Study of a nonlocal Chafee-Infante equation using semigroup theory

UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação Study of a nonlocal Chafee-Infante equation using semigroup theory Yessica Yulieth Julio Pérez Tese de Doutorado do Programa de Pós-Graduação em Matemática (PPG-Mat) SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: ______________________ Yessica Yulieth Julio Pérez Study of a nonlocal Chafee-Infante equation using semigroup theory Thesis submitted to the Instituto de Ciências Matemáticas e de Computação – ICMC-USP – in accordance with the requirements of the Mathematics Graduate Program, for the degree of Doctor in Science. EXAMINATION BOARD PRESENTATION COPY Concentration Area: Mathematics Advisor: Prof. Dr. Alexandre Nolasco de Carvalho (ICMC-USP, Brasil) and Prof. Dr. Tomás Caraballo (IMUS-US, España) USP – São Carlos March 2024 Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP, com os dados inseridos pelo(a) autor(a) Bibliotecários responsáveis pela estrutura de catalogação da publicação de acordo com a AACR2: Gláucia Maria Saia Cristianini - CRB - 8/4938 Juliana de Souza Moraes - CRB - 8/6176 J94s Julio Pérez, Yessica Yulieth Study of a nonlocal Chafee-Infante equation using semigroup theory / Yessica Yulieth Julio Pérez; orientador Alexandre Nolasco de Carvalho; coorientador Tomás Caraballo Garrrido. -- São Carlos, 2024. 104 p. Tese (Doutorado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2024. 1. non-local quasilinear parabolic problems without uniqueness. 2. Chafee-Infante problems. 3. multivalued attractor. 4. delay differential equation. 5. semigroup. I. Nolasco de Carvalho, Alexandre, orient. II. Caraballo Garrrido, Tomás, coorient. III. Título. Yessica Yulieth Julio Pérez Estudo de uma equação de Chafee-Infante não local usando teoria de semigrupos Tese apresentada ao Instituto de Ciências Matemáticas e de Computação – ICMC-USP, como parte dos requisitos para obtenção do título de Doutora em Ciências – Matemática. EXEMPLAR DE DEFESA Área de Concentração: Matemática Orientadores: Prof. Dr. Alexandre Nolasco de Carvalho (ICMC-USP, Brasil) and Prof. Dr. Tomás Caraballo (IMUS-US, España) USP – São Carlos Março de 2024 This thesis is dedicated to my parents, to whom I owe the gift of life. ACKNOWLEDGEMENTS I would like to begin by thanking God for granting me the health and wisdom to complete this academic cycle. To my family, especially my parents Xiomara and Edgar, I am grateful for their unwavering belief in me. Even from afar, I know they are always with me. To my partner in life’s battles Luisfer, who has been by my side for the last ten years, this triumph is as much his as it is mine. To my father-in-law Fernando, who has become like a second father to me, I will always be thankful for his sincere and selfless support, especially during the most challenging times. I also want to express my gratitude to all the professors I encountered during my academic journey. I owe my development as a mathematician to them. Special thanks to Professor Melba, who not only provided me with specific knowledge but also imparted invaluable life lessons through her unwavering support. My heartfelt thanks also go to my colleagues at ICMC in São Carlos and IMUS in Seville for their friendship and the countless hours spent discussing mathematics. A special shoutout to my dear friends Gina and William, whom I met during my master’s degree in Itajubá. Thank you for showing me the true value of friendship. I am indebted to my advisors for their guidance and wisdom throughout this journey. To Alexandre Nolasco, thank you for your availability, patience, and for introducing me to theories that reignited my passion for mathematics. To Tomás Caraballo, thank you for welcoming me in Seville and for the invaluable learning experience. Finally, I want to extend my gratitude to the universities that welcomed me during my doctoral studies: Universidade de São Paulo and Universidad de Sevilla. Thank you for providing the opportunity for foreign individuals like myself to receive quality education and to fulfill this dream. You have offered me more than just technical knowledge; you have provided me with a life-changing experience. Special thanks to Colombian Ministerio de Ciencia, Tecnología e Innovación (Minciencias) for funding my doctorate, and to the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) for financing my sandwich doctorate in Seville through CAPES grant: #88887.695331/2022−00. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. RESUMEN JULIO, Y. Estudo de uma equação de Chafee-Infante não local usando teoria de semigrupos. 2024. 105 p. Tese (Doutorado em Ciências – Matemática) – Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos – SP, 2024. En este trabajo, nos enfocamos en el análisis de variaciones no locales y cuasilineales del problema de Chafee-Infante, incluyendo una variante con retraso. Para abordar estos problemas, utilizamos una reparametrización en la escala del tiempo, lo que nos permite reformular los problemas como semilineales. Presentamos un enfoque abstracto para las ecuaciones semilineales, que ofrece una amplia aplicabilidad y facilita la extensión a modelos diversos. Nuestro análisis abarca varios aspectos clave. En primer lugar, abordamos la existencia (posiblemente no única) de soluciones debiles y su regularidad, cuando el termino no linear satisface un adecuado módulo de continuidad. Además, exploramos resultados de comparación utilizando la fórmula de variación de constantes. Aprovechando estas comparaciones, establecemos la existencia global de las soluciones, bajo la condición adicional de que la no linealidad cumpla con un criterio estructural y complementamos con limitación uniforme siempre que la no linearidad satisfaza una condición de disipatividad. Además, establecemos la existencia de atractor global/ pullback para el semigrupo/proceso multivaluado asociado. Palabras clave: Problemas parabólicos cuasilineales no locales sin unicidad, problemas de Chafee-Infante, atrator multivaluado, ecuación diferencial con retardo, semigupo, proceso de evolución. LIST OF FIGURES Figure 1 – ∆δ(a) = −Σσ(−a)............................... 30 Figure 2 – The Sectors ∆and Σwith δ=π−σ...................... 31 Figure 3 – Detail for Lemma 2.1.11 ........................... 32 Figure 4 – Curve Γ..................................... 33 Figure 5 – Continuity Module w(x) = (ln(|x|−1))−pand p(x) = x1 2........... 63 CONTENTS 1 INTRODUCTION ............................ 21 2 PRELIMINARY TOPICS ........................ 27 2.1 Semigroup theory and sectorial operators ................ 27 2.1.1 Strongly continuous semigroups of linear operators .......... 27 2.1.2 Analytic Semigroups and Fractional Powers of Closed Operators . . 30 2.2 Multivalued attractors ........................... 35 2.2.1 Basic results on multivalued semiflow .................. 36 2.2.2 Pullback attractors and m-processes ................... 39 2.3 Semilinear differential equations ..................... 42 2.3.1 Existence of solution for semilinear problems .............. 44 2.3.2 Comparison results ............................. 48 2.3.3 Global well-posedness and existence of pullback attractor ...... 50 3 A NONLOCAL CHAFEE-INFANTE PROBLEM ........... 53 3.1 Defining the problem ............................ 54 3.2 Existence of mild and strong solution .................. 55 3.3 Comparison results ............................. 64 3.4 Existence of global attractor ........................ 73 3.5 Results about pullback attractor ..................... 76 4 A DELAY PROBLEM .......................... 81 4.1 Local well-posedness ............................ 82 4.2 Global well-posedness ............................ 86 4.3 Global attractors of multivalued semiflows ............... 89 4.4 Pullback attractor .............................. 92 5 CONCLUSION: ENGLISH, PORTUGUESE AND SPANISH . . . . 97 5.1 Conclusion .................................. 97 5.2 Conclusão ................................... 98 5.3 Conclusión .................................. 98 BIBLIOGRAPHY ...................................101 21 CHAPTER 1 INTRODUCTION Recently there has been much interest in the study of reaction-diffusion equations with non-local terms. See, e.g.,(CHIPOT; LOVAT,1999a;CHIPOT; RODRIGUES,1992;MICHEL; LUE,2001;KAVALLARIS,2015;SZYMA ´ NSKA et al.,2009). The nonlocal term introduces additional complexity in the analysis and interpretation of solutions, making the study of these problems interesting and challenging. When studying the asymptotic behavior of a nonlocal differential equation, ensuring the existence of a solution can be a challenge, for which methods such as the Faedo-Galerkin method or the method of steps for equations with delay, as well as specialized theories, are often used. In our case, it will not be different, but first, we transform the problem into a semilinear one, thus simplifying the problem of finding a solution to a quasilinear equation. The prototype of the problem we aim to investigate is an initial value problem represented by the following form:                    ∂w ∂τ −a(l(w))∂w2 ∂x2=λf(w)+h(τ),τ>σ,x∈Ω, w(τ,0) = w(τ,1) = 0, w(σ,x) = w0(x), (1.1) where Ω= (0,1),λ>0 , w0∈H1 0(0,1),l is an operator from H1 0(0,1) to R+ and a , f and h are continuous functions. This equation is a one-dimensional nonlocal quasilinear problem, which was studied in (CHAFEE; INFANTE,1974a;CHAFEE; INFANTE,1974b) (local version) in the seventies and has been researched since then. See, e.g., (CABALLERO; MARÍN-RUBIO; VALERO,2021;CHANG; CHIPOT,2003;CHIPOT; LOVAT,1999b;CHIPOT; SIEGWART, 2003;MICHEL; LUE,2001). 