Gradient-like nonlinear semigroups with infinitely many equilibria and applications to cascade systems

GRADIENT-LIKE NONLINEAR SEMIGROUPS WITH INFINITELY MANY EQUILIBRIA AND APPLICATIONS TO CASCADE SYSTEMS EDER R. ARAGAO-COSTA1, ALEXANDRE N. CARVALHO2, PEDRO MAR´ IN-RUBIO3, AND GABRIELA PLANAS4 Abstract. We consider an autonomous dynamical system coming from a coupled system in cascade where the uncoupled part of the system satisfies that the solutions comes from −∞ and goes to ∞to equilibrium points, and where the coupled part generates asymptotically a gradient-like nonlinear semigroup. Then, the complete model is proved to be also gradient-like. The interest of this extension comes, for instance, in models where a continuum of equilibrium points holds, and for example a Lojasiewicz-Simon condition is satisfied. Indeed, we illustrate the usefulness of the theory with several examples. 1. Introduction The study of the structure of invariant sets for infinite-dimensional dynamical systems and its characterization has received a lot of attention in the last few decades. Actually, while the finite-dimensional case has been deeply understood, the same goal for the infinite-dimensional case used to be reduced to very particular examples, for instance, a gradient (or gradient-like) structure in the equations (or directly in the attractor), e.g. see [13, 21, 24] and the references therein. A deeper understanding of structure of attractors for autonomous and non-autonomous dynamical systems that generalize the above comes by some recent developments in [5, 6, 7] for dynamical systems where the equilibrium points are hyperbolic, and therefore there exists a finite number of them. It involves a new class, called of gradient-like nonlinear semigroups, which roughly speaking means that all complete trajectories in the attractor come from and go to equilibrium points (in particular the attractor is gradient-like) and there is no homoclinic structure among these points. However, in many situations, the set of equilibrium points is not finite but indeed they form a continuum and the above results cannot be applied. Nevertheless, in that context a new tool can be employed: the study of the asymptotic behaviour in the finite-dimensional setting of a gradient system associated to an analytic function in [15, 16] lead in [22] to the study of the asymptotic behaviour of semilinear parabolic equations when an additional condition, called Lojasiewicz-Simon’s inequality, holds. That condition ensures that any solution tends as t→ ∞ to a unique stationary solution (for instance see also [10, 25, 14] and the references therein). 1Partially supported by FAPESP 2008/50248-0, Brazil. 2Partially supported by CNPq 305230/2011-5, CAPES/DGU 238/2011 and FAPESP 2008/55516-3, Brazil. 3Partially supported by Ministerio de Ciencia e Innovaci´on (Spain) project MTM2008-00088, Junta de Andaluc´ıa grant P07-FQM-02468, and Ministerio de Educaci´on–DGPU (Spain) project PHB2010-0002-PC. 4Partially supported by CNPq 302865/2012-8, CAPES/DGU 238/2011 and FAPESP 2008/09342-3, Brazil. 1 2 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS Our main goal in this paper is to analyse the behaviour of a coupled system in cascade where the uncoupled part of the system satisfies that the solutions comes from −∞ and goes to ∞to equilibrium points, and where the coupled part generates asymptotically a gradient-like nonlinear semigroup. This leads to a new gradient-like nonlinear semigroup concept, and in particular allows to describe the attractor of the system as the union of unstable manifolds of all equilibria. The interest of this extension comes for instance in models where a continuum of equilibrium points holds, and for example a Lojasiewicz-Simon condition is satisfied. However, this condition has been applied to the study of the behaviour of a solution only when t→ ∞. We present an extension to deal also with the case t→ −∞. The convergence of solutions to equilibrium can also be proved assuming that the manifold of equilibria is normally hyperbolic (see [19, Th.6.1] or [20] for the case when the set of equilibria is normally stable). For this, instead of normal hyperbolicity, here we use the Lojasiewicz-Simon condition. The structure of the paper is as follows. In Section 2 we state the problem. In Section 3 we give our main result (cf. Theorem 21); that is, we give conditions ensuring that the coupled system generates a (new type of) gradient-like nonlinear semigroup, and therefore having a gradient-like attractor. In Section 4 we study the convergence to equilibria of any solution in the attractor, not only when t→ ∞ but also for t→ −∞ using the Lojasiewicz-Simon inequality (its proof is revised here, cf. Theorem 24). Problems of first and second order in time are analysed. In Section 5 we provide several examples to illustrate the applicability of the theory. Finally, in the Appendix we prove the backwards convergence to equilibrium stated in Section 4. 