us obtain a generalization for any k∈N. We first estimate the integral that appears in (7.27) using the L2−bound obtained for the solution in Proposition 7.24. We have Zt τ e2k(s−τ)kz(s)k2k L2(0,1)ds ≤Zt τ e2k(s−τ)(21 2)2ke−γ(s−τ)kz0kL2(0,1) +h2 γM2(s, τ, w0) + 2 γH2i1 22k ds ≤Zt τ e2k(s−τ)(21 2)2k(2)2kne−2kγ(s−τ)kz0k2k L2(0,1)ods +Zt τ e2k(s−τ)(21 2)2k(2)2ke−2k−1γ(s−τ)h2 γm2(w0) + 2 γH2eγ(s−τ)i2k 2ds 128 Chapter 7. Domains with a handle: Attractors ≤(21 2)2k(2)2ke2k(1−γ)(t−τ) 2k(1 −γ)kz0k2k L2(0,1) + (21 2)2k(2)2kZt τ e2k(2−γ)(s−τ)h2 γm2(w0) + 2 γeγ(s−τ)H2i2k 2ds ≤(21 2)2k(2)2ke2k(1−γ)(t−τ) 2k(1 −γ)kz0k2k L2(0,1) + (21 2)2k(2)2ke2k−1(2−γ)(t−τ) 2k−1(2 −γ)h2 γm2(w0) + 2 γeγ(t−τ)H2i2k 2 ≤(21 2)2k (2)2ke2k(1−γ)(t−τ) 2k(1 −γ)kz0k2k L2(0,1)+(21 2)2k(2)2ke2k(t−τ) 2k−1(2 −γ)(e−γ(t−τ))2k 2h2 γm2(w0)+2 γeγ(t−τ)H2i2k 2 ≤(21 2)2k(2)2ke2k(1−γ)(t−τ) 2k(1 −γ)kz0k2k L2(0,1) + (21 2)2k(2)2ke2k(t−τ) 2k−1(2 −γ)h2 γM2(t, τ, w0) + 2 γH2i2k 2. Therefore, replacing it in (7.27) and noticing that in the set (0,1), the embedding constants of L2k(0,1) ,→L2k−1(0,1) are less than 1, that is, k·kL2k−1(0,1) ≤ k·kL2k(0,1), we obtain kz(t)k2k L2k(0,1) ≤e−2k(t−τ)kz0k2k L2k(0,1) +c(2k)3 2(21 2)2k(2)2ke−2k(t−τ)e2k(1−γ)(t−τ) 2k(1 −γ)kz0k2k L2(0,1) +c(2k)3 2(21 2)2k(2)2k1 2k−1(2 −γ)h2 γM2(t, τ, w0) + 2 γH2i2k 2+h2 γM2(t, τ, w0) + 2 γH2i ≤e−2k(t−τ)kz0k2k L2k(0,1) +c(2k)3 2(21 2)2k(2)2ke−2kγ(t−τ) 2k(1 −γ)kz0k2k L2(0,1) +c(2k)3 2(21 2)2k(2)2k1 2k−1(1 −γ)h2 γM2(t, τ, w0) + 2 γH2i2k 2+h2 γM2(t, τ, w0) + 2 γH2i ≤e−2k(t−τ)kz0k2k L2k(0,1) +c(2k)3 2(21 2)2k(2)2ke−2kγ(t−τ) 1−γkz0k2k L2k(0,1) +c(2k)3 2(21 2)2k(2)2k1 1−γh2 γM2(t, τ, w0) + 2 γH2i2k 2 ≤c(2k)3 2(21 2)2k(2)2k2 1−γ(e−2kγ(t−τ)kz0k2k L2k(0,1) +h2 γM2(t, τ, w0) + 2 γH2i2k 2) and we assumed that c(2k)3 2(21 2)2k(2)2k2 1−γand 2 γH2are larger or equal than 1. Extracting the 2k−root on both sides and denoting ˜ C(k, γ) = nc(2k)3 22 1−γo1 2k, we obtain kz(t)kL2k(0,1) ≤˜ C(k, γ)(2)(21 2)(2 1 2k)(e−γ(t−τ)kz0kL2k(0,1) +2 γM2(t, τ, w0) + 2 γH21 2) =˜ D(2, γ)e−γ(t−τ)kz0kL2k(0,1) +h2 γM2(t, τ, w0) + 2 γH2i1 2, where ˜ D(k, γ) = 2k−1Qk i=1 21 2iQk i=2 ˜ C(i, γ). For k= 3, we obtain, in the same way as we did in Lemma 7.8, kz(t)kL2k(0,1) ≤˜ D(3, γ)e−γ(t−τ)kz0kL2k(0,1) +h2 γM2(t, τ, w0) + 2 γH2i1 2. 129 Chapter 7. Domains with a handle: Attractors In general, kz(t)kL2k(0,1) ≤˜ D(k, γ)e−γ(t−τ)kz0kL2k(0,1) +h2 γM2(w0) + 2 γH2i1 2, where ˜ C(i, γ) = nc(2i)3 22 1−γo1 2iand ˜ D(k, γ)=2k−1Qk i=1 21 2iQk i=2 ˜ C(i, γ). Vθ qestimate for z The estimates of the solution obtained so far allow us to derive estimates in better norms. To be precise, we can obtain estimates in each Vθ q=D(L(t)θ), for 0≤θ < 1, but in order to do that, we first need to evaluate kψ(t, z(t))kLq(0,1). Remark 