22 Chapter 1. Introduction Making a change in the time scale we obtained the nonlocal semilinear initial value problem:                    ut=uxx +λf(u)+h(α(t)) a(l(u)) ,t>σ,x∈Ω, u(t,0) = u(t,1) = 0, u(σ,x) = w0(x). (1.2) Here α:[σ,∞)→[σ,∞)is the function α(t) = σ+Rt σa(l(w(r)))dr,with inverse function α−1:[σ,∞)→[σ,∞)defined by α−1(t) =:σ+Zt σ a(l(u(r)))−1dr. Given a solution u of (1.2) , define τ=α(t) then w(τ):=u(t) = u(α−1(τ)) is a solution to the quasilinear equation (1.1) . Analogously, for a solution w of (1.1) , making t=α−1(τ) we obtain that u(t):=w(τ) = w(α(t)) is a solution to the initial value problem (1.2). Having established the relationship between the solutions of these two problems, we demonstrate abstract results of well-posedness which apply to problem (1.2), and consequently to problem (1.1). This time reparametrization technique has been used in other works (see, e.g., (LI et al.,2020;CARVALHO; MOREIRA,2021)) and its utilization presents notable advantages in contrast to the treatment of the original problem (1.1) with techniques specific to quasilinear problems. Particularly, it stands out when compared to approaches relying on semigroup theory (see (LUNARDI,1983;YAGI,2010)). More specifically, the transformed problem (1.2) imposes lower regularity requirements on the non-local diffusion component. Namely, lower bound condition and continuity. Consider the initial value problem.        du(t) dt +Au(t) = f(t,ut(t),ut(·)) t>σ, u(σ) = w0∈Xα, (1.3) where −A has compact resolvent and is the infinitesimal generator of an analytic semigroup {e−At :t⩾0} , which is exponentially decaying, Xα,α⩾0 is the fractional power spaces associated to A (KOMATSU,1966;HENRY,1981), and f:[σ,σ+T]×Xα×C([σ,σ+T],Xα)→X is a continuous function. Here, for u(·)∈C([σ,σ+T],Xα),ut(·) = u(·)|[σ,t],u(t) = ut(t). 23 In general, whenever possible, our results are abstract. Therefore, we analyze the problem (1.3) . The approach adopted for this equation can be applied to (1.2) , given that in this particular case, the operator involved is the Laplacian, which is sectorial. Our method deals with local and global existence separately, under different conditions. To examine the local existence of solution (possibly nonunique), we utilize techniques involving semigroup theory and we consider the problem in the space of fractional powers associated to the operator A; it represents a useful tool, because of estimates and embedding properties. To ensure that all solutions are globally defined we impose a structural condition. (S) Assume that there exist C0,C1∈Rsuch that u f (u)⩽−νC0u2+|u|C1 for all u∈Rand for both ν=m λand ν=M λ. The variation of constants formula, coupled with this condition as well as almost monotonicity on nonlinearity, allows for comparing solutions of problem (1.2) with those of the linear problem                wt+Aw(t) = −C0w+C1x∈Ω,t>σ, w=0,in x ∈∂Ω,t>σ, w(σ,|w0|) = |w0|, (1.4) obtaining global existence; it is a technique that has been employed in previous works. (See e.g., (ARRIETA; CARVALHO; BERNAL,2000)). Concerning the existence of global (pullback) attractor we assume, in addition to (S), a dissipativity condition on f. This condition is expressed as: (D) Assume that (S) holds for some C0such that the first eigenvalue λ1of A+C0νIis positive. Hypotheses of dissipativity are commonly necessary to guarantee uniform boundedness of the solutions. Using these bounds, we can obtain the existence of an absorbing set and consequently, of an attractor. Given that univalued theories for studying attractors cannot be used, as we cannot guarantee solution uniqueness for (1.2) , it is necessary to introduce theories of multivalued attractors. See (MELNIK; VALERO,2008;SIMSEN; VALERO,2016). We are also interested in studying the following quasilinear initial value problem with delay. 30 Chapter 2. Preliminary topics 2.1.2 Analytic Semigroups and Fractional Powers of Closed Operators If not stated otherwise, the properties and results presented here were taken from (PAZY, 1983) and (SELL; YOU,2013). Definition 2.1.7. A linear operator A:D(A)⊂X→X has compact resolvent if there exists ζ∈ρ(A) such that (ζ−A)−1 is a compact operator. That is, (ζ−A)−1BX 1(0) is relatively compact. A vector-valued function f:U→X where U⊂C is an open set, is said to be analytic mapping if the strong limit lim ξ→0 1 ξ[f(ξ0+ξ)−f(ξ0)] exist in X, for any ξ0∈U. ( ξ→0 by points in C ). As a example, if A is the infinitesimal generator of a C0 semigroup on X, then the resolvent operator (λI−A)−1 is an analytic mapping from ρ(A)into L(X)(HILLE; PHILLIPS,1957). Next, for δ,σ∈(0,π) we define some open sectors in the complex plane C , which are a highly useful tool for the definition of the analytic semigroup: ∆δ:={ξ∈C:|arg ξ|<δ,ξ=0}, ∆δ(a):=a+∆δ={ξ∈C:|arg (ξ−a)|<δ,ξ=a}, Σσ:={ξ∈C:|arg ξ|>σ,ξ=0}, Σσ(a):=a+Σσ={ξ∈C:|arg (ξ−a)|>σ,ξ=a} One can prove that ∆δ(−a) = −Σσ(a),when δ=π−σ,see Figure 1. x y a-a ∆δ(a)Σσ(−a) Figure 1 – ∆δ(a) = −Σσ(−a) . As a result, one has that Σσ(a)⊂ρ(A)⇐⇒ ∆δ(−a)⊂ρ(−A),where δ=π−σ,(2.1) for any linear operator A:D(A)⊂X→Xwhere D(A)is dense in X. Now, we shall define a class of semigroups for which the parameter domain can be expanded to encompass a region of the complex plane containing the non-negative real axis. It is evident that, to maintain the semigroup structure, the domain over which the complex parameter may vary must be an additive semigroup of complex numbers (a sector). 2.1. Semigroup theory and sectorial operators 31 a←∆δ(a)Σσ(a)→x y Figure 2 – The Sectors ∆and Σwith δ=π−σ. Definition 2.1.8. Let S(z),z∈∆δ be a bounded linear operator. The family {S(z):z∈∆δ} is an analytic semigroup if: i. z→S(z)is analytic in ∆δ. ii. S(0) = Iand lim z→0,z∈∆δ S(z)x=xfor all x∈X. iii. S(z1+z2) = S(z1)S(z2)for z1,z2∈∆δ∪0. An analytic semigroup is an extension of a C0 semigroup to a mapping defined in some sector ∆δ∪0 containing the non-negative real axis. Definition 2.1.9. We say that a linear operator A:D(A)⊂X→X is of positive type with constant M⩾1 if it is closed, densely defined, R+⊂ρ(−A)and ∥(s+A)−1∥L(X)⩽M 1+s,s∈R+. Definition 2.1.10. Consider A:D(A)⊂X→X . We say that A is sectorial if it is a closed densely defined operator such that there are constants M,a∈R , σ∈(0,π 2) such that one has Σσ(a)⊂ρ(A),and ∥(λ−A)−1∥L(X)⩽M |λ−a|for all λ∈Σσ(a).(2.2) By using (2.1), we can see that (2.2) is equivalent to ∥(λ+A)−1∥L(X)⩽M |λ+a|,for all λ∈∆δ(−a)⊂ρ(−A),where δ=π−σ.(2.3) Also, observe that (2.3) is equivalent to ∥(µ+B)−1∥L(X)⩽M |µ|,for all µ∈∆δ,(2.4) with B=A−aI.Furthermore, A is sectorial if and only if B is sectorial. This first technical result, whose proof can be found in (PAZY,1983, Theorem 1.7.7), shows that if an operator Ais sectorial, −Ais the infinitesimal generator of a C0semigroup. 32 Chapter 2. Preliminary topics Lemma 2.1.11. Let A be a sectorial operator on X. Then −A is the infinitesimal generator of a C0 semigroup e−At, and there is a constant M⩾1 such that ∥e−At∥⩽Me−at for all t⩾0. Additionally one has e−At =1 2πiZΓ eλt(λ+A)dλ,fort>0,(2.5) and e−At =I at t=0. Where Γ=Γ(η,θ) is a curve in the resolvent set ρ(−A) given by Γ=Γ1∪Γ2∪Γ3,where Γ1={λ=−a+re−i(π−θ):r⩾η}, Γ2={λ=−a+ηeiφ:|φ|⩽π−θ}, Γ3={λ=−a+rei(π−θ):r⩾η}, θ satisfies σ<θ<π 2,η is any positive constant. The integral in (2.5) exists in the uniform operator topology on L(X), for all t>0. -a Γ=Γ1∪Γ2∪Γ3 Figure 3 – Detail for Lemma 2.1.11 Definition 2.1.12. A linear operator K:X→Y is called compact if K(BX 1(0)) is a relatively compact subset of Y. Definition 2.1.13. A C0 semigroup S(t) is called compact if for every t>0,S(t) is a compact operator. Theorem 2.1.14. Let A:D(A)⊂X→X be an operator densely defined such that −A is sectorial with compact resolvent. Then the semigroup generated by Ais compact. The main motivation for introducing sectorial operators is to define fractional powers for closed operators, this class of operators offers prominent tools for studying the dynamics of infinite-dimensional systems. Let Abe a positive-type operator with constant M. Given s∈R+and λ∈Csatisfying |λ−s|⩽(1+s)/(2M), 2.1. Semigroup theory and sectorial operators 33 Since λ+A= (s+A)(1+(λ−s)(s+A)−1),follows that λ∈ρ(−A)and ∥(λ+A)−1∥L(X)⩽∥[1+(λ−s)(s+A)−1]−1∥L(X)∥(s+A)−1∥L(X) ⩽2M(1+s)−1⩽2M 1+|λ| 1+s+|λ−s| 1+s ⩽2M 1+|λ|1+1 2M=2M+1 1+|λ|. (2.6) From this, we deduce that ∆M:={z∈C:|argz|⩽arcsin1/(2M)}+{z∈C:|z|⩽1/(2M)} ⊂ ρ(−A). Therefore, any positive operator Ais sectorial. Assume A sectorial. Since the resolvent set of A is open and {0} ∈ ρ(A), we can choose r>0 (small) such that BX r(0)⊂ρ(A). Denote by Γ the curve defined by the boundary of −Σσ∪BX r(0) , as in Figure 4, oriented in the way that the imaginary part is increasing. With this, for all sectorial operator A and α∈C with Reα<0 , where Reα denote the real part of α, we define the operator Aα=1 2πiZΓ(−λ)α(λ+A)−1dλ.(2.7) Using (2.6) , (2.7) and the Cauchy’s Theorem it is possible to show that Aα is well defined in L(X). Figure 4 – Curve Γ. We summarize several of the well-known properties of these operators (See (HENRY, 1981;PAZY,1983) for details of the proofs). Lemma 2.1.15. (i) For α,β∈Cwith Reα,Reβ<0 we have AαAβ=Aα+β. (ii) For each α∈C, with Reα<0, the operator Aαis injective. As a consequence of the lemma, for β∈C , with Reβ>0 we are able to define de potential power of A,Aβ:D(Aβ)⊂X→X , where the domain is given by D(Aβ) = R(A−β) and Aβu= (A−β)−1u,for all u∈D(Aβ). 34 Chapter 2. Preliminary topics We can define a norm in D(Aα)given by ∥u∥α:=∥Aαu∥X. With this norm, D(Aα)becomes a Banach space. Definition 2.1.16. We define the space of fractional power of A, for α∈Cwith Reα>0, as Xα:= (D(Aα),∥·∥α). The spaces of fractional powers play a crucial role in establishing the existence of solutions for semilinear problems (involving sectorial operators). The following theorems outline pertinent properties that enhance our understanding of these spaces. Theorem 2.1.17. The fractional powers of the operator Asatisfies the following properties: (i) If α,β∈C,Reα,Reβ>0, then AαAβ=Aα+β. (ii) For α,β∈C , with Reβ>Reα>0 , we have D(Aβ)⊂D(Aα) and the space of fractional powers satisfies Xβ,→Xα,→X, where the symbol “,→” represents a continuous inclusion. Moreover, Xβis also dense in Xα. (iii) Additionally, if Ahas a compact resolvent, then the inclusions above are compact. For 0<α<1 we can deform the path of integration Γ into the upper and lower sides of the negative real axis and obtain A−α=sinπα πZ∞ 0s−z(s+A)−1ds 0<Rez <1.