2. Basic Facts and Notions In this section we introduce the basic facts and notions that are needed to state and prove our main results. Let Zbe a metric space with metric d:Z×Z→R+, where R+:= [0,∞). Given a subset A⊂Z, the −neighborhood of Ais the set O(A) = {z∈Z:d(z, a)<  for some a∈A}. Definition 1. An evolution process in Zis a two parameter family {T(t, τ) : t≥τ∈R}of continuous maps from Zinto itself such that (a)T(τ, τ) = IZ,with IZbeing the identity in Z, (b)T(t, σ)T(σ, τ) = T(t, τ),for all t≥σ≥τin R, (c)The map P ×Z3(t, τ, z)7→ T(t, τ)z∈Zis continuous, where P:= {(t, τ)∈R2:t≥ τ}. If T(t, τ) = T(t−τ, 0) for all t≥τ∈Rthe process {T(t, τ) : t≥τ∈R}is called an autonomous evolution process and the family {T(t) : t≥0},defined by T(t) := T(t, 0) for t≥0,is called a semigroup. Clearly (a0)T(0) = IZ, with IZbeing the identity in Z, (b0)T(t+s) = T(t)T(s), for all t, s ∈R+,and (c0)R+×Z3(t, z)7→ T(t)z∈Zis continuous. A continuous map z:R→Zis called a global solution for the evolution process {T(t, τ) : t≥τ}if it satisfies T(t, τ)z(τ) = z(t),for all t≥τ∈R. GRADIENT-LIKE NONLINEAR SEMIGROUPS 3 We note that the term global solution is often applied to refer solutions that are defined for all positive times. Here it is always used to indicate solutions defined for all real numbers. In particular, a global solution for a semigroup {T(t) : t≥0}is a continuous map ξ:R→Z with the property that T(t)ξ(s) = ξ(t+s) for all s∈Rand for all t∈R+. We say that ξ:R→Zis a global solution through z∈Zif it is a global solution and ξ(0) = zand that ξis a equilibrium point (or stationary solution) when the function ξ(t) = ξfor all t∈Ris a solution. The unstable set of an equilibrium z∗is the set Wu(z∗)=nz∈Z: there is a global solution ζthrough zsuch that lim t→−∞ dist(ζ(t), z∗)=0o. The notion of invariance plays a fundamental role in the study of the asymptotic behaviour of semigroups. Definition 2. A subset Aof Zis said invariant under the action of the semigroup {T(t) : t≥0} if T(t)A=Afor all t≥0. Now we will introduce the notions of attraction and absorption. For that we recall the definitions of Hausdorff semi-distance and distance. For A, B ⊂Z, the Hausdorff semi-distance from Ato Bis given by dist(A, B) := sup a∈A inf b∈Bd(a, b), and the Hausdorff distance between Aand Bis defined by dH(A, B) := max{dist(A, B), dist(B, A)}. Definition 3. Given two subsets A, B of Zwe say that Aattracts Bunder the action of the semigroup {T(t) : t≥0}if dist(T(t)B, A)t→∞ −→ 0,and we say that Aabsorbs Bunder the action of {T(t) : t≥0}if there is a tB>0such that T(t)B⊂Afor all t≥tB. With these elements we can introduce the notion of global attractors. Definition 4. A subset Aof Zis a global attractor for a semigroup {T(t) : t≥0}if it is compact, invariant and for every bounded subset Bof Zwe have that Aattracts Bunder the action of {T(t) : t≥0}. Next, we will introduce the notions of pullback attractor and generalized gradient-like process (see [6]). In order to do it, we first need the definition of invariance and of isolated global solution. Definition 5. Let {T(t, τ) : t≥τ}be an evolution process and {Ξ(t) : t∈R}a family of subsets of Z. We say that {Ξ(t) : t∈R}is invariant under the process {T(t, τ) : t≥τ},when T(t, τ)Ξ(τ) = Ξ(t)for t≥τ. Definition 6. Let ξ:R→Zbe a solution for a nonlinear evolution process {T(t, τ) : t≥τ∈ R}. The set Γ := {ξ(t) : t∈R} is called trace of ξ:R→Z. If ξ:R→Zis a solution and there exists δ > 0such that, any global solution ζ:R→Zwith ζ(R)⊂ Oδ(Γ) := {z∈Z:dist(z, Γ) < δ}must satisfy ζ(t) = ξ(t)for all t∈R, then we say that ξ:R→Zis an isolated global solution. Ξ={ξ1,··· , ξn}is said a set of isolated global solutions if each ξiis an isolated global solution 4 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS and there exists δ > 0such that Oδ(Γi)∩Oδ(Γj) = ∅,1≤i<j≤n, where Γiis the trace of ξi:R→Z. Definition 7. A pullback attractor for an evolution process {T(t, τ) : t≥τ∈R}is an invariant family {A(t) : t∈R}of compact sets with S t≤τA(t)bounded for each τ∈Rand such that, for each t∈Rand bounded subset Bof Z, we have that lim τ→−∞ dist(T(t, τ)B, A(t)) = 0. Remark 8. It must be pointed out that this is not the most general definition of a pullback attractor. Indeed, the assumption that ∪t≤τA(t)is bounded does not appear in many of the definitions in the literature. However, this suites our purposes in this paper. Let {T(t, τ) : t≥τ∈R}be a nonlinear evolution process with a pullback attractor {A(t) : t∈R}which contains a finite number of isolated global solutions Ξ={ξ1,··· , ξn}. Let Γibe the trace of ξi. Definition 9 (cf.