7.27. From this point forward we will group all the constants depending on θ, Ω, ρ, p, q, N, M and γin a single constants C∗=C∗(θ, Ω, ρ, p, q, N, M, γ)>0, enlarging it whenever necessary. We will lose precision on the estimates by doing this, but on the other hand, it will be easier to identify the main ideas in the sequel. Lemma 7.28. Let (w0, v0)∈L∞(Ω) ×L∞(0,1),z0=v0−ξ(τ, w0)∈L∞(0,1) and z(·, τ, z0)the solution of (7.20), then there exists C∗=C∗(Ω, ρ, p, q, N, M, γ)>0depending on Ω, ρ, p, q, N, M and γ(but not on τor w0) such that kψ(t, z)kLq(0,1) ≤C∗he−ργ(t−τ)kz0kρ L∞(0,1) + [M2(t, τ, w0) + 1]ρi. Proof. The nonlinearity ψ(t, z)is given by ψ(t, z) = −ξ(t)−ξt(t) + f(ξ(t) + z). From the growth of f and the fact that the embedding constants of Lr1(0,1) ,→Lr2(0,1) whenever r1> r2are all less then 1, we obtain kψ(t, z)kLq(0,1) =k−ξ(t)−ξt(t) + f(ξ(t) + z)kLq(0,1) ≤ kξ(t) + ξt(t)kLq(0,1) +C(1 + kξ(t) + z(t)kρ Lp(0,1)) ≤ kξ(t) + ξt(t)kL∞(0,1) +Ckξ(t)kρ L∞(0,1) +Ckz(t)kρ L2k0(0,1), where k0is the positive integer such that k0≥log2pand L2k0(0,1) ,→Lp(0,1). It follows from Proposition 7.20 that kξ(t) + ξt(t)kL∞(0,1) ≤G1 H1 M2(t, τ, w0) + G2 Ckξ(t)kρ L∞(0,1) ≤C1 ρG1 H1 M2(t, τ, w0) + C1 ρG2ρ . 130 Chapter 7. Domains with a handle: Attractors Enlarging the constants such that the terms inside the brackets is larger than 1, if necessary, we have kξ(t) + ξt(t)kL∞(0,1) +Ckξ(t)kρ L∞(0,1) ≤C∗[M2(t, τ, w0) + 1]ρ,(7.29) with C∗=C∗(θ, Ω, ρ, p, q, N, M, γ)>0. Moreover, it follows from Proposition 7.26 and the inequality |a+b|r≤2r(|a|r+|b|r), for r > 0, that Ckz(t)kρ Lp(0,1) ≤Ckz(t)kρ L2k0(0,1) ≤C˜ D(k, γ)ρ(2ρ)e−ργ(t−τ)kz0kρ L2k0(0,1) +C˜ D(k, γ)ρ(2ρ)h2 γM2(t, τ, w0) + 2 γH2iρ 2 ≤C˜ D(k, γ)ρe−ργ(t−τ)kz0kρ L∞(0,1) +C˜ D(k, γ)ρ2 γρ [M2(t, τ, w0) + H2]ρ 2, ≤C∗he−ργ(t−τ)kz0kρ L∞(0,1) + [M2(t, τ, w0) + 1]ρ 2i. (7.30) Therefore, the desired result follows from (7.29) and (7.30), adjusting the constant C∗. The results obtained earlier imply: Proposition 7.29. Let (w0, v0)∈L∞(Ω) ×L∞(0,1),z0=v0−ξ(τ, w0)∈L∞(0,1) and z(·, τ, z0)the solution of (7.20). Then, for any 0≤θ < 1, there exists C∗=C∗(θ, Ω, ρ, p, q, N, M, γ)>0depending on θ, Ω, ρ, p, q, N, M and γ(but not on τor w0) such that kz(t)kVθ q≤C∗he−γ(t−τ)(kz0kL∞(0,1) +kz0kρ L∞(0,1))+[M2(t, τ, w0) + 1]ρi, as long as the solution exists. Proof. From the variation of constants formula and Proposition 6.12, we obtain kz(t, τ, z0)kVθ q=kz(t, t −1, z(t−1, τ, z0))kVθ q ≤ kPR0(t, t −1)z(t−1, τ, z0)kVθ q+Zt t−1kPR0(t, s)ψ(s, z(s)))kVθ qds ≤C(θ)(t−(t−1))−θkz(t−1, τ, z0)kLq(0,1) +Zt t−1 C(θ)(t−s)−θkψ(s, z(s))kLq(0,1)ds. Note that kz(t−1, τ, z0)kLq(0,1) is estimated in the same way as in (7.30): kz(t)kρ Lq(0,1) ≤C∗he−ργ(t−τ)kz0kρ L∞(0,1) + [M2(t, τ, w0) + 