(2.8) Theorem 2.1.18. If A is a positive type operator and α∈[0,1], there exists a constant K depending on A,such that ∥Aαx∥X⩽K[(1−α)µα∥x∥X+αµα−1∥Ax∥X]for all µ∈(0,∞) or equivalently ∥Aαx∥X⩽K∥Ax∥α X∥x∥1−α X,for 0 ⩽α⩽1,x∈D(A). Theorem 2.1.19. Let A:D(A)⊂X→X be a sectorial operator and {S(t);t⩾0} the analytic semigroup generated by −A. Assume that (−∞,0]⊂ρ(A).Then (i) S(t):X→D(Aα)for every t>0 and α⩾0. (ii) For every x∈D(Aα)we have S(t)Aαx=AαS(t)x. 2.2. Multivalued attractors 35 (iii) For α⩾0,R(S(t)) ⊂D(Aα)and ∥AαS(t)∥L(X)⩽Mαt−αe−ωt,t>0, α7−→ Mαis continuous in [0,∞). (iv) For α>0,tαAαS(t)xt→0+ −−−→ 0,for all x∈X. (v) For 0 <α⩽1,t⩾0,and x∈D(Aα),∥(S(t)−I)x∥⩽M1−αα−1tα∥x∥α. In order to ensure the existence and regularity to a solution of the problems (1.1) and (1.5) , we employ an abstract result that is sufficiently general, encompassing our specific cases, namely, semilinear problems with principal part given by the Laplacian operator defined on H2(0,1)∩H1 0(0,1)⊂L2(0,1) . We are aware that under the boundary conditions, the set of eigenvalues of the involved Laplacian operator is discrete, contributing to its compactness and, consequently, to the compactness of the generated semigroup. In the next example, we resume some properties of such an operator. Example 2.1.20. Consider A:D(A)⊂L2(0,π)→L2(0,π) the operator given by Au =uxx , for u∈D(A) = H2(0,π)∩H1 0(0,π). Define the problem        ut=Au,x∈(0,1),t>0, u(t,0) = u(t,1) = 0,t⩾0, u(0,·) = u0∈H1 0(0,1). The operator −A is sectorial with (0,∞)⊂ρ(−A) , see (CARVALHO; LANGA; ROBINSON,2013). Also, for all element u0∈L2(0,π)with u0⩾0 we have (λ+A)−1u0⩾0,∀λ>0. this is, −A has positive resolvent. −A has compact resolvent then, by Theorem 2.1.14 the semigroup S(t) generated by −A is compact. We can conclude using Lemma 2.1.11 that S(t) is a C0semigroup. The above problem defines a semigroup {e−At :t⩾0} , for X=L2(0,π) . We can consider Xα , α∈(0,1] , as the fractional powers defined by −A (see Section 6.4.2, (CARVALHO; LANGA; ROBINSON,2013), for more details) and holds the following inequality: ∥eAtu∥α⩽Mαt−αe−ωt∥u∥α,for all t>0.(2.9) 2.2 Multivalued attractors We acknowledge the theory of nonlinear semigroups of operators in Banach spaces and the corresponding version for the study of non-autonomous systems, are important tools for 36 Chapter 2. Preliminary topics studying the qualitative behavior of infinite-dimensional dynamical systems. These approaches have developed in recent years, and several results on attractors have been obtained. However, these results cannot be applied to the class of initial value problems where the solution is not unique. To guarantee the existence of attractors for such systems, a corresponding theory for multi-valued semigroups and processes must be developed. The multivalued theory has a particular importance to us as it allows us to study the asymptotic dynamics of problems (1.1) and (1.5) . In Chapter 3, we demonstrate the existence of a solution for (1.3) using Schauder fixed-point theorem. This result is then employed in Chapter 4to ensure the existence of a solution for (1.5) . Therefore, the multivalued theory is utilized to investigate both (1.1) and (1.5) in both autonomous and non-autonomous cases. Similar to the univalued case, the analysis of autonomous problems is approached separately from non-autonomous ones. This section is divided into two parts: in the first part we present the theory that assists us in approaching autonomous systems, so we introduce the multivalued semiflows theory, and in the second one we introduce multivalued processes theory, which we employ in the study of the non-autonomous case. 2.2.1 Basic results on multivalued semiflow Set R+= [0,∞) , X be a metric space with metric ρ:X×X→R+ and P(X) ( B(X) , C(X) , K(X) ) be the set of all (bounded, closed, compact) subsets of X . The presentation is based on (MELNIK; VALERO,1998). Definition 2.2.1. A map G:R+×X→P(X) is called a multivalued semiflow (m-semiflow) if the following are satisfied: (1) G(0,·) = Iis the identity map; (2) G(t1+t2,x)⊂G(t1,G(t2,x)),∀t1,t2∈R+,∀x∈X, where, if B⊂X,G(t,B) = Sx∈BG(t,x),B⊂X. If G(t1+t2,x) = G(t1,G(t2,x)), for all x∈X and t1,t2∈R+ we say that the m-semiflow is strict. For x∈X , A,B⊂X we set dist(x,B) = inf{ρ(x,y):y∈B}. The Hausdorff semi-distance from Ato Bis defined by distH(A,B) = supx∈Adist(x,B). Definition 2.2.2. We say that A⊂X attracts B⊂X under the action of the m-semiflow G if distH(G(t,B),A)→0 as t→+∞ . A set M is said to be an attracting set for G if M attracts each set B∈B(X) under the action of the m-semiflow G . If there exists a bounded attracting set M for Gwe say that Gis bounded dissipative. 2.2. Multivalued attractors 37 Consider A⊂X , define γ+ t(A) = Sτ⩾tG(τ,A) , t⩾0 . The set γ+(A) = γ+ 0(A) is called the positive orbit of A and the set ω(A) = Tt⩾0γ+ t(A) is called the omega limit ( ω -limit) set of A. For ε>0,B∈B(X). Definition 2.2.3. We say that the map x(·):R→X is a trajectory of the m-semiflow G through x0if x(t+τ)∈G(t,x(τ)),for all t,s∈Rwith t⩾0. The following lemma provides a characterization of ω-limit set. Lemma 2.2.4. The set ω(A) consists of the limits of all converging sequences {ξn}, where ξn∈G(tn,A),tn→∞. Definition 2.2.5. The set Ais said to be negatively semi-invariant if A⊂G(t,A),∀t∈R+. Lemma 2.2.6. Let M⊂B(X) be a negatively semi-invariant set with respect to the m-semiflow G. If Ghas an attracting set Zthen M⊂Z. Definition 2.2.7. It is said that the set A⊂X attracts the set B⊂X under the action of the m-semiflow G if distH(G(t,B),A)→0 as t→+∞. A set M is said to be attracting for G if it attracts each B∈B(X). Definition 2.2.8. The set A is called a global attractor of the m-semiflow G if it satisfies the next conditions: (1) Aattracts any B∈B(X); (2) Ais negatively semi-invariant, i.e. A⊂G(t,A),∀t∈R+. Remark 2.2.9. If G has a bounded attractor A, then A⊂Z for any closed set Z attracting each B⊂B(X); with this, if A is closed, it is minimal among all closed sets attracting each B∈B(X). Define R⊂C(R+,X). Consider the following conditions: (C1) For any z∈Xthere exists at least one φ∈Rsuch that φ(0) = z. (C2) φτ(·):=φ(·+τ)∈Rfor any τ⩾0 and φ∈R(translation propriety). (C3) Let φ,ψ∈Rbe such that φ(0) = ψ(s)for some s>0. Then, the function θdefined by θ(t):=(φ(t),0⩽t⩽s, ψ(t−s),s⩽t, belongs to R(concatenation propriety). (C4) For any sequence {φj}j∈N⊂R such that φj(0)→z in X , there exists a subsequence {φµ}µ∈Nof {φj}j∈Nand φ∈Rsuch that φµ(t)→φ(t)for each t⩾0. 38 Chapter 2. Preliminary topics For each t⩾0 we denote the m-map G(t,·):X→P(X)as follows: G(t,w0) = {y∈X:y=z(t),z(·)∈R,z(0) = w0}.(2.10) Observe that R fulfilling (C1) and (C2) gives rise to an m-semiflow G through (2.10) and if besides (C3) holds, then this m-semiflow is strict (see (CARABALLO; MARIN-RUBIO; ROBINSON,2003), Proposition 2 or (KAPUSTYAN; PANKOV; VALERO,2012), Lemma 9). The functions φ∈Rare called solutions. Definition 2.2.10. The m-map G(t,·):X→P(X) is upper semicontinuous if for every sequence xn→x∈Xthen limn→∞distH(G(t,xn),G(t,x)) = 0. Definition 2.2.11. The m-semiflow G is called pointwise (bounded) dissipative if ∃B∈B(X) attracting any x∈X(B∈B(X)). Definition 2.2.12. A multivalued semiflow G is said to be asymptotically upper semicompact if for all B∈B(X) such that, for some T(B)>0 , γ+ T(B)(B)∈B(X) , any sequence ξn∈G(tn,B) with tn n→∞ −→ ∞has a convergent subsequence. Theorem 2.2.13. (MELNIK; VALERO,1998) If G is an asymptotically upper semicompact multivalued semiflow which is bounded dissipative and such that G(t,·):X→P(X) is upper semicontinuous, then Ghas the global attractor A. Theorem 2.2.14. (MELNIK; VALERO,1998) Let G be an asymptotically upper semicompact msemiflow. Suppose that G(t,·):X→P(X) is upper semicontinuous for any t∈Γ+. If ∀B∈B(X) , ∃T(B)∈Γ+such that γ+ T(B)(B)∈B(H1 0(Ω)),then Ghas the global attractor Adefined by A=[ B∈B(X) ω(B). Moreover, for any closed set Zattracting each B∈B(X),A⊂Z. We introduce the following definition that is necessary to enunciate a standard wellknown result for describing the attractor as the union of bounded complete trajectories reads in the multivalued case as follows. Definition 2.2.15. A map γ:R→H1 0(Ω) is called global solution of R (resp. of G ) if γ(·+ h)|[0,∞)∈Rfor all h∈R(resp. if γ(t+s)∈G(t,γ(s)) for all s∈Rand t⩾0). Theorem 2.2.16. (TORO et al.,2019) Consider R satisfying (C1) and (C2), and either (C3) or (C4). Assume also that Gpossesses a compact global attractor A. Then A={γ(0):γ∈K}=[ t∈R {γ(t):γ∈K} where Kdenotes the set of all bounded global solutions in R. 2.2. Multivalued attractors 39 Theorem 2.2.17. (MELNIK; VALERO,1998) Let G(t,·):X→P(X) be an upper semicontinuous map. If there exists a compact set K⊂Xsuch that for all B∈B(X) dist(G(t,B),K)→0 as t→∞, the m-semiflow G has the global compact attractor R⊂X. It is the minimal closed set attracting each BB(X). 2.2.2 Pullback attractors and m-processes In this section, we provide an introduction to the theory of multivalued processes to study pullback attractors. This theory can be employed for the asymptotic analysis of non-autonomous problems where solution uniqueness is not guaranteed. We define the necessary objects to present necessary and sufficient conditions for the existence of a pullback attractor. As before, let X be a metric space with metric ρ:X×X→R+ and P(X)B(X) be the set of subsets and bounded subsets of X , respectively. Reference (SIMSEN; VALERO,2016) Let us denote Wτ=C([τ,∞),X) and let G={G(τ)}t∈R consists of maps φ∈Wτ . Let us consider the following conditions: (C1) For each τ∈Rand z∈Xthere exists at least one ϕ∈G(τ)with ϕ(τ) = z; (C2) If ϕ∈G(τ)and s⩾0, then ϕ+s∈G(τ+s), where ϕ+s:=ϕ|[τ+s,∞); (C3) (Concatenation) If ϕ,ψ∈G with ϕ∈G(τ),ψ∈G(r) and ϕ(s) = ψ(s) for some s⩾τ , then θ∈G(τ), where θ(t):=(ϕ(t),t∈[τ,s]; ψ(t),t∈(s,∞). (C4) If {ϕj}j∈N⊂G(τ) and ϕj(τ)→z , then there exists a subsequence {ϕµ}µ∈N of {ϕj}j∈N and ϕ∈G(τ)with ϕ(τ) = zsuch that ϕµ(t)→ϕ(t)for each t⩾τ. We define a multivalued map U(t,τ):X→P(X) associated with the family G in the following way: U(t,τ)x:={ϕ(t);ϕ∈G(τ),ϕ(τ) = x}. If (C1)-(C2) hold, then the map Uis a multivalued process, that is: •U(t,t)x=xfor all t∈R,x∈X; •U(t,τ)x⊂U(t,s)U(s,τ)x for all −∞<τ⩽s⩽t<∞,x∈X; where U(t,s)C=Sx∈CU(t,s)x for any C⊂X. The concept of complete trajectory plays a crucial role in studying the structure of the pullback attractor. 46 Chapter 2. Preliminary topics We want to show that Fis a contraction. Denote B=supt∈[t0,T]∥f(t,u0)∥.Since Ais sectorial, it holds ∥AαeAt∥⩽Mαt−αe−ωt,t>0,(2.16) and, for 0 <α≤1 and x∈Xα, ∥eAtx−x∥⩽M1−α αtα∥x∥α,t>0.