[11]).Let δbe as in Definition 6 and fix 0∈(0, δ). For ξ∈Ξand ∈(0, 0), an −chain from ξto ξis a sequence of natural numbers `i∈ {1,··· , n}, a sequence of real numbers, τi< σi< ti, and a sequence ziin Z,1≤i≤k, such that zi∈ O(Γ`i),T(σi, τi)zi/∈ O0(∪n j=1(Γj)) and T(ti, τi)zi∈ O(Γ`i+1 ),1≤i≤k, with ξ=ξ`k+1 =ξ`1. We say that ξ∈Ξ is chain recurrent if there is an 0∈(0, δ)and −chain from ξto ξfor each ∈(0, 0). We are now ready to define gradient-like evolution processes. Definition 10. Let Zbe a metric space and {T(t, τ) : t≥τ}be a nonlinear evolution process in Z. Let {A(t) : t∈R}be the pullback attractor for {T(t, τ) : t≥τ}. We say that {T(t, τ) : t≥τ}is a generalized gradient-like process if the following two hypotheses are satisfied: (H1) {A(t) : t∈R}contains a finite number of isolated global solutions Ξ={ξ1,··· , ξn} with the property that any global solution ξ:R→ Z in {A(t) : t∈R}satisfies lim t→−∞ dist(ξ(t), ξi(t)) = 0 and lim t→∞ dist(ξ(t), ξj(t)) = 0, for some 1≤i, j ≤n. (H2) Ξ={ξ1,··· , ξn}does not contain any chain recurrent isolated solution. Next we seek to introduce the notion of gradient-like semigroups (see [6]). To that end we first need the definition of homoclinic structure. Definition 11. Let {T(t) : t≥0}be a semigroup which has a set Eof equilibrium points. A homoclinic structure associated to Eis a finite subset {z∗ 1,··· , z∗ p}of Etogether with a set of global solutions {ξ1,··· , ξp}such that z∗ j t→−∞ ←− ξj(t)t→∞ −→ z∗ j+1,1≤j≤p where z∗ p+1 := z∗ 1. Now, for the autonomous case, we recall the definition of a gradient-like semigroup (which indeed is slightly different from that in [6]). GRADIENT-LIKE NONLINEAR SEMIGROUPS 5 Definition 12. Consider a metric space (Z, d)and a nonlinear semigroup {T(t) : t≥0}in Z, which has a global attractor Aand a set of equilibrium points E(possible infinite). We say that {T(t) : t≥0}is a gradient-like semigroup when (G1) For any bounded global solution ξ:R→Zfor {T(t) : t≥0},there exist two equilibrium points z∗ 1and z∗ 2in Esuch that lim t→−∞ d(ξ(t), z∗ 1) = 0 and lim t→∞ d(ξ(t), z∗ 2) = 0. (G2) The attractor Adoes not contain homoclinic structures. Remark 13. We remark that, in the context of this definition, it is possible that a gradient system is not gradient-like, since the condition (G1) above, may not hold for a general gradient system. In [1], the notion of generalized gradient-like semigroups is proved to coincide with the notion of generalized gradient semigroups (replacing equilibria by isolated invariant sets). It is fairly difficult to find examples of gradient semigroups which do not satisfy (G1) (one example can be found in [17, p.15]). Our goal is to describe the asymptotic dynamics of the following partially coupled (autonomous) model:    x0=Ax +g(x), t > 0, y0=By +f(x, y), t > 0, x(0) = x0∈X, y(0) = y0∈Y, (1) where Xand Yare Banach spaces, the operators A:D(A)⊂X→Xand B:D(B)⊂Y→Y generate strongly continuous semigroups of linear operators and f, g are suitable nonlinearities. Let also assume that there exists a vectorial subspace D(C) in D(A)×D(B) (possibly all D(A)×D(B)) dense in Z=X×Y, and we consider the linear operator C:D(C)⊂Z→Z given by Cz =C(x, y) := (Ax, By) for z= (x, y)∈D(C) and suppose that Cgenerates a C0-semigroup in Z(or even a singular semigroup). Then define h:Z→Zas h(z) = h(x, y) := (g(x), f(x, y)) for z= (x, y)∈Z. So, problem (1) can be reformulated as z0=Cz +h(z), t > 0, z(0) = z0∈Z. (2) Finally, assume that f:X×Y→Yand g:X→Xare so that the systems (2) and x0=Ax +g(x), t > 0, x(0) = x0∈X, generate, respectively, a nonlinear semigroup {T(t) : t≥0}in Zwith global attractor ACand set of equilibrium points EC, and a nonlinear semigroup {S(t) : t≥0}in X, which is gradient with Lyapunov functional E:X→R, global attractor A,and set of equilibrium points EA. We aim to establish conditions so that the semigroup {T(t) : t≥0}related to (2) is gradientlike in the sense of Definition 12 . In order to obtain this, we introduce the notion of hyperbolic equilibrium and recall two theorems which will be useful for our results. Definition 14. An equilibrium solution x∗ 0for the problem x0=Dx +m(x), t > 0 x(0) = x0∈X 6 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS is hyperbolic when the spectrum of e D:= D+m0(x∗ 0)does not intersect the imaginary axis, the set σ+={λ∈σ(e D) : Reλ > 0}is compact and, if γis a smooth closed simple curve in ρ(e D)∩{λ∈C: Reλ > 0}oriented counterclockwise and enclosing σ+, Q=Q(σ+) := 1 2πi Zγ (λ−e D)−1dλ, Qe˜ D t =e˜ D tQ,R(Q)⊂D(˜ D)and there are constants M1⩾1,β > 0, such that kee DtQkL(X)⩽M1eβt, t ⩽0, kee Dt(I−Q)kL(X)⩽M1e−βt, t ⩾0. Next we define exponential dichotomy for a linear evolution process and introduce the concept of global hyperbolic solutions, which is the analogous non-autonomous of the concept of hyperbolic equilibrium. Definition 15. We say that a linear evolution process {U(t, s) : t≥s}⊂L(X)in a Banach space Xhas exponential dichotomy with exponent ωand constant Mif there exists a family of projections {Q(t) : t∈R} ⊂ L(X)such that i) Q(t)U(t, s) = U(t, s)Q(s), for all t⩾s; ii) The restriction U(t, s)|R(Q(s)) ,t⩾sis an isomorphism from R(Q(s)) into R(Q(t)) and its inverse is denoted by U(s, t) : R(Q(t)) →R(Q(s)); iii) for some ω > 0 kU(t, s)(I−Q(s))k⩽Me−ω(t−s)t⩾s, kU(t, s)Q(s)k⩽Meω(t−s)t⩽s. We will say that a global solution ζ:R→Xof a nonlinear evolution process {T(t, τ) : t≥τ} in X, generated by a semilinear equation y0=By +f(t, y)t > τ, y(τ) = y0, is a global hyperbolic solution when the linearization around it, i.e., the linear equation y0=By +fy(t, ζ (t))y t > τ, y(τ) = y0, has solution operator with exponential dichotomy. With the above definitions and notation we can state the following result. Theorem 16 (cf.