1]ρ 2i, whereas kψ(s, τ)kLq(0,1) was estimated in Lemma 7.28. From those results, we obtain kz(t, τ, z0)kVθ q≤C(θ)C∗(21 ρ)he−γ(t−1−τ)kz0kL∞(0,1) + [M2(t−1, τ, w0) + 1]1 2i 131 Chapter 7. Domains with a handle: Attractors +Zt t−1 C(θ)(t−s)−θC∗he−ργ(s−τ)kz0kρ L∞(0,1) + [M2(s, τ, w0) + 1]ρids ≤C∗(21 ρ)he−γ(t−τ)kz0kL∞(0,1) + [M2(t, τ, w0) + 1]1 2i +C∗he−ρ(t−1−τ)kz0kρ L∞(0,1) + [M2(t−1, τ, w0) + 1]ρi ≤C∗he−γ(t−τ)(kz0kL∞(0,1) +kz0kρ L∞(0,1))+[M2(t, τ, w0) + 1]ρi, where we used the relation presented in Remark 7.22, item (2), to replace M2(t−1, τ, w0)by M2(t, τ, w0) and we adjusted C∗to incorporate the other constants. 7.2.4 Global well-posedness and Pullback attracting set for z(t) The results on the previous section allow us to obtain global well-posedness and the existence of attracting set for the solution z(t, τ, z0)of the problem zt+L(t)z=ψ(s, z), t > τ;z(τ) = z0, which can be transferred to v(t)trough the relation v(t) = z(t) + ξ(t). Before we state the results, we recall some essential points: 1. The constant M2(t, τ, w0)given in (7.25) satisfies M2(t, τ, w0)t−τ→∞ −−−−→ 0,uniformly for w0in bounded sets of L∞(Ω). 2. ξ(t, r; (τ, w0)), estimated in Proposition 7.20, satisfies |ξ(t, r; (τ, w0))| ≤ G1e−γ(t−τ)kw0kL∞(Ω) +kw0kρ L∞(Ω)+G2, and |ξ(t, r; (τ, w0))|t−τ→∞ −−−−→ G2uniformly for w0in bounded sets of L∞(Ω). The L2k−estimate obtained in Proposition 7.26 implies that kz(t, τ, z0)kLp(0,1) is bounded in each bounded interval [τ, T ]. Consequently, v(t)is bounded in any interval [τ, T]. Therefore, z(t, τ, z0)and v(t, τ)are globally defined in time (Theorem 6.8). We prove with details this statement in the next proposition. Proposition 7.30. Let z(·, τ, z0)be the solution of (7.20) associated to the initial conditions (w0, v0)∈ L∞(Ω) ×L∞(0,1) and z0=v0−ξ(τ, w0)∈L∞(0,1). Then z(·, τ, z0)is globally defined. Moreover, 132 Chapter 7. Domains with a handle: Attractors there exists a closed ball in Vθ qcentered in zero and with a radius C∗=C∗(θ, Ω, ρ, p, q, N, M, γ)>0 (independent of τ,w0, or z0), BVθ q[0, C∗], such that z(t, τ, z0)is pullback attracted by this set in the topology of Vθ q, uniformly for z0in bounded sets of Lp(0,1) and for w0in bounded sets of L∞(Ω). Proof. Let z0∈Lp(0,1). It follows from Remark 7.23 that after any arbitrarily small evolution in time, z∗=z(t∗, τ, z0)∈L∞(0,1). Therefore, if we start the evolution at instant t∗and at the point z∗, Proposition 7.29 implies that kz(t, t∗, z∗)kVθ qis finite in any bounded interval [t∗, T]. For θ > 1 2qρ−1 ρ, this boundedness implies Lp(0,1) boundedness and global existence of the solution (see Figure 7.2). Moreover, let B⊂Lp(0,1) be a bounded set such that kz0kLp(0,1) ≤Lfor any z0∈B. It also follows from Remark 