(2.17) Let u(·)∈M. It is easy to see that F(·)∈C([t0,T],Xα). Now, using (2.16) and (2.17) for t∈[t0,T]we have ∥F(u)(t)−u0∥α⩽∥[eA(t−t0)−I]u0∥α+Zt t0 ∥eA(t−s)f(s,u0)∥αds +Zt t0 ∥eA(t−s)[f(s,u(s))−f(s,u0)]∥αds ⩽M1−α α(t−t0)α∥u0∥α+MαBZt t0 e−ω(t−s)(t−s)−αds +MαLZt t0 e−ω(t−s)(t−s)−α∥u(s)−u0∥αds ⩽M1−α α(t−t0)α∥u0∥α+Mα(B+LR)(T−t0)1−αZ1 0e−ωrr−αdr. We may assume that T−t0 was chosen small enough so that ∥F(u)(t)−u0∥α⩽R, for all t∈[t0,T].Then, Fmaps Minto itself. We may also assume that T−t0is chosen small enough such that such that MαLZt t0 e−ω(t−s)(t−s)−αds ⩽η<1. If u,v∈Mthen for t∈[t0,T], ∥F(u)(t)−F(v)(t)∥α⩽Zt t0 ∥eA(t−s)[f(s,u(s))−f(s,v(s))]∥αds ⩽MαLZt t0 e−ω(t−s)(t−s)−αds|||u−v||| ⩽η|||u−v|||. (2.18) Hence, F defines a contraction and there is u∈M such that Fu=u. By Lemma 2.3.3 this is the unique solution of (2.11) on (t0,T)with initial value x(t0) = x0. One can check that for each (t0,x0)∈R+×Xα there is a maximal solution of the Cauchy problem (2.11) . In fact, if there are two solutions xj:[t0,τj)→X of (2.11) , with xj(t0) = x0,j=1,2. Then x1(t) = x2(t), for t∈[t0,min{τ1,τ2}). With this, for each (t0,x0)∈U we can define τmax =τmax(t0,x0) = sup{τ>t0; there is x:[t0,τ)→Xwith x(t0) = x0}. For each n∈N, we can find a solution xn:[t0,τn)→X with xn(t0) = x0, such that t0<τn→τmax as n→+∞and define the maximal solution x:[t0,τmax)→Xas x(t) = xn(t),for t∈[t0,τn). 2.3. Semilinear differential equations 47 Theorem 2.3.6. Assume the conditions of the Theorem 2.3.5. For each (t0,x0)∈U, if its maximal solution is bounded, it follows that τmax = +∞. Theorem 2.3.7. (HENRY,1981, Corollary 3.3.5) Assume A is sectorial, U= (τ,+∞)×Xα,f locally Hölder continuous in t, locally Lipschizian in x for (t,x)∈U. Additionally, f satisfies the next condition f(t,x)⩽K(t)(1+∥x∥α), where K(·) is continuous on (τ,+∞). If t0>τ and x0∈Xα, the unique solution of (2.11) u(·,t0,x0)exist for all t⩾t0. We can define a process using global solutions: If a semilinear initial value problem as (2.11) has a global solution that we denote x(·,s,x0) (solution that passes through x0 at time s ) and continuity on initial data, then S(t,s)x0=x(t,s,x0) for t⩾sdefine a process. The following theorem is a local well-posedness result for the problem (2.12). Theorem 2.3.8. (ARRIETA; CARVALHO; BERNAL,1999) Assume that 1<q<∞ and f satisfies ( Gq ). Then for each t0∈R and u0∈BX1 q r(u0) for any r>0 , there exists T>0 and a unique local solution u(·,t0,x0)∈C([t0,t0+T),X1 q) of (2.12) with u(t0,t0,u0) = u0 in X1 q. This is a classical solution for t>t0 and depends continuously on the initial condition u0. Also, the following regularizing effect takes place: if u0∈X1 q. then u(t,t0,u0)∈X3/2 q for any q⩾1 and t∈(t0,t0+T). It also holds the next result. Theorem 2.3.9. (ARRIETA; CARVALHO; BERNAL,1999) Assume that 1<q<∞ and f satisfies ( DGq ). Then for each t0∈R and u0∈BX1 q r(u0) for any r>0 , there exists T>0 and a unique local solution u(·,t0,x0)∈C([t0,t0+T),X1 q) of (2.12) with u(t0,t0,u0) = u0 in X1 q. This is a classical solution for t>t0 and is continuously differentiable with respect to the initial condition u0. Consider the next condition: For α∈(0,1] , the nonlinear term f:R×X1→Xα is such that, for each R>0, there are constant C=C(R)such that for ∥x∥,∥y∥⩽R,we have that ∥f(t,x)−f(t,y)∥α⩽C(R)h(t)∥x−y∥; ∥f(t,x)∥α⩽C(R)h(t),(2.19) where h(t):=(1,α=1, ψ(t−t0)|t−t0|−α,α<1. with ψ:[0,∞)→[0,∞)an increasing continuous function such that ψ(0) = 0. 48 Chapter 2. Preliminary topics With these hypotheses, we have the following local well-posedness result. Theorem 2.3.10. (CARVALHO; LANGA; ROBINSON,2013, Theorem 6.27) Assume that (i) Ais sectorial and α∈(0,1],or (ii) −Agenerates a strongly continuous semigroup and α=1. If f satisfies (2.19) . Then for any r>0, there exists a time t1>0 such that, for any initial value condition x0∈BX1 r(0), exist a unique mild solution x∈C([t0,t0+t1],X1) of (2.11) . This solution depends continuously on the initial data: if x0,z0∈BX1 r(0), then ∥x(t,t0,x0)−x(t,t0,z0)∥X1⩽C∥x0−z0∥X1for all t∈[t0,t0+t1]. When Ais sectorial and x0,z0∈BX1 r(0)we have x∈C((t0,t0+t1],X1+θ)for all 0 ⩽θ<α, (t−t0)θ∥x(t,t0,x0)∥X1+θ→0 as t→t0,0<θ<α, and, for 0 ⩽θ⩽θ0<α, (t−t0)θ∥x(t,t0,x0)−x(t,t0,z0)∥X1+θ⩽Cθ0∥x0−z0∥X1for all t∈[t0,t0+t1]. If additionally, t7→ (t,x) is locally Hölder continuous as a map from [t0,∞) into Xα , uniformly on bounded subsets of X1 , then x∈C1((t0,t0+t1],Xα)TC((t0,t0+t1],X1+α) and x(·,t0,x0) satisfies (2.11) in Xα for all t∈(t0,t0+t1]. If the map f:R×X1→Xα is Lipschitz continuous in both variables uniformly in bounded subsets of R×X1 , then x∈C((t0,t0+ t1],X1+β)for any β<α. 2.3.2 Comparison results Results on monotonicity and comparison are of great utility for the asymptotic analysis of solutions; thus, comparing our problem with another becomes an important tool for obtaining the existence of global/pullback attractor. In the following, we present some results on comparison, positivity, and monotonicity. To this end, we restrict our attention to semilinear problems defined in an ordered Banach space like X=L2(0,1) or X=H1 0(0,1) . This presentation follows (ARRIETA; CARVALHO,2000). Definition 2.3.11. We say that the pair (X;⩽) is an ordered Banach space if X is a Banach space and ⩽is and order relation in Xsatisfying: (i) x⩽yimplies that x+z⩽y+z,x,y,z∈X; (ii) x⩽yimplies that λx⩽λy,for x,y∈X,λ∈R; (iii) The positive cone C={x∈X;x⩾0}is closed in X. 2.3. Semilinear differential equations 49 We recall the following lemma, which is necessary for the proof of some comparison results, both in the single and multivalued cases. Lemma 2.3.12. Let f∈L1((t0,t1,X)be such that f(t)⩾0 for almost t∈(t0,t1).Then Zt1 t0 f(s)ds ⩾0. Definition 2.3.13. Let (X;⩽) be an ordered Banach space and −A be the generator of a C0 semigroup. We say that −A has positive resolvent in X if there exist λ0∈R such that (λI+A)−1 is increasing for every λ>λ0. Consider the following semilinear problem (˙x=Ax +f(t,x) x(t0) = x0.(2.20) Here (i) Ais sectorial; (ii) for some α>0,f:[t0,t1]×X1→Xα is locally Hölder continuous in t and locally Lipschitz in x; (iii) for each x0∈X1 there exists a solution x(t,x0)∈C([t0,τ],X1) for some τ(x0) and x depends continuously on the initial condition x0. Again, we consider the scale of fractional power spaces associated to A with X0=X and Xα=D(Aα),⩾0.We consider in Xαthe order induced by X. The following comparison results inspired the comparison results obtained for (1.2) , which are presented in Chapter 3. Theorem 2.3.14. (ARRIETA; CARVALHO,2000) Assume that for every r>0 there is a constant κ=κ(r)>0 such that for every [t∈t0,t1],f(t,·)+κI is positive on the elements in BX1 r(0).Then if 0 ⩽x0,then the solution of (2.20)x(t,x0,f)is positive as long as it exist. Theorem 2.3.15. (ARRIETA; CARVALHO,2000) Assume that f,g,h satisfies the conditions required for fin the Theorem 2.3.14. (i) If for every r>0 there is a constant γ=γ(r)>0 such that f(t,·) + γI is increasing in BX1 r(0), for all t∈[t0,t1] and u0,u1∈X1 with u0⩾u1, then uf(t,s,u0)⩾uf(t,s,u1) as long as both solutions exist. (ii) If f(t,·)⩾g(t,·) for all t∈R and u0∈X1 then uf(t,s,u0)⩾ug(t,s,u0) as long as both solutions exist. 50 Chapter 2. Preliminary topics (iii) If f,g are such that for every r>0 there exist a constant γ=γ(r)>0 and an increasing function h(t,·)such that, for every t∈[t0,t1] f(t,·)+γI⩾h(t,·)⩾g(t,·)+ γI in BX1 r(0) and u0,u1∈X1 with u0⩾u1, then uf(t,s,u0)⩾ug(t,s,u1) as long as both exist. 2.3.3 Global well-posedness and existence of pullback attractor In this section, we need to consider structural and dissipativity conditions on the nonlinearity f. The first is used to obtain the global existence of solutions, and the second is to achieve bounds on the attractor. See (CARVALHO; LANGA; ROBINSON,2013;ARRIETA; CARVALHO; BERNAL,1999). (S) Assume that there exist C0∈Rand,C1⩾0 such that u f (x,t,u)⩽−C0u2+|u|C1 for all u∈Rand x∈Ω. (D) Assume that (S) holds for some C0such that the first eigenvalue λ1of A+C0Iis positive. Proposition 2.3.16. Let 1 <q<∞and assume that fsatisfies (Gq). (i) If (S) holds, then the solutions of (2.12) with initial condition u0∈X1 q are globally defined. (ii) If (D) holds and u0∈X1 q , then there are constants K∞ and Kα r , for 2α<1+1/r,r⩾q , such that limsup s→−∞ ∥u(t,s,u0)∥L∞(Ω)⩽K∞,(2.21) limsup s→−∞ ∥u(t,s,u0)∥α⩽Kα r,(2.22) and these limits are uniform for u0in bounded subsets of X1 q. (iii) In addition, if (D) holds, u0∈X1 q,and φis the solution of (−div(a(x)∇u)+(c(x)+C0)φ=C1,in Ω, φ=0,in ∂Ω,(2.23) then 0⩽φ∈L∞(Ω),limsups→−∞|u(t,s,u0)(x)|⩽φ(x) uniformly in x∈¯ Ω and for u0 in bounded subsets of Xα q . Also, if |u0(x)|⩽φ(x) for all x∈¯ Ω , then |u(t,s,u0)(x)|⩽φ(x) for all x∈¯ Ωand t⩾s. The following results provide sufficient conditions for the existence of a pullback attractor for (2.12). 2.3. Semilinear differential equations 51 Theorem 2.3.17. Let 1<q<∞ and assume that f satisfies ( Gq ) and (D). Then (2.12) has a pullback attractor, A(·)∈X1 qsuch that [ t∈R Aq(t)⊂Σ(φ):={u∈L∞(Ω),|u(x)|⩽φ(x),a.e. x∈Ω}, and St∈RAq(t)⊂Eα r for any r⩾q and any α<1 is compact in this space and pullback attracts bounded sets of X1 qin the norm of Eα r. Theorem 2.3.18. Let 1<q⩽r<∞ and assume that ( Gq ) and (D) hold. Then there exists a pullback attractor Ar(·). Moreover, Aq(t) = Ar(t) for all t∈R and Ar(t) pullback attracts bounded subsets of X1 qin the topology of X1 q. 53 CHAPTER 3 A NONLOCAL CHAFEE-INFANTE PROBLEM In this chapter, we shall prove results concerning the equation (3.1) . In the first part, we delve into the findings regarding the existence and regularity of the solution. The autonomous and non-autonomous scenarios are tackled concurrently since the time-independent problem almost automatically follows from the time-dependent case. The second part is dedicated to elucidating the results about comparison. It is worth noting that the solution satisfies the variation of constants formula. In this context, we rigorously establish estimations that underpin the assurance of the existence of a global and a pullback attractor to the respective cases. This endeavor occupies the third and ultimate segment of the chapter. This chapter is based on (CARABALLO; CARVALHO; JULIO,2024b) We are interested in studying the nonlocal initial value problem:                    ∂w ∂τ −a(l(w))∂w2 ∂x2=λf(w)+h(τ),τ>σ,x∈Ω, w(τ,0) = w(τ,1) = 0, w(σ,x) = w0(x), (3.1) where Ω= (0,1),λ>0 , w0∈H1 0(0,1) , l a continuous operator from H1 0(0,1) to R+ and a , f and h are continuous functions satisfying some additional conditions depending on what are the properties that we require for the solutions. 