[8]).Consider Ya Banach space, {fη:R×Y→Y}η∈(0,1] a family of applications with each element having continuous partial derivative with respect to the second variable in R×Y, f0:Y→Ya continuously differentiable map and B:D(B)⊂Y→Ythe generator of a strongly continuous semigroup. Consider the following problems y0=By +fη(t, y)t > τ, y(τ) = y0,(3) y0=By +f0(y)t > τ, y(τ) = y0.(4) GRADIENT-LIKE NONLINEAR SEMIGROUPS 7 Assume that {fη:R×Y→Y}η∈(0,1] and f0:Y→Yare such that (3) and (4) generate nonlinear evolution processes {Tη(t, τ) : t≥τ}η∈(0,1],and a nonlinear semigroup {T0(t) : t≥0}, respectively. Suppose also that T0has a global attractor with a finite number of equilibrium points E={y∗ 1, . . . , y∗ n},all of them being hyperbolic. If for each r > 0it holds that lim η→0+sup t∈R sup kykY≤r{kfη(t, y)−f0(y)kY+k(fη)y(t, y)−f0 0(y)kL(Y)}= 0, then there exists η0∈(0,1] such that for each η≤η0there exist ξ∗ i,η :R→Y,i= 1, . . . , n, global solutions for (3), that are hyperbolic, and satisfying lim η→0sup t∈Rkξ∗ i,η(t)−y∗ ikY= 0,∀i= 1, . . . , n. Theorem 17 (cf.[6]).Consider Ya Banach space, and {Tη(t, τ) : t≥τ}a nonlinear evolution process on Ywith a pullback attractor {Aη(t) : t∈R}for η∈[0,1]. Assume that the following conditions hold: (a)∪ η∈[0,1] ∪ t∈RAη(t)is compact. (b){T0(t, τ) : t≥τ}is an autonomous process, i.e., T0(t, τ) = S(t−τ)for all t≥τ, where {S(t) : t≥0}is a semigroup, which additionally is gradient-like with a finite number of equilibrium points E={y∗ 1,0, . . . , y∗ n,0}. (c)For each η∈(0,1],{Tη(t, τ) : t≥τ∈R}has nglobal isolated solutions ξ∗ i,η :R→Y i= 1, . . . , n, such that, if Γi,η is the trace of ξ∗ i,η :R→Y, for i= 1, . . . , n and η∈(0,1], then one has that lim η→0+sup 1≤i≤n dH(y∗ i,0,Γi,η)=0. (d)For each T > 0and compact set K⊂Y, it holds lim η→0+sup τ∈R sup t∈[0,T] sup y∈KkTη(t+τ, τ)y−T0(t+τ, τ)ykY= 0. (e)There exist δ > 0and η1∈(0,1] such that if ξη:R→Yis a bounded solution of {Tη(t, τ) : t≥τ∈R}with η≤η1so that there exists t0∈Rand some i∈ {1, . . . , n} with sup t≤t0 dist(ξη(t),Γi,η)< δ (resp. sup t≥t0 dist(ξη(t),Γi,η)< δ), then lim t→−∞kξη(t)− ξ∗ i,η(t)kY= 0 (resp. lim t→∞kξη(t)−ξ∗ i,η(t)kY= 0). Then, there exists η0∈(0, η1]such that for each η∈(0, η0]{Tη(t, τ) : t≥τ∈R}is a non-autonomous gradient-like evolution process. 3. Gradient Like Cascade Systems Combining Theorem 16 and Theorem 17, we will be able to establish our first result, concerning the asymptotic behaviour of system (1). But firstly let us give one last definition. Definition 18. Let f:R×Y→Ybe an application such that the semilinear problem y0=By +f(t, y)t > τ, y(τ) = y0(5) generates a nonlinear evolution process {T(t, τ) : t≥τ∈R}which has a pullback attractor {A(t) : t∈R}. 8 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS Denote, for any ν > 0and µ > 0, the applications fν, fµ:R×Y→Yas fν(t, y) = f(t, y), if t ≤ −ν, f(−ν, y), if t > −ν and fµ(t, y) = f(t, y), if t ≥µ, f(µ, y), if t < µ. We will say that fis compatible from the left (respectively from the right) with respect to the system (5) if there exists ν0(resp. µ0)such that for all ν≥ν0(resp. µ≥µ0) the problem y0=By +fν(t, y), t > τ, y(τ) = y0, (resp. the problem y0=By +fµ(t, y), t > τ, y(τ) = y0) generates a nonlinear evolution process {Tν(t, τ) : t≥τ}which has a pullback attractor {Aν(t) : t∈R}with S ν≥ν0S t∈RAν(t)relatively compact (resp. generates a nonlinear evolution process {Tµ(t, τ) : t≥τ}which has a pullback attractor {Aµ(t) : t∈R}with S µ≥µ0S t∈RAµ(t)relatively compact). Lemma 19. With the above notation, suppose that the uncoupled equation of system (1), x0= Ax +g(x),generates a nonlinear semigroup {S(t) : t∈R}which has a global attractor A, set of equilibria E,and such that any global solution in the attractor converges to equilibrium points when t→ ±∞.Assume also that for any equilibrium point x∗∈ E the autonomous semilinear problem y0=By +f(x∗, y), t > τ, y(τ) = y0 defines a semigroup {Sx∗(t) : t≥0}which is gradient-like, has global attractor Ax∗and a finite number of equilibrium points Ex∗,all of them hyperbolic. Let ξ= (ϕ, ψ) : R→X×Y be a (global) solution in the global attractor related to (2), and denote f(t, y) := f(ϕ(t), y)for (t, y)∈R×Y. Assume that the non-autonomous problem y0=By +f(t, y), t > τ, y(τ) = y0(6) generates a nonlinear evolution process {S(t, τ) : t≥τ}which has a pullback attractor {A(t) : t∈R}, and that fis compatible from the left and from the right with respect to the system (6). According to the above assumption, there exist equilibrium points x∗ −, x∗ +∈ E such that lim t→−∞kϕ(t)−x∗ −kX= 0 and lim t→∞kϕ(t)−x∗ +kX= 0. If for each r > 0,it holds that lim t→−∞ sup kykY≤r{kf(t, y)−f(x∗ −, y)kY+kfy(t, y)−fy(x∗ −, y)kL(Y)}= 0 (7) and lim t→∞ sup kykY≤r{kf(t, y)−f(x∗ +, y)kY+kfy(t, y)−fy(x∗ +, y)kL(Y)}= 0, GRADIENT-LIKE NONLINEAR SEMIGROUPS 9 then there exist equilibrium points y∗ −∈ Ex∗ −and y∗ +∈ Ex∗ +such that lim t→−∞ kξ(t)−z∗ −kX×Y= 0 and lim t→∞ kξ(t)−z∗ +kX×Y= 0, where z∗ −= (x∗ −, y∗ −)and z∗ += (x∗ +, y∗ +). Proof. We will only deal with the case t→ −∞,since the case t→ ∞ can be treated analogously. For each ν > 0,define the map fν:R×Y→Yas before. Now we consider the family of semilinear problems y0=By +fν(t, y), t > τ, y(τ) = y0.