7.23 that after an arbitrarily small evolution in time, t∗> τ, the elements z∗=z(t∗, τ, z0) are bounded in L∞(0,1), that is kz∗kL∞(0,1) ≤˜ L. We denote by B∗={z∗=z(t∗, τ, z0); z0∈B}. Therefore, Proposition 7.29 states that there exists a constant C∗=C∗(θ, Ω, ρ, p, q, N, M, γ)>0such that kz(t)kVθ q≤C∗he−γ(t−τ)(kz0kL∞(0,1) +kz0kρ L∞(0,1))+[M2(t, τ, w0) + 1]ρi. In particular, for z∗∈B∗, we obtain kz(t, t∗, z∗)kVθ q≤C∗he−γ(t−t∗)(˜ L+˜ Lρ)+[M2(t, τ, w0) + 1]ρi and since M2(t, τ, w0)t−τ→∞ −−−−→ 0,uniformly for w0in bounded sets of L∞(Ω), we have dist(z(t, t∗, z∗), BVθ q[0, C∗]) t−t∗→∞ −−−−−→ 0, uniformly for z∗∈B∗and w0in bounded sets of L∞(Ω). In order to transfer the results to v(t), we must estimate ξ(t, r)in the k·kVθ q-norm, since kv(t, ·)kVθ q≤ kz(t, ·)kVθ q+kξ(t, ·)kVθ q. This is done in the next lemma. Lemma 7.31. Let w0∈L∞(Ω),τ < t −1< t,E1, E2the constants given in (7.13) and H1given in (7.24) (they only depend on Ω,ρ,N,M,γ), then kξ(t, r; (τ, w0))kVθ q≤CE1 H1 M2(t, τ, w0) + CE2. Proof. Recall that ξ(t, r) = w(t, τ, w0)(p0)X0(t, r) + w(t, τ, w0)(p1)X1(t, r), where X0(t, r) =     Z1 r 1 a(t, θ)dθ Z1 0 1 a(t, θ)dθ     ,X1(t, r) =     Zr 0 1 a(t, θ)dθ Z1 0 1 a(t, θ)dθ     . 133 Chapter 7. Domains with a handle: Attractors Differentiating X0(t, r)in rtwo times, we obtain ∂2 rX0(t, r) = ∂ra(t, r) (a(t, r)2) Z1 0 1 a(t, θ)dθ and |∂2 rX0(t, r)| ≤ C, which is bounded due to Assumption (A.3) (this estimate does not dependent of t,τ,w0or v0). The same holds for ∂2 rX1(t, r). Therefore, kX0(t, r)kVθ qand kX1(t, r)kVθ qare bounded by a constant Cfor any 0≤θ < 1. From Lemma 7.19, we obtain kξ(t, r; (τ, w0))kVθ q≤ |w(t, τ, w0)(p0)|kX0(t, r)kVθ q+|w(t, τ, w0)(p1)|kX1(t, r)kVθ q ≤C(|w(t, τ, w0)(p0)|+|w(t, τ, w0)(p1)|) ≤CE1e−γ(t−τ)kw0kL∞(Ω) +kw0kρ L∞(Ω)+CE2. =CE1 H1 M2(t, τ, w0) + CE2. As a consequence of Lemma 7.31, Proposition 7.29 and fact that v(t, r) = ξ(t, r) + z(t, r), we obtain Proposition 7.32. Let (w0, v0)∈L∞(Ω) ×L∞(0,1),z0=v0−ξ(τ, w0)∈L∞(0,1),z(·, τ, z0)the solution of (7.20) and v(t) = ξ(t)+z(t). Then, for 0≤θ < 1, there exists C∗=C∗(θ, Ω, ρ, p, q, N, M, γ)>0 depending on θ, Ω, ρ, p, q, N, M and γ(but not on τ,w0or z0) such that kv(t)kVθ q≤C∗he−γ(t−τ)(kz0kL∞(0,1) +kz0kρ L∞(0,1))+[M2(t, τ, w0) + 1]ρi. Proof. From Proposition 7.29 and Lemma 7.31 kv(t)kVθ q≤ kz(t)kVθ q+kξ(t)kVθ q≤C∗he−γ(t−τ)(kz0kL∞(0,1) +kz0kρ L∞(0,1))+[M2(t, τ, w0) + 1]ρi +CE1 H1 M2(t, τ, w0) + CE2. By adjusting the constant C∗the result follows. 7.3 Pullback attractor in Ω0= Ω ∪R0 We finally return to the equation (w, v)t(t) = −A0(t)(w, v)(t) + F0((w, v)(t)), t > τ; (w, v)(τ) = (w0, v0)∈U0 p, whose solution (w, v)(t):(τ, τM(w0, v0)) →U0 pis given by (w, v)(t) = U0(t, τ)(w0, v0) + Zt τ U0(t, s)F0(t, s)ds.