54 Chapter 3. A Nonlocal Chafee-Infante Problem 3.1 Defining the problem Consider the semilinear initial value problem:                    ut=uxx +λf(u)+h(α(t)) a(l(u)) ,t>σ,x∈Ω, u(t,0) = u(t,1) = 0, u(σ,x) = w0(x). (3.2) Here α:[σ,∞)→[σ,∞) is the function α(τ) = σ+Rτ σa(l(w(r)))dr, with inverse function α−1:[σ,∞)→[σ,∞)defined by α−1(t) =:σ+Zt σ a(l(u(r)))−1dr. Observe that a(l(u))ut=wτ,if τ=α(t). Then, the solutions of (3.1) and (3.2) are related as follows: Given a solution uof (3.2) and defining τ=α(t)we obtain a solution for the non-local initial value problem (3.1) setting w(τ) = u(t). On the other hand, given a solution w of (3.1) making t:=α−1(τ)=σ+Rτ σa(l(w(r)))−1dr we have that it also solves the problem                    ∂w ∂τ −a(l(w))∂w2 ∂x2=λf(w)+h◦α◦α−1(τ),τ>σ,x∈Ω, w(τ,0) = w(τ,1) = 0, w(σ,x) = w0(x), (3.3) and the function udefined by u(t) = w(τ)satisfies (3.2). Note that with this change in the time scale, for each solution of (3.1) , we can construct a corresponding one for (3.2) and vice versa. For this reason, we cannot assert that we are dealing with a change of variable. What is clear, however, is that this change in the time scale transforms the quasilinear problem (3.1) into a semilinear one, thereby simplifying the task of finding a solution for (3.1) . Additionally, one can observe that the problem (3.2) remains non-local in space, and now, it is also non-local in time since h(α(t)) depends on the solution from the initial time σto the final time t. With the preceding considerations, we shall establish the existence and regularity of solutions for the semilinear problem: 3.2. Existence of mild and strong solution 55        du(t) dt +Au(t) = f(t,ut(t),ut(·)),t>σ, u(0) = w0∈Xα, (3.4) where −A has compact resolvent and is the infinitesimal generator of an exponentially decaying analytic semigroup {e−At :t⩾0} , Xα,α⩾0 is the fractional power spaces associated to A (KOMATSU,1966;HENRY,1981), and f:[σ,σ+T]×Xα×C([σ,σ+T],Xα)→X is a continuous function. Here, u(·)∈C([0,T],Xα),ut(·) = u(·)|[0,t],u(t) = ut(t). For our specific problem, for each σ, we have g(t,ut(t),ut(·)) = 1 a(l(u(t))) λf(u(t))+h(σ+Zt σ a(l(u(r)))dr).(3.5) Note that the problem (3.2) fits within these conditions (see Example 2.1.20). 3.2 Existence of mild and strong solution In this section, we shall establish the existence of a mild solution and (classical) solution for the problem (3.4). In the next result we extend the theory in (PAZY,1983, Theorem 6.2.1), which establishes the existence of a mild solution for (3.4) when −A is the infinitesimal generator of a compact C0 semigroup ( α=0 ), to the case when −A is the infinitesimal generator of a compact analytic semigroup. We observe that often, when passing to applications, generators of compact semigroups occur when −A has a compact resolvent and generates an analytic semigroup {S(t):t⩾0} . We follow along the same lines as a such similar result proving the existence of a (possibly non-unique) mild solution. Theorem 3.2.1. Let X be a Banach space, A a sectorial operator such that (λ+A)−1 is compact for all λ∈ρ(−A) , and let {S(t):t⩾0} be the analytic semigroup generated by −A . If f: [σ,σ+T]×Xα×C([σ,σ+T],Xα)→X,0⩽α⩽1, is a continuous map then for each w0∈Xα there exists a T1=T1(w0)∈(0,T] such that the initial value problem (3.4) has a mild solution u∈C([σ,σ+T1];Xα) . Furthermore, T1 may be chosen uniformly for w0 in bounded subsets of Xα. Proof. We assume without loss of generality the initial time σ=0 . For the general case, simply consider a new time variable t′=t−σ. We know {S(t):t⩾0} is a compact exponentially decaying analytic semigroup. Then by Theorem 2.1.19 ∥AαS(t)∥⩽Mαt−αe−ωt,t>0,(3.6) 62 Chapter 3. A Nonlocal Chafee-Infante Problem uniformly for t∈[0,T1]and, for h<δ, ∥Zt+h δ AS(t+h−s)(g(s)−g(t+h))ds −Zt δ AS(t−s)(g(s)−g(t))ds∥ =∥Zt δ−h AS(t−s)(g(s+h)−g(t+h))ds −Zt δ AS(t−s)(g(s)−g(t))ds∥ ⩽∥Zt δ AS(t−s)(g(s+h)−g(t+h)−g(s)+g(t))ds∥ +Zδ δ−h AS(t−s)(g(s+h)−g(t+h))ds∥. (3.19) Now, since Zt δ∥AS(t−s)(g(s+h)−g(t+h)−g(s)+g(t))∥ds ⩽2Zt δ(t−s)−1(w(Cδ−β(t−s)β)+w((t−s)β)ds <∞, (3.20) and, for each s∈(0,t], we have ∥AS(t−s)(g(s+h)−g(t+h)−g(s)+g(t))∥h→0+ −−−→ 0, and it follows by Lebesgue’s Dominated Convergence Theorem that Zt δ∥AS(t−s)(g(s+h)−g(t+h)−g(s)+g(t))∥ds h→0+ −−−→ 0, and since ∥Zδ δ−h AS(t−s)(g(s+h)−g(t+h))ds∥ ⩽M1Zδ δ−h (t−s)−1(w(Cδ−β(t−s)β)+w((t−s)β))ds ⩽M1Zδ−1C1/β(t−δ+h) δ−1C1/β(t−δ) w(uβ) udu h→0+ −−−→ 0, (3.21) we conclude that the second integral on the right-hand side of (3.17) is a continuous function of t . Thus, AG1(t) is continuous and therefore AG(t) is a continuous function on (0,T1] . An analogous result holds for functions f that only depend on time and f∈L1(0,T;X) (see (PAZY,1983, Theorem 4.3.2) for details). We extend this result to the case when f also depends on state. Also holds the next result which is an autonomous version of the above theorem and follows as consequence. 3.2. Existence of mild and strong solution 63 Corollary 3.2.4. If X,A and {S(t):t⩾0} are as in Theorem 3.2.1 and f:Xα→X,0⩽α<1, is a continuous map, the initial value problem        du(t) dt +Au(t) = f(u)t>0, u(0) = w0∈Xα, (3.22) possesses a mild solution for each w0∈Xα. If fis such that for each u∈C([0,T],Xα) ∥f(u(t)) −f(u(s))∥X⩽w(∥u(t)−u(s)∥α),(3.23) where w:[0,∞)→[0,∞)is a continuous increasing function, w(0) = 0 and Zt 0u−1w(uβ)du <∞,(3.24) then (3.22) has a strong solution for each w0∈Xα. Remark 3.2.5. The condition imposed on the function w is strictly weaker than holder continuity. In fact, consider the function w:(0,e−1)→R defined by w(x) = (ln(|x|−1))−p , x>0 , p>1 . This example satisfies the conditions of the theorem since is strictly increasing and Ze−1 0 1 x(ln(x−1)pdx =Z∞ 1u−pdu <∞. and, for r>0, lim x→0 xr w(x)=lim x→0 xr (ln(|x|−1))−p=lim x→0(xr/p(−ln|x|))p=0. Figure 5 – Continuity Module w(x) = (ln(|x|−1))−pand p(x) = x1 2 Now consider the initial boundary value problem.                    ∂v ∂t=∂v2 ∂x2+1 a(l(v)) λf(v)+ h(σ+Rt σa(l(v))dθ),t>σ,x∈Ω, v(t,σ,0) = v(t,σ,1) = 0, v(σ,σ,x) = w0(x). (3.25) 64 Chapter 3. A Nonlocal Chafee-Infante Problem In the following result, we present the appropriate conditions to apply Theorem 3.2.3 to (3.25). Corollary 3.2.6. Assume that a,h∈C(R+),a(s)⩾m>0 , l:H1 0(Ω)→R+ is a continuous functional and f:H1 0(Ω)→L2(Ω), a continuous function. Then for each w0∈H1 0(0,1) the initial value problem (3.25) possesses a mild solution. If additionally for each w0∈Xαthere is a neighborhood Vw0of w0in Xαsuch that ∥f(u1)−f(u2)∥⩽w(∥u1−u2∥α), ∥a(l(u1))−a(l(u2))∥⩽w(∥u1−u2∥α),for all u1,u2∈Vw0,and ∥h(t)−h(s)∥⩽w(|t−s|β),for some β∈(0,1−α), (3.26) where w:[0,∞)→[0,∞)is increasing, continuous, w(0) = 0 and Zt 0u−1w(uβ)du <∞, the problem (3.25) has a strong solution which turns out to be (by simple bootstrapping arguments) classical. Remark 3.2.7. Making the time reparametrization t=s+Rτ sa(l(v))−1dθ we have that w(τ,x) = v(t,x) is a solution of (3.1). Thus we have proved the existence and regularity of a solution to the problem (3.1) thanks to the results obtained for (3.25). 3.3 Comparison results Usually, it is highly beneficial to compare a problem with another to conduct an asymptotic analysis of the solutions. For this to be possible, our semilinear system must be defined in a space with a partial ordering. Typically these spaces are linear subspaces of the space of functions defined in a domain Ω⊂RN and taking value in R which are locally integrable ( L1 loc(Ω) ) like X=L2(Ω)or X1/2=H1 0(Ω)which admit the following partial ordering u⩽v⇔u(x)⩽v(x)a.e. for xin Ω. We are interested in providing comparison results for our prototype model (3.2) and similar models without uniqueness. The main results we present in this section are abstract and based on (ARRIETA; CARVALHO; RODRÍGUEZ-BERNAL,2000) (see also (CARVALHO; LANGA; ROBINSON,2012)) and extended to include semilinear problems without uniqueness. As before we assume that −Ahas compact resolvent, it is the infinitesimal generator of an analytic semigroup {e−At :t⩾0}and that α∈(0,1). We also assume that X has a partial ordering ⩽ and that the positive cone in Xα , that is, Cα={x∈Xα: 0 ⩽x},α⩾0 3.3. Comparison results 65 is closed. We also require that −Ahas positive resolvent, that is, (−A)−1Cα⊂Cα. We prove comparison results for the initial value problems of the form (˙u=Au +g±(u),t>0, u(0) = w0∈Xα(3.27) where g±:Xα→X are continuous maps. We use the comparison results obtained for this autonomous problem to obtain those for our non-autonomous problem (3.25). Also, by using the comparison results in this section, we are able to ensure that the solutions of (3.25) are globally defined, therefore those of (3.1) are also globally defined. In the following theorem, we prove that if g−⩽g+ and, for some choice of k∈R , g+(u)+ ku is an increasing function, for each solution u+(t,w0) exist u−(t,w0) such that u−(t,w0)⩽ u+(t,w0). Theorem 3.3.1. Let g±:Xα→X be continuous maps such that g−⩽g+ and g+(u)+ku is an increasing function. Then (i) For each solution u(t,w0,g+) exist u(t,w0,g−) such that u(t,w0,g−)⩽u(t,w0,g+) as long as both are defined. (ii) Given a solution u(t,u+,g+) and u−∈Xα with u−⩽u+ , there exists a solution u(t,u−,g+) such that u(t,u−,g+)⩽u(t,u+,g+)as long as both are defined. An analogous result holds if g−+kI is an increasing function. Proof. (i) We write the initial value problem (3.27) (with forcing term g+) as        du(t) dt =−(A+kI)u+g+(u)+ku,t>0, u(0) = w0∈Xα. (3.28) Note that k∈R may be chosen such that (g++kI)(·) is increasing and A+kI is the generator of an exponentially decaying semigroup. Since g+ is continuous, follows that g+(u)+ku is an continuous function, it follows from Theorem 3.2.1 that the initial value problem (3.28) has a mild solution u+(·). Let S(t)be the semigroup generated by −A−kI, then (3.6) and (3.7) hold. ∥AαS(t)∥⩽Mαt−αe−ωtfor t>0. 