(8) From the compatibility assumption we know that fνis such that there exists a family of evolution processes {Tν(t, τ) : t≥τ}with ν∈(ν0,∞), for some ν0>0,such that each of them has a pullback attractor {Aν(t) : t∈R}with S ν≥ν0S t∈RAν(t) relatively compact on Y. It is not difficult to see that for t≤ −νit holds that Tν(t, τ) = S(t, τ) for all τ≤t. We also have that if ζ:R→Yis a global solution for {S(t, τ) : t≥τ}then ζ(t) = S(t, τ)ζ(τ) = Tν(t, τ)ζ(τ) provided that τ≤t≤ −ν. So, defining for each ν≥ν0,ζν:R→Yas ζν(t) = ζ(t),if t≤ −ν, Tν(t, −ν)ζ(−ν),if t > −ν, then ζν:R→Yis a global solution for {Tν(t, τ) : t≥τ}. Now, using (7), one has that lim ν→∞ sup t∈R sup kykY≤r {kfν(t, y)−f(x∗ −, y)kY+k(fν)y(t, y)−fy(x∗ −, y)kL(Y)}= 0.(9) Therefore, from Theorem 16 we deduce that there exists ν1>0 such that for each ν≥ν1 problem (8) has nglobal hyperbolic solutions ξ∗ i,ν :R→Y, i = 1, . . . , n which satisfy lim ν→∞ sup t∈Rkξ∗ i,ν(t)−y∗ i,−kY= 0,∀i= 1, . . . , n, (10) where Ex∗ −={y∗ 1,−,··· , y∗ n,−}. From (9) we deduce that for each T > 0 and each compact set K⊂Y, it holds lim ν→∞sup τ∈R sup t∈[0,T] sup y∈KkTν(t+τ, τ)y−Sx∗ −(t+τ, τ)ykY= 0. Thus, from the above and the hyperbolicity of the solutions ξ∗ i,ν :R→Y, i = 1, . . . , n, the assumptions in Theorem 17 are fulfilled. Therefore, there exists ν2≥ν1such that for each ν≥ν2,{Tν(t, τ) : t≥τ}is a non-autonomous gradient-like process. Finally, consider ψ:R→Yas in the statement. This is a bounded solution of {S(t, τ) : t≥τ}. This implies that the functions ψν:R→Yassociated to ψ:R→Yare solutions for {Tν(t, τ) : t≥τ}and they are in {Aν(t) : t∈R},as long as they are bounded because they coincide with ψfor t≤ −ν. But in {Aν(t) : t∈R}there exists a (unique) value i0∈ {1, . . . , n} such that for νbig enough it holds that lim t→−∞ kψ(t)−ξ∗ i0,ν(t)kY= lim t→−∞ kψν(t)−ξ∗ i0,ν(t)kY= 0. 16 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS such that the orbit γu(u0) = {u(t)∈V:t∈R}is relatively compact in V. Then, there are ϕ, ψ ∈ E such that lim t→−∞ku(t)−ϕkV= 0 and lim t→∞ku(t)−ψkV= 0. Next we adapt, following [14], the previous results to deal with an abstract problem of second-order in time. Consider −A:D(A)⊂H→Ha sectorial operator in H, with compact resolvent. Let f:V→Hbe a smooth enough map, β > 0 and consider the following semilinear problem of second order    u00 +βu0=Au +f(u), u(0) = u0∈V, u0(0) = u1∈H. (22) We assume that there exists G∈C2(V;R) with M:= G0:V→V0satisfying for u∈D(A) −M(u) = Au +f(u). Problem (22) can be rewritten as a semilinear problem of first order (without lost of generality we assume β= 1 making if necessary the change of variables t=s/β): z0=Cz +f0(z), z(0) = z0∈Z, (23) where z∈Z:= V×H, equipped with the inner product ((v1, v2),(w1, w2))Z:= (v1, w1)V+ (v2, w2)H, C:D(C)⊂Z→Zbeing the linear operator with D(C) := D(A)×Vgiven by C:= 0I A−I, Ithe identity in V, and f0:Z→Zdefined by f0(z) := f0(u, v) := (0, f(u)),for z= (u, v)∈ Z=V×H. With the above notation and assumptions, we have that the linear operator Cgenerates a C0 semigroup of contractions in Zand (23) has an associated nonlinear semigroup {S(t) : t≥0} in Zas the solution operator. With this, the following result holds (again, a proof of the backwards convergence to equilibrium is included in the Appendix). Theorem 27. Assume that the hypotheses of Theorem 24 are fulfilled for any ϕ∈ EA:= {ϕ∈D(A) : −Aϕ =f(ϕ)}. Then, if u:R→Vis a global (strong) solution for (22) with its orbit {(u(t), u0(t)) : t∈R} relatively compact in V×H, and with kM0(u(t))u0(t)kV0≤aku0(t)kV0for all t∈Rand some a > 0, then there exist ϕand ψ∈ EAsuch that lim t→−∞ ku(t)−ϕkV= 0,lim t→−∞ ku0(t)kH= 0, and lim t→∞ ku(t)−ψkV= 0,lim t→∞ ku0(t)kH= 0. GRADIENT-LIKE NONLINEAR SEMIGROUPS 17 5. Examples In this section, we present some examples for which the results in Section 4 can be applied. Observe that we can couple any of these examples with a gradient-like problem such that we may take advantage of Theorem 21 to describe the attractor of the semigroup associated to a coupled system in cascade. In this way, we also present some examples of coupled system (1) to illustrate the results in Section 3. There are a large variety of examples of the form x0=Ax +f(x), x(0) = x0∈X arising in partial differential equations such that Xis a Hilbert space, for each x0∈Xthe solution x(t, x0) of the above initial value problem is defined for all t≥0 and the semigroup {S(t) : t≥0}defined by S(t)x0:= x(t, x0), t≥0, x0∈Xhas a global attractor, is gradient and