66 Chapter 3. A Nonlocal Chafee-Infante Problem Fix a mild solution u+(·)∈C([0,T1],Xα) of (3.28) and choose T1∈(0,t1] and ρ>0 such that ∥S(t)w0−w0∥α⩽ρ/2 for 0 ⩽t⩽T1, ∥g−(v)+kv∥X⩽NT1for all v ∈BXα ρ(w0) = {v0:∥v0−w0∥α⩽ρ}, NT1MαZt 0u−αe−ωudu ⩽ρ/2 for 0 ⩽t⩽T1and MαNT1ρ/2≥T1. Set Y=C([0,T1];Xα) and Y0={u−:u−∈Y,u(0) = w0,u−(t)∈BXα ρ(w0),u−(t)⩽u+(t),0⩽ t⩽T1}.We define a mapping Fw0:Y0→Yby (Fw0u−)(t) = S(t)w0+Zt 0S(t−s)g−(u−(s))+ku−(s))ds. Note that, for all u−∈Y0, ∥(Fw0u−)(t)−w0∥α⩽∥S(t)w0−w0∥α+Zt 0∥S(t−s)(g−(u−(s))+ku−(s))∥αds. ⩽ρ, and since g+(u)+ku is an increasing function we have (Fw0u−)(t)−u+(t) = Zt 0S(t−s)(g−(u−(s))+ku−(s))ds −Zt 0S(t−s)(g+(u+(s))+ku+(s))ds ⩽Zt 0S(t−s)(g+(u−(s))+ku−(s)−g+(u+(s))−ku+(s))ds ⩽Zt 0S(t−s)(g+(u+(s))+ku+(s)−g+(u+(s))−ku+(s))ds =0. It follows that Fw0 maps Y0 into Y0 . Also, the continuity of g−(u−)+ku− implies that Fw0 is continuous. Moreover, as before, we conclude that Fw0(Y0)is compact into Y0. Schauder fixed point Theorem implies that F has a fixed point in Y0 and it is a mild solution of (3.27) (with forcing term g− ) on C([0,T1],H1 0(Ω)) . This procedure successively repeated at the end time of the interval proves the result. (ii) Fix u+∈Xαand write equation (3.27) as ut=−(A+βI)u+g+(u)+βu. 3.3. Comparison results 67 Given a mild solution u(·,u+,g+)∈C([0,T1],Xα) we recursively build the following sequence u1(t)=Fu−(u(t,u+,g+))(t)=S(t)u−+Zt 0S(t−s)[g+(u(s,u+,g+))+βu(s,u+,g+)]ds; u2(t) =Fu−(u1)(t) = S(t)u−+Zt 0S(t−s)[g+(u1(s))+βu1(s)]ds; . . . un+1(t) =Fu−(un)(t) = S(t)u−+Zt 0S(t−s)[g+(un(s))+βun(s)]ds; (3.29) where S(t) is the semigroup generated by −A−βI . Since S(t)u+⩾S(t)u− we have that u1(t)⩽u(t,u+,g+) , due to g+(·) + βI is an increasing function it follows that u2(t) = Fu−(u1(t)) ⩽Fu−(u(t,u+,g+)) = u1(t)⩽u(t,u+,g+) and repeating the same argument we conclude that un(t)⩽u(t,u+,g+). Moreover, as before, we conclude that {un:n∈N} is compact in Y0 . Then there exists a subsequence that we denote {un}n∈N(relabeled the same) that converges to u∗and since lim n→∞(Fu−un)(t) = lim n→∞un+1(t) then (Fu−u∗)(t)=u∗(t) and since (Fu−un)(t)⩽u(t,u+,g+) it follows that u∗(t) = u(t,u−,g+) ⩽u(t,u+,g+). Corollary 3.3.2. Let g±R→Rbe continuous functions such that g−≤g+and liminf t∈[−R,R],R>0 f+(t)−f+(s) t−s≥ −k.(3.30) Them for each solution u(t,w0,g−)exists u(t,w0,g−)such that u−(t,w0)≤u+(t,w0). Proof. we note that (g+(u)+ k)(·)is an increasing function. In fact, for t≤swe have g+(t)+kt −g+(s)−ks =k(t−s) + g+(t)−g+(s) =k(t−s)+ g+(t)−g+(s) t−s(t−s) ≥k(t−s)+limt→s g+(t)−g+(s) t−s(t−s) ≥k(t−s)−k(t−s)≥0 68 Chapter 3. A Nonlocal Chafee-Infante Problem The hypothesis (3.30) is a restriction on the regions where the nonlinearity g+ is decreasing. An analogous result holds when g−satisfies (3.30). In what follows we consider g instead of g+ or g− in (3.27) in order to be able to use it when either (g−+kI)(·)is increasing or when (g++kI)(·)is increasing. Theorem 3.3.3. Assume that for every r>0 there exists a constant β=β(r)>0 such that g(·)+βI is positive in the positive elements in BXα r(0) and that g is a continuous function. Then, if 0 ⩽w0∈Xα, the solution of (3.27)u(t,w0,g)is positive as long as it exists. Proof. For w0∈Xα, we write the equation as ut=−(A+βI)u+g(u)+βu and recursively define the sequence {ui:i∈N} u1(t) = F(w0)(t) = e−(A+βI)tw0+Zt 0e−(A+β)(t−s)[g(w0)+βw0]ds u2(t) = F(u1)(t) = e−(A+βI)tw0+Zt 0e−(A+β)(t−s)[g(u1(s))+βu1(s)]ds . . . ui+1(t) = F(ui)(t) = e−(A+βI)tw0+Zt 0e−(A+β)(t−s)[g(ui(s))+βui(s)]ds (3.31) Note that ui⩾0 , i∈N since w0⩾0 and g(·)+βI is positive in the positive elements in BXα r(0) . Let T1>0 be such that ∥S(t)w0−w0∥Xα⩽ρ/2 for 0 ⩽t⩽T1, ∥g(ui)+kui∥X⩽NT1for all i ∈N, NT1MαZt 0u−αe−ωudu ⩽ρ/2 for 0 ⩽t⩽T1and T1⩽MαNT1ρ/2. Set Y=C([0,T1],Xα)and Y0={v∈C([0,T1],Xα):v(0) = w0and v(t)⩾0}. Clearly ui+1= (Fui)(t)≥0 for all t∈[0,T1] . As before, {ui:i∈N} is compact in Y0 and there exists a subsequence that we denote {ui}i∈N (relabeled the same) that converges to some function u∈Y0uniformly in [0,T1]and since lim i→∞F(ui)(t) = lim i→∞ui+1(t), it follows that Fu =uand that u(t)⩾0. With a continuation argument, we can show that u(t)⩾0 as long as the solution is defined. 3.3. Comparison results 69 In a similar way we can obtain the following result. Theorem 3.3.4. (i) Assume that g(Cα)⊂C0 . Then w0⩾0 implies that any solution u(·) of (3.27) (with ginstead of g±)must satisfy u(t)⩾0 as long as the solution exists. (ii) Let u+⩾u− be initial conditions in Xα , g+,g− continuous functions such that g+⩾g− and either g+(·)+βI or g−(·)+βI is an increasing function. If u(t,u+,g+),u(t,u−,g−) are solutions of the respective initial value problem in (3.27) , then there are solutions u∗(t,u+,g+) , u∗(t,u−,g−) such that u(t,u+,g+)⩾u∗(t,u−,g−) and u∗(t,u+,g+)⩾u(t,u−,g−) . (iii) If g−,g+ are such that for every r>0 there exist a constant β=β(r)>0 and an increasing function h(·)such that g−(·)+βI⩽h(·)⩽g+(·)+ βI in BXα r(0) ; u−,u+∈Xα , u−⩽u+, and u(t,u+,g+),u(t,u−,g−) solutions of the respective initial value problem in (3.27) , then there are solutions u∗(t,u+,g+) , u∗(t,u−,g−) such that u∗(t,u+,g+)⩾u(t,u−,g−)and u(t,u+,g+)⩾u∗(t,u−,g−). Proof. (i) Since w0⩾0 and g(u(t)) ⩾0 we have S(t)w0 and S(t−s)g(u(s)) ⩾0 then u(t)⩾0 for all t∈[0,T1] . A continuation argument shows that this inequality is true as long as the solution exists. (ii) Let h+βI be increasing with either h=g+ or h=g− . Given u(·,u−,g−) , from Theorem 3.3.1, there is a u(·,u−,h) with u(t,u−,h)⩾u(t,u−,g−) and using that h+βI is increasing there is a u(·,u+,h) such that u(t,u+,h)⩾u(t,u−,h)⩾u(t,u−,g−) and finally there is a u∗(·,u+,g+) such that u∗(t,u+,g+)⩾u(t,u+,h)⩾u(t,u−,h)⩾u(t,u−,g−) . In a similar way, using that h+βI is increasing, from Theorem 3.3.1, we obtain u(·,u+,h) , u(·,u−,h) and u∗(·,u−,g−)such that u(t,u+,g+)⩾u(t,u+,h)⩾u(t,u−,h)⩾u∗(t,u−,g−). (iii) After adding and subtracting the term βu in the equation (3.27) the result follows from (ii). The following corollary is a key result for ensuring that the solutions of (3.25) are globally defined as well as to ensure that they are bounded. Consider the auxiliary initial boundary value problems:                wt=wxx +λf(w)−K M,t>σ,x∈Ω, w(t,σ,0) = w(t,σ,1) = 0, w(σ,σ,x) = w0(x), (3.32) 70 Chapter 3. A Nonlocal Chafee-Infante Problem and                zt=zxx +λf(z)+K m,t>σ,x∈Ω, z(t,σ,0) = z(t,σ,1) = 0, z(σ,σ,x) = w0(x). (3.33) Corollary 3.3.5. Assume that a∈C(R+,[m,M]) for some choice of m,M∈(0,∞) and that f:R→R is a continuous function. Assume also that f is such that for every r>0 there exists a constant γ=γ(r)>0 such that γI+λf(·) is increasing in (−r,r) . Let u0,u1,u2∈H1 0(Ω),u0⩽ u1⩽u2. If g−:=λf(·)−K M and g+:=λf(·)+K m where K:=sups|h(s)| , for any solution v(t,σ,u1,g) of (3.25) there are solutions w(t,σ,u0,g−) , w(t,σ,u1,g−) , z(t,σ,u1,g+) , z(t,σ,u2,g+) such that w(t,σ,u0,g−)⩽w(t,σ,u1,g−)⩽v(t,σ,u1,g) v(t,σ,u1,g)⩽z(t,σ,u1,g+)⩽z(t,σ,u2,g+). Proof. Observe that, since H1 0(0,1)⊂L∞(0,1) with continuous embedding, given r>0 it is easy to see that there exists γ(r)>0 such that for ν∈R+and ∥u∥H1 0(0,1)⩽r γu+λf(u)−K M⩽γu+λf(u)+h(s) a(ν)⩽γu+λf(u)+K m,u∈BH1 0(0,1) r(0)(3.34) with γu+λf(u)−K Mand γu+λf(u)+K mbeing increasing functions in the variable u∈BH1 0(0,1) r(0). The result now follows by successive applications of Theorem 3.3.4 with g−(t,u) = λf(u)−K M, g+(t,u) = λf(u)+K m, g(t,u) = λf(u)+h(σ+Rt σa(l(u))dθ) a(l(u)) . (3.35) Remark 3.3.6. One can modify the hypothesis γ+f(·) to be increasing in Corollary (3.3.5) by imposing condition (3.30), and still obtain the same result. To prove that all solutions of (3.2) are globally defined, we need to impose the structural condition (S) on the non-linearity f. 3.3. Comparison results 71 To prove the existence of a pullback attractor we assume in addition to (S) the dissipativity condition (D) on f. For the rest of this section Au =uxx with Dirichlet boundary conditions in the interval (0,1) and we use the notation of Corollary 3.3.5 for g,g+ and g− . We apply the comparison results of Theorem 3.3.4 to g−and gor gand g+with hbeing, respectively, g−or g+. We commence by revisiting the following lemma from (ARRIETA; CARVALHO; RODRÍGUEZ-BERNAL,2000). Lemma 1. Let λ1 be the first eigenvalue of A+C0νI , which we assume to be positive. If α0>0 then there is a constant M=M(A,λ1,α0)such that for every α0⩾α⩾β⩾−α0 ∥e−(A+C0νI)tu0∥α⩽Me−λ1tt−(α−β)∥u0∥βt>0,u0∈Xβ. Furthermore, given q>1 there exist a δ0∈(0,1)and a constant ˜ M=˜ M(A,λ1)such that ∥e−(A+C0νI)tu0∥L∞(Ω)⩽˜ Mt−δ0e−λ1t∥u0∥Lq(Ω)t>0,u0∈Lq(Ω) In the next proposition, we are going to prove the global existence of solutions and a pointwise bound on the global attractor assuming conditions (S)and (D)as needed. Proposition 3.3.7. Let w0∈H1 0(0,1) , σ∈R and let u(·,σ,w0) be a solution of (3.2) . Assume that f is such that for every r>0 there exist a constant γ=γ(r)>0 such that γI+λf(·) is increasing in (−r,r). (i) If (S) holds, then u(·,σ,w0)is globally defined. (ii) If (D) holds, there are constants K∞and Kα,α∈(0,1)such that limsup t−→∞ ∥u(t+σ,σ,w0)∥L∞(Ω)⩽K∞,(3.36) limsup t−→∞ ∥u(t+σ,σ,w0)∥α⩽Kα,(3.37) and these limits are uniform for w0in bounded subsets of H1 0(Ω)and for σ∈R. (iii) In addition, if (D) holds, w0∈H1 0(Ω)and φis the solution of      (A+C0ν)φ=C1,in Ω, φ=0,in ∂Ω, (3.38) then 0⩽φ∈L∞(Ω),limsupt→∞|u(t+σ,σ,w0)(x)|⩽φ(x) uniformly in x∈¯ Ω and for w0 in bounded subsets of H1 0(Ω) . Also, if |w0(x)|⩽φ(x) for all x∈¯ Ω , then |u(t+ σ,σ,w0)(x)|⩽φ(x)for all x∈¯ Ωand t⩾0. 78 Chapter 3. A Nonlocal Chafee-Infante Problem since X1/2+β⊂⊂ H1 0(Ω) , it follows that {uj(t):uj(·)∈G(τ),j∈N} is precompact in H1 0(Ω) . ∥φj(t2)−φj(t1)∥1 2⩽    (S(t2−τ)−S(t1−τ))φj(τ)+Zt2 τ S(t2−s)g((s,φs j(s),φs j(·))ds −Zt1 τ S(t1−s)g((s,φs j(s),φs j(·))ds ±Zt1 τ S(t2−s)g((s,φs j(s),φs j(·))ds   1 2 ⩽∥(S(t2−t1)−I)S(t1−τ)φj(τ)∥1 2+NZt1 τ∥S(t2−s)−S(t1−s)∥L(X,X1 2)ds +Zt2 t1 ∥S(t2−s)g((s,φs j(s),φs j(·))∥1 2ds ⩽∥(S(t2−t1)−I)A−βS(t1−τ)Aβφj(τ)∥1 2 +NZt1 τ∥(S(t2−t1)−I)A−βAβ+1 2S(t1−s)∥L(X)ds+NM′Zt2 t1 (t2−s)−1 2ds ⩽C(t2−t1)β+NK Zt1 τ(t2−t1)β(t1−s)−1 2−βds +NK′Zt2 t1 (t2−s)−1 2ds. This last expression is independent of φj,j∈N , and tends to zero as t2−t1→0 since S(t) is strongly continuous at t=0. Note also that sup t∈[τ,∞),j∈N ∥φj(t)∥H1 0<C. Then, (C4) follows. Theorem 3.5.2. (SIMSEN; VALERO,2016) Let (C1)-(C2), (C4) hold. Then U is pullback asymptotically compact and monotonically pullback bounded dissipative if and only if it possesses the unique backwards bounded global pullback attractor B={B(s):s∈R} defined by A(t) = [ B∈B(X) ω(t,B). If (C3) is also satisfied, then Ais invariant. Theorem 3.5.3. Let f,a◦l,h be continuous functions such that (D) . Assume also that f is such that, for every r>0 , there is a constant γ=γ(r)>0 such that γI+λf(·) is increasing in [−r,r] . Then the multivalued process U(t,τ) generated by G defined in Proposition 3.5.1 has a pullback attractor B={B(s):s∈R}. Proof. Properties (C1)−(C4) have been checked in Proposition 3.5.1. To prove that G is pullback asymptotically compact, pullback bounded dissipative and monotonically pullback bounded dissipative we proceed as in Theorem 3.4.2 using the results of Proposition 3.3.7 and the compactness of the embedding of X1 2+β into X1 2=H1 0(0,1) for 0<β<1 2 . That is, defining B0(t):=BH1 0 Kα(0) for all t∈R follow by Proposition 3.3.7 and by Corollary 3.3.5 , that U is monotonically pullback bounded dissipative and the result follows using Theorem 3.5.2. 