for which either the set of equilibria of the uncoupled equation is finite or, otherwise, the Lojasiewicz-Simon inequality is satisfied. With this, {T(t) : t≥0}(as in Lemma 19, see also equation (2)) is gradient-like in the sense of Definition 12. Our first example is a simple system with one-sided coupling for which we can apply the abstract theory developed here to conclude the gradient-like structure. We note that, in this example, the nonlinearity is not assumed to be analytic (C2is enough) enhancing the applicability of Theorem 24. For the other examples, we can assume that, either the set of equilibria for the uncoupled equation is finite or, otherwise, the nonlinear term in it is analytic to use the Theorem 24 (for the analytic case, see [23]). Example 28. Let g:R→Rbe a C2(R)function such that g(s)6= 0 for all s∈Rand consider the function G:R2→Rgiven by G(s, t) := t2g(s).It follows that G∈C2(R2,R)with M(s, t) := G0(s, t) = (t2g0(s),2tg(s)) and G00(s, t) = t2g00(s) 2tg0(s) 2tg0(s) 2g(s)for all (s, t)∈R2, particularly, G00(s, t)is a symmetric matrix for all sand t, so the orthogonal projection coincides with the spectral one. It is clear that M−1(0) = R× {0}and therefore, since G00(s, 0) = 0 0 0 2g(s),we have that dim ker(G00(x∗)) = 1 for each x∗∈M−1(0).Now, the map h:R→R×Rgiven by h(s) := (s, 0) is a homeomorphism between Rand M−1(0).Thus, the map Gsatisfies hypothesis (ii)in Theorem 24. Let Ybe a Banach space and f:R2×Y→Ya smooth map such that for every bounded set B⊂R2the restriction of fto B×Yis bounded. Also, let −A:D(A)⊂Y→Ybe a sectorial operator such that the analytic semigroup generated for it, {eAt :t≥0},satisfies  eAt L(Y)≤Me−δt for every t > 0and some constants M≥1and δ > 0. Under these assumptions, by using the results from [3] on existence of pullback attractors, it is not difficult to see that, for any bounded solution ϕ:R→R2of x0+M(x) = 0,the map fϕ:R×Y→Y, given by fϕ(t, y) := f(ϕ(t), y),is compatible from the left and from the right 18 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS respect to the system y0=Ay +fϕ(t, y)t > τ y(τ) = y0∈Y. If for every equilibrium point x∗∈M−1(0),the equilibria of the problem y0=Ay +f(x∗, y) t > 0are all hyperbolic, then    x0+M(x)=0 t > 0 y0=Ay +f(x, y)t > 0 x(0) = x0∈R2, y(0) = y0∈Y is gradient-like. The next examples are applications of Theorem 24. More exactly, we consider applications to differential systems and to PDE, in particular, the reaction-diffusion equation, the damped wave equation and the Cahn-Hilliard equation. Example 29. Let φ∈C2(Rn,R)be such that φ(u)→ ∞ as kukRn→ ∞ and consider the ordinary differential equations x0=−∇φ(x), x(0) = x0∈Rn,(24) or, for β > 0, x00 +βx0=−∇φ(x), x(0) = x0∈Rn, x0(0) = v0∈Rn.(25) Then, (24) and (25) have global attractors A1in Rnand A2in Rn×Rnrespectively. If E= {x∈Rn:∇φ(x)=0}, it follows from the Lojasiewicz-Simon condition (which we assume) and from the results in Section 4 that A1=[ x∈E Wu(x),and A2=[ x∈E Wux 0. We remark that we are not assuming that the set of equilibria Eis finite and without the Lojasiewicz-Simon condition we only have that A1=Wu(E),and A2=WuE 0. Example 30. Let f∈C2(R,R)be such that lim sup |u|→∞ f(u) u≤ −δ < 0 (26) and |f00(u)| ≤ c(1 + |u|p) (27) with p∈(0,∞)to be specified. Consider the initial boundary value problems ut= ∆u+f(u),in Ω, ∂u ∂n = 0,in ∂Ω, u(0) = u0∈H1(Ω) (28) GRADIENT-LIKE NONLINEAR SEMIGROUPS 19 with p+ 2 ≤n+2 n−2and n≤6or, for β > 0, utt +βut= ∆u+f(u),in Ω ∂u ∂n = 0,in ∂Ω, u(0) = u0∈H1(Ω), ut(0) = v0∈L2(Ω), (29) with p+ 2 ≤n n−2and n≤4. Then, it is well-known that (28) and (29) have global attractors A3in H1(Ω) and A4in H1(Ω) ×L2(Ω) respectively (see [3]). If ˜ E={u∈H2(Ω) : ∆u+f(u) = 0 and ∂u ∂n = 0}, it follows from the Lojasiewicz-Simon condition (which we assume) and from the results in Section 4 that A3=[ u∈˜ E Wu(u),and A4=[ u∈˜ E Wuu 0. We remark that we are not assuming that the set of equilibria ˜ Eis finite. Example 31. Let f∈C2(R,R)be such that (26) and (27) with p+ 2 ≤n+2 n−2,n≤6hold. Consider the initial boundary value problem ut=−∆(∆u+f(u)),in Ω, ∂u ∂n =∂∆u ∂n = 0,in ∂Ω, u(0) = u0∈H1(Ω) (30) Then (30) has a global attractor A5in H1(Ω) (see [4]). If ¯ E={u∈H4(Ω) : ∆u+f(u) = 0,∂u ∂n = 0 and ∂∆u ∂n = 0}, it follows from the Lojasiewicz-Simon condition (which we assume) and from the results in Section 4 that A5=[ u∈¯ E Wu(u). We remark that the set of equilibria ¯ Eis always infinite. Finally, we consider the situation of cascade system (1). The following examples are applications of Theorem 21. Example 32. Let φ∈C2(Rn,R)and f∈C1(R,R)be such that φ(u)→ ∞ as kukRn→ ∞, f(s) + x0·x∗= 0 ⇒f0(s) + x0·x∗6= 0 for all x∗∈ {x∈Rn:∇φ(x) = 0}and fsatisfies (26). Consider the ordinary differential equations x0=−∇φ(x), s0=f(s) + x0·x, s(0) = s0∈R, x(0) = x0∈Rn, (31) or, for β > 0, x00 +βx0=−∇φ(x), s00 +βs0=f(s) + x0·x, s(0) = s0∈R, x(0) = x0∈Rn, x0(0) = v0∈Rn. (32) 20 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS Then, (31) and (32) have global attractors A6in Rn×Rand