3.5. Results about pullback attractor 79 Corollary 3.5.4. Let f,a◦l,h be continuous functions such that (D) . Assume also that f is such that, for every r>0 , there is a constant γ=γ(r)>0 such that γI+λf(·) is increasing in [−r,r] . Then A−,B(t),A+are subsets of Σ(φ):={u∈L∞(Ω),|u(t,u0)(x)|⩽φ(x),a.e. x∈Ω}. for all t∈R , where A−( resp. A+) is the global attractor of multivalued semiflow G−( resp. G+) defined by the solutions of (3.32) (resp. (3.33) ) and B={B(t):t∈R} is the multivalued pullback attractor of the multivalued process U(t,τ)associated to the solutions of (3.25) . Proof. The existence of the attractors is guaranteed by Theorems 3.4.2 and 3.5.2 and the comparison between them follows using Corollary 3.3.5 and the uniform bounds follow as in Theorem 3.4.3. 81 CHAPTER 4 A DELAY PROBLEM In this chapter, which is based on (CARABALLO; CARVALHO; JULIO,2024a), we investigate the following variation of the Chafee-Infante equation; it is a non-local version with delay.            ∂w ∂τ −a(l(w))∂w2 ∂x2=λf(w)+γw(τ−ρ)+h(τ),τ>0,x∈Ω, w(τ,0) = w(τ,1) = 0, w(τ) = φ(τ),τ∈[−ρ,0]. (4.1) Here Ω= (0,1) , γ,λ>0 , l a continuous operator from H1 0(Ω) into R+ , a∈C(R) with a(R)⊂ [m,M]⊂(0,∞) , f∈C(R,R) , h∈C(R,L2(0,1)) is bounded and φ∈C([−ρ,0],H1 0(Ω)) . The results presented here are based on (CARABALLO; CARVALHO; JULIO,2024a). We make a change in the time scale that transforms (4.1) into a non-local (in time and in space) non-autonomous semilinear problem, similar to what was done in Chapter 3, now with some additional considerations. Given a solution w:[−ρ,∞)→H1 0(0,1) of the problem (4.1) making t:=α−1(τ) = Zτ 0a(l(w(r)))−1dr , τ∈[−ρ,∞) , we have that the function u defined by u(t) = w(τ) will be a solution of the problem                    ut=uxx +λf(u)+γu(α(t)−ρ) a(l(u)) ,t>0,x∈Ω, u(t,0) = u(t,1) = 0, u(t) = φ(α0(t)),t∈[α−1(−ρ),0], (4.2) where α−1(−ρ)=Z−ρ 0a(l(φ(r)))−1dr<0 and, since a(R)⊂[m,M]⊂(0,∞) , ρ m⩽−α−1(−ρ)⩽ ρ M. 82 Chapter 4. A Delay Problem We note that, if τ⩾0, α(t) = Zt 0a(l(u(r))dr and α0:[α−1(−ρ),0]→[−ρ,0]is the only solution of the integral equation α(s) = Zs 0a(l(φ◦α(r)))dr,s∈[α−1(−ρ),0]. On the other hand, given φ∈C([−ρ,0],H1 0(Ω)) we define α0 and α−1(ρ) as above. If u:[α−1(−ρ),∞)→H1 0(0,1) is a solution of the semilinear delay differential problem (4.2) , making τ=α(t) = Zt 0a(l(u(r)))dr the function w(τ):=u(t), will be a solution of the problem (4.1). Observe that given φ∈C([−ρ,0],H1 0(Ω)) , the delay is always the same, and determined by the initial function φ . In fact, a remarkable aspect of this model is that α0 and α−1(−ρ) are uniquely determined by φ only and that, to solve the (4.1) with initial function φ , corresponds to solve the nonlocal non-autonomous semilinear delay differential problem (4.2) with delay determined by φ. 4.1 Local well-posedness In this section, we establish conditions to ensure local well posedness of (4.1) , which involve delays depending on the initial function φ . We use the findings presented in Chapter 3, however, this task necessitates additional assumptions not only on f,a,l and h as outlined in Theorem 3.2.3, but also on the initial function φ . These additional assumptions are crucial to ensure the well posedness and regularity of solutions, particularly in problems involving delays. We use the method of steps which consists of solving the problem iteratively, in intervals of time of length α−1(ρ) = Z0 −ρ a(l(φ(r)))−1dr . In each step we transform (4.1) into a semilinear problem and apply the Theorem 3.2.1 (3.2.3) in order to guarantee the existence of mild (strong) solution for the semilinear non-local (in time and in space) non-autonomous problem obtained. To guarantee that the solution is defined in each whole interval we need the additional structural condition (S) which we recall below. (S) Assume that there exist C0,C1∈Rsuch that u f (u)⩽−νC0u2+|u|νC1, for all u∈Rand for both ν=m λand ν=M λ. 4.1. Local well-posedness 83 Theorem 4.1.1. Let X=L2(0,1) , D(A) = H2(0,1)∩H1 0(0,1) and A:D(A)⊂X→X be the operator defined by Au =uxx . Then −A is positive and self-adjoint (hence sectorial), with fractional power spaces Xα:=D(−Aα) with the graph norm, α⩾0 , X1 2=H1 0(0,1) , (λ+A)−1 has compact resolvent, λ∈ρ(−A) and the semigroup {S(t):t⩾0} generated by A is a compact and exponentially decaying analytic semigroup. If l∈C(H1 0(Ω)),R+ , a∈C(R) and f:R→R is a continuous function satisfying (S), then given φ∈C([−ρ,0],H1 0(0,1)) the initial value problem (4.1) has a mild solution u∈C([0,∞);H1 0(0,1)) , furthermore u∈C([0,∞);Xα) , for all α∈(0,1). We obtain the next regularity result. Theorem 4.1.2. Under the assumptions of Theorem 4.1.1 and that f,h,a,lsatisfy |f(s)−f(s′)|⩽w(|s−s′|), |a(l(u))−a(l(v))|⩽w(∥u−v∥H1 0(0,1)), |h(s)−h(s′)|⩽w(|s−s′|β), and∥φ(t)−φ(t′)∥X⩽w(|t−t′|β),for some β∈(0,1−α), (4.3) where w:[0,∞)→[0,∞)is a continuous increasing function, w(0) = 0 and Z1 0u−1w(uβ)du <∞, the initial value problem (4.1) has a strong solution w∈C([0,∞);H2(0,1)∩H1 0(0,1)). Proof. We employ the method of steps. Let us revisit the logic and the necessary details to implement Theorem 3.2.3. We know that w(τ) = φ(τ) for τ∈(−ρ,0] . Then, for τ∈(0,ρ],τ−ρ∈(−ρ,0] and then we have w(τ−ρ) = φ(τ−ρ) , so that the equation (4.1) becomes the following non-autonomous quasilinear non-local scalar one-dimensional parabolic partial differential equation            ∂w ∂τ =a(l(w))∂w2 ∂x2+λf(w)+γφ(τ−ρ),τ>0,x∈Ω, w(τ,0) = w(τ,1) = 0, w(0) = φ(0). (4.4) One can make a change in the time scale, like in Chapter 3, in order to obtain the semilinear problem            ut=uxx +λf(u)+γφ(α(t)−ρ) a(l(u)) ,t>0,x∈Ω, u(t,0) = u(t,1) = 0, u(0) = φ(0), (4.5) 84 Chapter 4. A Delay Problem where τ=Zt 0a(l((u(r)))dr =:α(t). Set f1:[0,α−1(ρ)]×C([0,α−1(ρ)],Xα)→Xdefined by f1(t,ut(·)) = λf(u(t))+γφ(α(t)−ρ) a(l(u(t))) ,α(t) = Zt 0a(l(u(θ)))dθ, where for u∈C([0,α−1(ρ)],H1 0(0,1)), we denote ut(·) = u(·)[0,t] , t∈[0,T] . Since [0,α−1(ρ)] ∋ t7→ φ(α(t)−ρ)∈H1 0(0,1) is continuous, follows by Theorem 3.2.1 that the initial value problem (4.5) has at least a mild solution that we will call g1 . Under the assumption (S), this solution exists in the whole interval [0,α−1(ρ)] (see Proposition 3.3.7). Thus we just need to check regularity. In fact, for g1(t),g1(s)∈V,for Va neighborhood of φ(0),we have that ∥f1(t,gt 1(·))−f1(s,gs 1(·))∥ ⩽    λf(g1(t))+γφ(α(t)−ρ) a(l(g1(t))) −λf(g1(s))+γφ(α(s)−ρ) a(l(g1(s))) ±λf(g1(s))+γφ(α(s)−ρ) a(l(g1(t)))     ⩽m−1∥λ(f(g1(t)) −f(g1(s)))∥+m−1∥γ(φ(α(t)−ρ)−φ(α(s)−ρ))∥ +∥λf(g1(s))+γφ(α(s)−ρ)∥ a(l(g1(s)))−a(l(g1(t))) a(l(g1(t)))a(l(g1(s)))  . (4.6) Since for t>swe have |α(t)−α(s)|=Zt s a(l(g1(θ)))dθ ⩽M|t−s|. (4.7) It follows that ∥f1(t,gt 1(·))−f1(s,gs 1(·))∥⩽K1w(∥g1(t)−g1(s)∥α)+K2w(c|t−s|β),(4.8) for some positive numbers K1,K2and c, and since Zt 0u−1w(uβ)du <∞,f or 0<β<1−α, we can apply Theorem 3.2.3 and conclude that g1 is a strong solution to the equation (4.5) in the interval [0,α−1(ρ)]. Also w(τ) = g1(τ)is a solution of (4.4) in the interval [0,ρ]. Now, for τ∈[ρ,2ρ],τ−ρ∈[0,ρ] and we have that w(τ) = u(α−1(τ)) = g1(τ) so the equation (4.1) becomes                    ∂w ∂τ =a(l(w))∂w2 ∂x2+λf(w)+γg1(τ−ρ),τ>ρ, w(τ,0) = w(τ,1) = 0, w(ρ) = g1(ρ). (4.9) 4.1. Local well-posedness 85 Again, by making a change in the time scale we have the initial value problem                    ∂u ∂t=∂u2 ∂x2+λf(u)+γg1(α(t)−ρ) a(l(u)) ,t∈(α−1(ρ),α−1(2ρ)], u(t,0) = u(t,1) = 0, u(α−1(ρ)) = g1(α−1(ρ)). (4.10) Since g1∈C((0,α−1(ρ)],Xα), we have that the application f2(t,u(t),u(·)) = λf(u(t))+γg1((α(t)−ρ)) a(l(u(t)) is continuous as long as α(t)∈[ρ,2ρ] which means that t∈[α−1(ρ),α−1(2ρ)] . Thus, again we just need to check the regularity of the solution. Hence, we need to see that ∥f2(t,ut(·))−f2(s,us(·))∥⩽w(∥u(t)−u(s)∥α)+w(|t−s|β), but that follows exactly as before. To proceed we make g2(τ) = u(α−1(τ)) for τ∈[ρ,2ρ]. Continuing in this way, if gn−1(τ) = u(α−1(τ)) for τ∈[(n−1)ρ,nρ] we have, for n⩾3 ,            ∂w ∂τ −a(l(w))∂w2 ∂x2=λf(w)+γgn−1(τ−ρ),τ∈((n−1)ρ,nρ], w(τ,0) = w(τ,1) = 0, w((n−1)ρ) = gn−1((n−1)ρ). (4.11) has a mild solution gn:[(n−1)ρ,nρ]→H1 0(0,1). Therefore, w:[−ρ,∞)→H1 0(0,1)given by w(τ) = (φ(τ),τ∈[−ρ,0]; gn(τ),τ∈((n−1)ρ,nρ],n⩾1.