A7in Rn×Rn×Rrespectively. If ˆ E={(x∗, s)∈Rn×R:∇φ(x∗) = 0 and f(s) + x0·x∗= 0}, it follows from the Lojasiewicz- Simon condition (which we assume for x0=−∇φ(x)or x00 +βx0=−∇φ(x)) and from the results in Section 3 and Section 4 that A6=[ z∈ˆ E Wu(z),and A7=[ z∈ˆ E Wuz 0. In addition, all solutions in A6(or A7) are forwards asymptotic to equilibria and A6(or A7) does not contain homoclinic structures. The last example corresponds to the limiting problem of reaction-diffusion problems in a dumbbell domain (see [2]). Example 33. Let Ωbe a bounded smooth domain in Rnand P0, P1∈¯ Ω. Assume that f, g ∈ C2(R,R),fsatisfies (27) with p+ 2 ≤n+2 n−2and that both fand gsatisfy (26). Consider the initial boundary value problem ut= ∆u+f(u),in Ω, ∂u ∂n = 0,in ∂Ω, u(0) = u0∈C(¯ Ω), vt=vxx +g(v),in (0,1), v(t, 0) = u(t, P0), vx(t, 1) = 0, v(0,·) = v0∈H1(0,1). (33) Then, the initial boundary value problem (33) defines a semigroup {S(t) : t≥0}in C(¯ Ω) × H1(0,1) which has a global attractor A8. Assume that for all c∈[α, ω]with α= inf{ξ:f(ξ) = 0}and ω= sup{ξ:f(ξ) = 0}, the solutions of vxx +g(v+c)=0,in (0,1), v(0) = vx(1) = 0,(34) are hyperbolic (which may be accomplished, for example, if |α|and |ω|are suitably small as in [9]). If the Lojasiewicz-Simon condition holds for the equation in Ω, it follows from Theorem 21 that the solutions in A8are backwards and forwards asymptotic to equilibria. Appendix A. This appendix is dedicated to the proof of Theorems 26 and Theorem 27 in the case t→ −∞. To do that, we will make use the following two technical lemmas, which helpful in the proof of convergence to an equilibrium point when the Lojasiewicz-Simon inequality holds. Lemma 34. Let u: (−∞,0] →Hbe differentiable. Assume that there exist constants a > 0 and γ > 0such that for some T > 0and all t∈[−T, 0], Zt −∞ ku0(s)k2 Hds ≤aeγt.(35) GRADIENT-LIKE NONLINEAR SEMIGROUPS 21 Then, for −T≤τ≤t≤0, ku(t)−u(τ)kH≤b√aeγ 2t, where b=eγ 2/(eγ 2−1). Proof. Consider −T≤τ≤t≤0.First assume that t−τ≤1. Then, from H¨older inequality and (35), one has ku(t)−u(τ)kH=   Zt τ u0(s)ds   H≤√t−τZt τku0(s)k2 Hds1 2 ≤√aeγ 2t. Now, if t−τ > 1,let nbe the least natural number such that (t−τ)/n ≤1.Then, from above we deduce ku(t)−u(τ)kH≤Zt−n+1 τku0(s)kHds + n−2 X k=0 Zt−k t−k−1ku0(s)kHds ≤√aeγ 2(t−n+1) + n−2 X k=0 √aeγ 2(t−k)≤√aeγ 2teγ 2 eγ 2−1, which concludes the proof.  The next result is analogous to the above lemma for R+instead of R−. Its proof can be found in [14, Lem.2.2]. Lemma 35. Let u: [0,∞)→Hbe differentiable. Assume that there exist constants a > 0and γ > 0such that for some T > 0and all t∈[0, T], Z∞ tku0(s)k2 Hds ≤ae−γt. Then, for 0≤t≤τ≤T, ku(t)−u(τ)kH≤b√ae−γ 2t, where b=eγ 2/(eγ 2−1). Proof of Theorem 26: We only sketch the proof of the case t→ −∞,since it is similar to the case t→ ∞ which is proven in [14, Th.1.1]. Observe that for any global (strong) solution ξ:R→Vof (21), the following inequality holds for all t∈R: d dt(G◦ξ)(t) = hM(ξ(t)), ξ0(t)i=−kξ0(t)k2 H=−kM(ξ(t))k2 H≤0. Therefore, the function R3t7→ G(ξ(t)) ∈Ris non-increasing. From this and the fact that d dt (G◦ξ)(t) = −kξ0(t)k2 H,we conclude that G:V→Ris a Lyapunov functional for the nonlinear semigroup in Vassociated to (21). Then, the α-limit of u0relative to a global solution u, that is, αu(u0) := nϕ∈V:∃(tn)n∈N, tn→ −∞ such that lim n→∞ku(tn)−ϕkV= 0o, is a subset of E,which is nonempty from the relatively compactness of γu(u0). Consider a point ϕ∈αu(u0).Then, there exists a sequence (tn)n∈Nin Rwith tn→ −∞ such that lim n→∞ku(tn)−ϕkV= 0. 22 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS Hence, lim n→∞G(u(tn)) = G(ϕ) and from the monotonicity of R3t7→ G(u(t)) ∈Rit holds that lim t→−∞G(u(t)) = G(ϕ) with G(u(t)) ≤G(ϕ) for all t∈R. As before we have that d dt [(G◦u)(t)−G(ϕ)] = hM(u(t)), u0(t)i=−ku0(t)k2 H=−kM(u(t))k2 H.(36) So, for a fixed value t < 0,integrating on (−∞, t),we have Zt −∞ ku0(s)k2 Hds = [G(ϕ)−G(u(t))] .(37) On the other hand, for each natural number jwe take nj∈Nstrictly increasing such that ku(tnj)−ϕkV<1 jand G(ϕ)−G(u(tnj))1/2<1 j.(38) Choose σ > 0 as in Theorem 24 associated to the equilibrium point ϕand j0such that 1/j < σ if j≥j0. Define for j≥j0 tj:= inf τ < tnj:ku(t)−ϕkV< σ ∀t∈τ, tnj. From the Lojasiewicz-Simon inequality and the continuous injection of Hin V0we deduce the existence of a constant γ > 0 depending on ϕsuch that (36) implies, for t∈tj, tnj, that d dt [(G◦u)(t)−G(ϕ)] ≤γ[G(u(t)) −G(ϕ)] . This yields d dt e−γt [(G◦u)(t)−G(ϕ)]≤0. Integrating over (tj, tnj) we obtain e−γtnjG(u(tnj)) −G(ϕ)−e−γt [G(u(t)) −G(ϕ)] ≤0, which, combined with (37), implies that Zt −∞ ku0(s)k2 Hds ≤eγt e−γtnjG(ϕ)−G(u(tnj)),∀t∈(tj, tnj), j ≥j0.(39) Claim: There exists a value j1≥j0such that tj1=−∞. If G(ϕ) = G(u(tnj)) for some jthe claim is clearly true. On the other hand, if G(ϕ)> G(u(tnj)) and tj>−∞, for all j≥j0, then, taking into account (39), we may apply Lemma 34 with τ=tj,t=tnjand a=e−γtnjG(ϕ)−G(u(tnj))and conclude that ku(tj)−u(tnj)kH< beγ 2(tj−tnj)qG(ϕ)−G(u(tnj))<b j, for all j≥j0,where b > 0 is a constant depending on γ. From (38) and the continuous embedding V ,→H, it holds that ku(tnj)−ϕkH<C0 jfor all j≥j0,and some constant C0. From above we have that, for all j≥j0, ku(tj)−ϕkH≤ ku(tj)−u(tnj)kH+ku(tnj)−ϕkH<b j+C0 j, GRADIENT-LIKE NONLINEAR SEMIGROUPS 23 hence lim j→∞ku(tj)−ϕkH= 0. This also means, from the relatively compactness of γu(u0) in V that for a subsequence (relabelled the same) (u(tj))j∈Nconverges to ϕin V. In particular, there exists a value j∗such that ku(tj∗)−ϕkV< σ, which is a contradiction with the definition of tj∗.Therefore, there must exists a j1≥j0such that tj=−∞ for all j≥j1and the claim is proved. Now, from (39), Zt −∞ ku0(s)k2 Hds ≤eγt e−γtnjG(ϕ)−G(u(tnj)) ∀t∈(−∞, tnj), j ≥j1. Lemma 34 implies that, for any j≥j1and t∈(−∞, tnj), ku(t)−u(tnj)kH< beγ 2(t−tnj)qG(ϕ)−G(u(tnj))<b j. Therefore, ku(t)−ϕkH≤ ku(t)−u(tnj)kH+ku(tnj)−ϕkH<b j+C0 j. From this we have that lim t→−∞ku(t)−ϕkH= 0. The relatively compactness of γu(u0) in Vimplies that lim t→−∞ku(t)−ϕkV= 0, completing the proof. Proof of Theorem 27: As in the previous section, we will pay only attention to the case t→ −∞,being the case t→ ∞ analogous. Define the continuous functional E:V×H→Rby E(u0, u1) := 1 2ku1k2 H+G(u0). It is not difficult to check that Eis a Lyapunov functional for the semigroup {S(t) : t≥0}. From the relatively compactness of the orbit of u:R→Vwe have that the α-limit of u, α(u) = n(z, w)∈V×H:∃tn→ −∞,lim n→∞ ku(tn)−zkV= 0 and lim n→∞ ku0(tn)−wkH= 0o is nonempty, compact, invariant, and attracts the solution (u, u0) in V×Hwhen t→ −∞. Moreover, being S(·) a gradient semigroup, α(u) is contained in E∗={(z∗, w∗)∈V×H: S(t)(z∗, w∗)=(z∗, w∗) for all t≥0}. As far as equilibrium points of problem (23) are also strong solutions, then any (z∗, w∗)∈ E∗ must satisfy w∗= 0 and M(z∗) = 0, i.e., E∗⊂ EA×{0}.Moreover, the equality E∗=EA×{0} holds since the other inclusion E∗⊃ EA×{0}is always true. In particular, this implies that lim t→−∞ku0(t)kH= 0. Now we prove that there exists a unique ϕ∈ EAsuch that α(u) = {(ϕ, 0)},which will conclude the proof. Consider a positive value ε > 0,which will be fixed later on. Define for t≤0 Eε(t) := E(u(t), u0(t)) + ε(M(u(t)), u0(t))V0. 24 E. ARAGAO-COSTA, A. CARVALHO, P. MAR´ IN-RUBIO, AND G. PLANAS Since a strong solution satisfies (22) in a classic sense, we can derive and obtain E0 ε(t) = −ku0(t)k2 H+ε[(M0(u(t))u0(t), u0(t))V0+ (M(u(t)), u00(t))V0] ≤ −ku0(t)k2 H+εkM0(u(t))u0(t)kV0ku0(t)kV0−kM(u(t))k2 V0+kM(u(t))kV0ku0(t)kV0]. From the relatively compactness of the orbit of u, the hypothesis kM0(u(t))u0(t)kV0≤aku0(t)kV0 for all t∈Rand the continuity of the injection H ,→V0, one can estimate the right hand side above by −ku0(t)k2 H+εCku0(t)k2 H−kM(u(t))k2 V0+1 2kM(u(t))k2 V0+C 2ku0(t)k2 H =ε(C+C 2)−1ku0(t)k2 H−ε 2kM(u(t))k2 V0, for some constant C > 0 depending on u. Therefore, we conclude E0 ε(t)≤ε(C+C 2)−1ku0(t)k2 H−ε 2kM(u(t))k2 V0. So, fixing ε > 0 small enough, we obtain that for a constant C2>0 and all t≤0,it holds d dt [Eε(t)−G(ϕ)] ≤ −C2ku0(t)k2 H+kM(u(t))k2 V0.(40) In particular, Eε: (−∞,0] →Ris non-increasing. By the relatively compactness of the orbit of u, Eεis bounded from below, thus it exists the limit lim t→−∞Eε(t). Now, take any point (ϕ, 0) ∈α(u),so there exists a sequence (tn)n∈Nwith tn→ −∞ and lim n→∞ku(tn)−ϕkV= 0,and, from above, it also holds lim n→∞Eε(tn) = lim n→∞ 1 2ku0(tn)k2 H+G(u(tn)) + ε(M(u(tn)), u0(tn))V0=G(ϕ). Actually, we deduce that lim t→−∞Eε(t) = G(ϕ) with Eε(t)≤G(ϕ) for all t≤0. On the other hand, from (40) there exists some constant e C > 0 such that E0 ε(t)≤ −1 e Cku0(t)k2 H, whence integrating over (−∞, t),for t∈(−∞,0),it turns out Zt −∞ ku0(s)k2 Hds ≤e C[G(ϕ)−Eε(t)] .(41) Now we take a sequence (nj)j∈Nas in Theorem 26 with ku(tnj)−ϕkV<1 jand G(ϕ)−Eε(tnj)1 2<1 j. Analogously, take σ > 0 associated to the equilibrium point ϕaccording to the statement of Theorem 24; fix j0∈Nsuch that 1/j0< σ and define for j≥j0 tj= inf τ < tnj:ku(t)−ϕkV< σ for all t∈τ, tnj. GRADIENT-LIKE NONLINEAR SEMIGROUPS 25 Then, for each j≥j0and t∈tj, tnj,the Lojasiewicz-Simon inequality gives Eε(t)−G(ϕ) = 1 2ku0(t)k2 H+ε(M(u(t)), u0(t))V0+ [G(u(t)) −G(ϕ)] ≥1 2ku0(t)k2 H+ε(M(u(t)), u0(t))V0−ckM(u(t))k2 V0, which, jointly with Young inequality with some δ > 0,implies Eε(t)−G(ϕ)≥1 2ku0(t)k2 H−ckM(u(t))k2 V0−ε 2δkM(u(t))k2 V0−εδ 2ku0(t)k2 V0. Taking a value δ > 0 big enough, we obtain the existence of some constant C1>0 such that Eε(t)−G(ϕ)≥ −C1ku0(t)k2 H+kM(u(t))k2 V0,∀t∈tj, tnj. This, combined with (40), implies that for t∈tj, tnj d dt [Eε(t)−G(ϕ)] ≤γ[Eε(t)−G(ϕ)] , with γ:= C2/C1.Now we just have to argue in the same way as in the proof of Theorem 26, to obtain that for all t∈tj, tnj [G(ϕ)−Eε(t)] ≤G(ϕ)−Eε(tnj)eγ(t−tnj). This, combined with (41), implies that for t∈tj, tnj, it holds that Zt −∞ ku0(s)k2 Hds ≤e CG(ϕ)−Eε(tnj)eγ(t−tnj). Now we can apply Lemma 34 and conclude the proof similarly as done for Theorem 26. Acknowledgments. 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