(4.12) is a strong solution of (4.1). Using the variation of constants formula we have g1(τ) = u(t) = S(t)φ(0)+Zt 0S(t−s)λf(g1(α(s)))+γφ(α(s)−ρ) a(l(g1(α(s)))) ds,0⩽t⩽α−1(ρ) g2(τ) = u(t) = S(t−α−1(ρ))g1(ρ)+Zt α−1(ρ) S(t−s)(λf(g2(α(s)))+γg1(α(s)−ρ)) a(l(g2(α(s)))) ds, for t∈[α−1(ρ),α−1(2ρ)]. We know that g1(ρ) = S(α−1(ρ))φ(0)+Zα−1(ρ) 0S(α−1(ρ)−s)λf(g1(α(s)))+γφ(α(s)−ρ) a(l(g1(α(s)))) ds, 86 Chapter 4. A Delay Problem then, we obtain that S(t−α−1(ρ))g1(ρ) = S(t)φ(0)+Zα−1(ρ) 0S(t−s)λf(g1(α(s)))+γφ(α(s)−ρ) a(l(g1(α(s)))) ds and g2(τ) = u(t) = S(t)φ(0)+Zα−1(ρ) 0S(t−s)λf(g1(α(s)))+γφ(α(s)−ρ) a(l(g1(α(s)))) ds +Zt α−1(ρ) S(t−s)(λf(g2(α(s)))+γg1(α(s)−ρ)) a(l(g2(α(s)))) ds. Similarly, we obtain that for n>1 we have gn(τ) = u(t) = S(t)φ(0)+ n−1 ∑ m=1Zα−1(mρ) α−1((m−1)ρ) S(t−s)(λf(gm(α(s)))+γgm−1(α(s)−ρ)) a(l(gm(α(s)))) ds +Zt α−1((n−1)ρ) S(t−s)(λf(gn(α(s)))+γgn−1(α(s)−ρ)) a(l(gn(α(s)))) ds. Remark 4.1.3. In each step, we solve the equation within the interval (nρ,(n+1)ρ) and consider the initial condition depending solely on one endpoint of the interval, namely (n+1)ρ , to concatenate the solutions. This poses no issue as in each step the problem has no delay. Remark 4.1.4. The proof of Theorem 4.1.2 contains the proof of existence of solution for (4.1) . That is, it also shows the Theorem 4.1.1. In the proof of the theorem, it becomes clear why it is necessary to add the hypothesis (S) to ensure the existence of a mild solution. Observe that in the step one, we need to guarantee the continuity of the map [0,α−1(ρ)] ∋t7→ φ(α(t)−ρ)∈L2(0,1) but that follows from continuity of φand from the fact |α(t)−α(t′)|=|Zt t′a(l(u(r)))dr|t→t′ −→ 0. The continuity of [α−1((n−1)ρ),α−1(nρ)] ∋t7→ gn(α(t)−ρ)∈L2(0,1)n⩾1 is obtained in an analogous way. 4.2 Global well-posedness Throughout this section we consider “ ⩽” a partial ordering in H1 0(Ω), that is: u⩽vin H1 0(Ω)⇔u(x)⩽v(x)a.e. for x in Ω. 4.2. Global well-posedness 87 We assume that φ is the initial function of the problem (4.1) and exists K>0 such that −K⩽ φ(t)⩽K. To guarantee the global existence we consider the auxiliary initial value problems : ∂u ∂t=Au+λf(u)−K M,t>0,x∈Ω, u(0) = −K, u|∂Ω=0. (4.13) ∂u ∂t=Au+λf(u)+K m,t>0,x∈Ω, u(0) = K, u|∂Ω=0. (4.14) and we write f−(u) = λf(u)−γK M,u f −(u)⩽C0u2+γK M+C1|u| f+(u) = λf(u)+γK m,u f +(u)⩽C0u2+γK m+C1|u| g(t,ut(·)) = λf(u(t))+γu(Rt 0a(l(u(θ)))−ρ)dθ a(l(u(t))) . where A:D(A)→L2(Ω),D(A) = H2(Ω)∩H1 0(Ω)is the linear operator defined by, Au =uxx,u∈D(A), f:R→R and g:[0,T]×C([0,T],Xα)→X are continuous functions. For u(·)∈C([0,T],Xα),ut(·) = u(·)|[0,t]. In the next theorem we see that, if for each r>0 there is a κ=κ(r) such that u7→ κu+f(u) is an increasing function in [−r,r] , them for each n∈N there is a constant Kn such that there are u(t,Kn),u(t,−Kn), solutions of (4.14) and (4.13) with K replaced by Kn , such that u(t,−Kn)⩽u(t,φ)⩽u(t,Kn),t∈[α−1((n−1)ρ),α−1(nρ)]. If (S) holds, this results in global existence to solutions of (4.2) thanks to the fact that the solutions of (4.13) and (4.14) are globally defined (see Proposition 3.3.7). Theorem 4.2.1. Assume that (S) holds for a continuous function f:R→R such that for every r>0 there exist a constant κ=κ(r)>0 such that s7→ κs+f(s) is increasing in [−r,r] . Let φ:[−ρ,0]→H1 0(0,1) and K>0 such that −K⩽φ(τ)(x) = φ(α(t))(x)⩽K for all τ∈[−ρ,0] and x∈[0,1]. If u(t,φ)is a solution of        ut=uxx +g(t,ut(·)),t>0,x∈Ω, u(t,0) = u(t,1) = 0, u(r) = φ(α(r)),r∈[α−1(−ρ),0], (4.15) there are u(t,−K) , u(t,K) solutions of (4.13) and (4.14) such that u(t,−K)⩽u(t,φ)⩽u(t,K) , for t∈[0,α−1(ρ)]. As an immediate consequence of this u(·,φ)is defined for all t⩾0. 94 Chapter 4. A Delay Problem N}is equicontinuous. In fact, for r>0t1>σ, if t1+r,t1∈(σ+(n−1)ρ,σ+nρ]we have ∥ξj(t1+r)−ξj(t1)∥H1 0⩽∥(S(t1+r−σ)−S(t1−σ))φj(0)∥H1 0 +Zt1+r σ+α−1((n−1))ρ S(t1+r−s)(λf(ξj(s))+γwj(σ+α(s)−ρ)+h(α(s)) a(l(ξj(s))) ds −Zt1 σ+α−1((n−1)ρ) S(t1−s)(λf(ξj(s))+γwj(σ+α(s)−ρ)+h(α(s)) a(l(ξj(s))) ds +∥ n−1 ∑ m=1Zσ+α−1(mρ) σ+α−1((m−1)ρ) S(t1+r−s)(λf(ξj(s))+γwj(σ+α(s)−ρ)+h(α(s)) a(l(ξj(s))) − n−1 ∑ m=1Zσ+α−1(mρ) σ+α−1((m−1)ρ) S(t1−s)(λf(ξj(s))+γwj(σ+α(s)−ρ)+h(α(s)) a(l(ξj(s))) ∥H1 0 ⩽∥(S(t1−σ)(S(r)−I)φj(0)∥H1 0 +NZt1 σ+α−1((n−1)ρ)∥S(t1+r−s)−S(t1−s)∥L(L2,H1 0)ds +Zt1+r t1 ∥S(t1+r−s)(λf(ξj(s))+γwj(σ+α(s)−ρ)+h(α(s)) a(l(ξj(s))) ∥H1 0ds +N n−1 ∑ m=1Zσ+α−1(mρ) σ+α−1((m−1)ρ)∥S(t1+r−s)−S(t1−s)∥L(L2,H1 0)ds ⩽M∥(S(r)−I)φj(0)∥H1 0 +NZt1 σ+α−1((n−1)ρ)∥(S(r)−I)A−β∥L(L2)∥Aβ+1 2S(t1−s)∥L(L2)ds +NM′Zt1+r t1 (t1+r−s)−1 2e−a(t1+r−s)ds +N n−1 ∑ m=1Zσ+α−1(mρ) σ+α−1((m−1)ρ)∥(S(r)−I)A−β∥L(L2)∥Aβ+1 2S(t1−s)∥L(L2)ds ⩽M∥(S(r)−I)φj(0)∥H1 0 +NK Zt1 σ rβ(t1−s)−1 2−βe−a(t1−s)ds +NK′Zt1+r t1 (t1+r−s)−1 2e−a(t1+r−s)ds, ⩽M∥(S(r)−I)φj(0)∥H1 0+rβNK Z∞ 0s−1 2−βe−asds +NK′Zr 0s−1 2e−asds, This last expression is independent of ξj,j∈N and tends to zero as r→0 since S(t) is a C0 semigroup. From Corollary [(ii)]4.2.2 sup t∈[σ,∞),j∈N ∥ξj(t)∥H1 0<C, then (C3) follows from Arzela-Ascoli Theorem applied to the family wj∈C(σ) with wj(τ)(·) = ηj(τ+·)∈X,τ∈[σ,∞). Theorem 4.4.2. Let X=C([−ρ,0],H1 0(0,1)) , f , a , l , h and φ are continuous functions and assume that f satisfies (D2) and is such that, for every r>0 , there is a constant κ=κ(r)>0 such that κI+f(·) is increasing in [−r,r] . Then the multivalued process {T(t,σ):t⩾σ} generated by C defined in Proposition 4.4.1 has a unique pullback attractor B={B(s):s∈R}. 4.4. Pullback attractor 95 Proof. Fix t∈R , we are going check that the family {φj∈T(τ,σj)B,j∈N} for any B∈B(X) and σj→ −∞ is compact. In fact, since φj∈T(τ,σj)B, there is ηj∈C(σj) such that ηj(σj)∈B and φj=ηj(τ), for each j ; in addition, for each σj⩽τ<∞ and j∈N we define ξj(t) = wj(τ) = ηj(τ)(0) were, for each j , τ=σj+αj(t) with α−1(τ) = Rτ σj(a(l((ηj(τ)))−1dτ have that for 0<β+1 2<1, 1 ⩽n<∞, ∥Aβξj(t)∥H1 0⩽∥AβS(t−σj)φj(0)∥H1 0 +Zt σj+α−1((n−1)ρ)∥AβS(t−s)λf(ξj(s))+γwj(α(s)−ρ))+h(α(s) a(l(ξj(s))) ∥H1 0ds + n−1 ∑ m=1Zσj+α−1(mρ) σj+α−1((m−1)ρ)∥AβS(t−s)λf(ξj(s))+γwj(α(s)−ρ))+h(α(s)) a(l(ξj(s)) ∥H1 0ds ⩽Mβ(t−σj)−βe−at∥φj(0)∥H1 0+NM Zt σj+α−1((n−1)ρ)(t−s)−1 2−βe−a(t−s)ds +MN n−1 ∑ m=1Zσj+α−1(mρ) σj+α−1((m−1)ρ)(t−s)−1 2−βe−a(t−s)ds ⩽Mβ(t−σj)−βe−at∥φj(0)∥H1 0+NM Z∞ σj (t−s)−1 2−βe−a(t−s)ds <∞. Since X1 2+β⊂⊂ X1 2 follows that {ηj(σj)(·) = ξj(t+·)} is precompact in C([−ρ,0],H1 0(Ω)) . Hence there exists a subsequence of {ξj} (relabeled the same) that converges to ξ and we conclude that Cis pullback asymptotically compact. On the other hand, defining B0(t):=BX(0,Kα) for all t∈R follow by Corollary 4.2.2 and by Theorem 4.2.1, that T is monotonically pullback bounded dissipative and the result follow using Theorem 2.2.27. 97 CHAPTER 5 CONCLUSION: ENGLISH, PORTUGUESE AND SPANISH 5.1 Conclusion The abstract approach to semilinear problems is a powerful tool, as exemplified in Theorem 3.2.3, which was initially designed for asymptotic analysis of (3.1) . However, it could also be applied to study (4.2) by adding a structural condition on the nonlinearity. Sometimes, it is not feasible to conduct purely abstract analysis as it is necessary to determine a specific space to achieve certain results. Note that this occurs in the comparison results for (3.1) when h is nonzero, although this is different in the autonomous case. The difficulty is that we need to work in an ordered space. Extending the results to ensure the existence of classical solutions for problems of the type (3.4) poses a challenge. In Theorem 3.2.3, we observed how extending the class of functions for which the problem has a regular solution results in the loss of uniqueness. However, for our specific problems, this was not a difficulty due to the use of multivalued theory for the study of asymptotic dynamics. It was interesting to apply techniques similar to those used in (ARRIETA; CARVALHO; BERNAL,1999), (where uniqueness of solution is guaranteed), to obtain the existence of attractor. Note that, as in Theorem 2.3.8, in our case, the continuity modulus imposed on the nonlinearity allows us to obtain a regularization effect on the solutions, namely, Hölder continuity, which is a very useful tool for applicability purposes. This facilitated, for example, demonstrating the regularity of solutions for (4.2) in Theorem 4.1.2. Next steps include developing a more comprehensive characterization of the attractor and expanding the class of nonlinearities where regularity with uniqueness of solution is achieved. 98 Chapter 5. conclusion: English, portuguese and spanish 5.2 Conclusão A abordagem abstrata de problemas semilineares é uma ferramenta poderosa, como exemplificado no Teorema 3.2.3, que foi inicialmente projetado para análise assintótica de (3.1) . No entanto, também pode ser aplicado ao estudo de (4.2) , adicionando uma condição estrutural à não linearidade. Às vezes, não é viável conduzir uma análise puramente abstrata, pois é necessário determinar um espaço específico para alcançar certos resultados. Note que isso ocorre nos resultados de comparação para (3.1) quando h é não nulo, embora isso seja diferente no caso autônomo. A dificuldade reside na necessidade de trabalhar em um espaço ordenado. Estender os resultados para garantir a existência de soluções clássicas para problemas do tipo (3.4) apresenta um desafio. No Teorema 3.2.3, observamos como estender a classe de funções para as quais o problema tem uma solução regular resulta na perda de unicidade. No entanto, para nossos problemas específicos, isso não foi uma dificuldade devido ao uso da teoria multivaluada para o estudo da dinâmica assintótica. Foi interessante aplicar técnicas semelhantes às usadas em (ARRIETA; CARVALHO; BERNAL,1999), onde a unicidade da solução é garantida, para obter a existência de atrator. Os próximos passos incluem desenvolver uma caracterização mais abrangente do atrator e expandir a classe de não linearidades onde se obtém regularidade com unicidade de solução. Note que, como no Teorema 2.3.8, em nosso caso, o módulo de continuidade imposto à não linearidade nos permite obter um efeito de regularização nas soluções, ou seja, continuidade de Hölder, que é uma ferramenta muito útil para fins de aplicabilidade. Isso facilitou, por exemplo, demonstrar a regularidade das soluções para (4.2) no Teorema 4.1.2. 5.3 Conclusión El abordaje abstracto de problemas semilineales es una herramienta poderosa, esto puede verse ejemplificado en el Teorema 3.2.3, el cual fue pensado inicialmente para realizar un análisis asintótico de (3.1) , sin embargo, pudo aplicarse para el estudio de (4.2) , adicionando una condición estructural sobre la no linealidad. En ocasiones no es posible realizar un análisis netamente abstracto ya que a veces es necesario determinar un espacio específico para conseguir ciertos resultados, note que esto sucede en los resultados de comparación para (3.1) cuando h no es nula, no obstante, esto es diferente en el caso autónomo. La dificultad radica en la necesidad de trabajar en un espacio ordenado. Extender los resultados para garantizar existencia de solución clásica para problemas del tipo (3.4) constituye un reto. En el Teorema 3.2.3 vimos cómo extender la clase de funciones para la cual el problema tiene solución regular representa la perdida de unicidad de la misma, sin embargo, para nuestros problemas específicos no fue una dificultad, debido al uso de la teoría multivaluada para el estudio de la dinámica asintótica. Lo interesante fue conseguir aplicar 5.3. Conclusión 99 técnicas parecidas a las utilizadas en (ARRIETA; CARVALHO; BERNAL,1999), donde existe unicidad de solución, para garantizar existencia de atractor. 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