Attractors for non-autonomous retarded lattice dynamical systems

©2015 Tomás Caraballo et al., licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Nonauton. Dyn. Syst. 2015; 2:31–51 Research Article Open Access Tomás Caraballo, Francisco Morillas, and José Valero* Attractors for non-autonomous retarded lattice dynamical systems DOI 10.1515/msds-2015-0003 Received January 22, 2015; accepted May 9, 2015 Abstract: In this paper we study a non-autonomous lattice dynamical system with delay. Under rather general growth and dissipative conditions on the nonlinear term, we define a non-autonomous dynamical system and prove the existence of a pullback attractor for such system as well. Both multivalued and single-valued cases are considered. Keywords: lattice dynamical systems, non-autonomous systems, differential equations with delay, set-valued dynamical systems, pullback attractor MSC: 34K05, 34K31, 35B40, 35B41, 35K55, 35K40, 37L30, 58C06 1Introduction Lattice dynamical systems often arise as an approximative system of infinite differential equations of a partial differential equation in an unbounded domain, although they also appear as models of a variety of phenomena such as image processing, pattern recognition, brain science, among others. In the last years many authors have been interested in the asymptotic behaviour of solutions of such systems. As a result, a sheer number of papers have been published concerning the existence and properties of global attractors in the autonomous, nonautonomous and stochastic cases; with or without uniqueness; in weighted or unweighted spaces. Usually, the models under consideration are obtained by a spatial discretization of a parabolic or a hyperbolic equation (see e.g. [1], [2], [4], [5], [8], [11] [12], [15], [16], [19], [20], [22], [23], [26], [28], [29]). The addition of a delay in the system, which appears naturally in real models, gives rise to new difficulties. Retarded autonomous lattice dynamical systems were studied from the point of view of dynamical systems in [25], [27], [24]. These results were improved later on by Caraballo et. al. [13]. Our main aim in this paper is to analyze the asymptotic behavior of the following nonautonomous retarded lattice differential equation        dui dt −(ui−1−2ui+ui+1)+λui+fi(t,uit)= 0,t>τ,i∈Z, ui(s)=ψi(s),∀s∈[τ−h,τ], (1) Tomás Caraballo: Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080- Sevilla, Spain, E-mail: [email protected] Francisco Morillas: Department d’Economia Aplicada, Facultat d’Economia, Universitat de Valéncia, Campus del Tarongers s/n, 46022-Valéncia, Spain, E-mail: [email protected] *Corresponding Author: José Valero: Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. de la Universidad, s/n, 03202-Elche, Spain, E-mail: [email protected] Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM 32 |Tomás Caraballo, Francisco Morillas, and José Valero where λ∈R. This model is obtained after a spatial discretization of the scalar retarded reaction-diffusion equation:        du dt −∂2u ∂x2+λu +f(t,ut) = 0,t>τ,x∈R, u(s)=ψ(s),∀s∈[τ−h,τ]. Here u= (ui)i∈Z∈`2,Zdenotes the integers set and for a continuous function u: [τ−h,T]→Y(where Y is some space), utdenotes the segment of the solution, i.e., the element in C[−h,0],Ydefined by ut(s)= u(t+s),s∈[−h,0]. The existence and uniqueness of solutions for problem (1) were addressed in [13]. It is worth pointing out that rather general assumptions on the nonlinear functions fi(just continuity and growth conditions) are imposed, not ensuring any kind of compactness properties in the space `2for the corresponding Nemytskii operator, which are necessary in order to apply the solvability results stated in other papers (see [14], [17], [21]). Also, in the autonomous case, when fdoes not depend explicitely on t, the existence of global attractors was established in both the multivalued and single-valued settings for a particular type of functions fi. In the present paper we extend the results carried out in [13] to the nonautonomous case. For this aim we apply the well-known theory of pullback attractors [7], [9]. The paper is organized in two parts. In Section 2 we recall briefly the general solvability theorems proved in [13] and apply them to problem (1) under rather general assumptions on the nonlinear term f. In Section 3 we consider the particular case of a lattice dynamical system with a nonlinear term of the form fi(t,uit) = F0,i(ui(t)) +F1,iuit−ρ(t)+ 0 Z −h bi(t,s,ui(t+s)) ds, with ρ(·)∈C1(R)and ρ(t)∈[0,h]for all t∈R.Under some dissipative and sublinear growth conditions on the maps F0,i,F1,i,bi, we define for this problem a multivalued process Uand prove the existence of a pullback attractor. Additionally, with extra Lipschitz conditions we obtain uniqueness of the Cauchy problem, so that Uis in fact a single-valued process. 2Existence of solutions of a lattice differential equation with delay 2.1 Some results on the existence of solutions of differential equations with delay in Banach spaces Let us first recall some abstract results which were proved in [13] and which will be useful in the present case. Let Ebe a real Banach space with dual E*,and let E0=C([−h,0],E), with norms k·k,k·k*and k·kE0, respectively, where kφkE0= maxt∈[−h,0] kφ(t)k. Also, BX(y0,r) = {y∈X:ky−y0kX≤r}, where X=Eor E0, and (·,·)will denote the pairing between Eand E*. Let us consider the following Cauchy problem for a functional differential equation in a Banach space:        du dt =F(t,ut), uτ=ψ∈E0, (2) where F:R×E0→E.Also, for any u∈C([τ−h,+∞),E), the function ut∈E0,t≥τ,is defined by ut(s)=u(t+s),s∈[−h,0]. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM Attractors for non-autonomous retarded lattice dynamical systems |33 Let Ewbe the space Eendowed with the weak topology. We consider the space E0,w=C([−h,0],Ew). Let un,u∈E0,w. We say that un→uin E0,wif un(sn)→u(s)in Ewfor all sn→s∈[−h,0]. We will say that the function Fis sequentially weakly continuous in bounded sets if tn→t,un→uin E0,w and kunkE0≤M,for all n, imply F(tn,un)→F(t,u)in Ew. On the other hand, we will say that the function Fis bounded if it maps bounded subsets of R×E0onto bounded subsets of E. Definition 1. The map u: [τ−h,T]→Eis called a solution of problem (2) if uτ=ψ,u(·)is continuous, once weakly continuously differentiable in [τ,T]and satisfies u(t)=u(τ)+ t Zτ f(s,us)ds,for all t∈[τ,T]. Remark 2. It follows from this definition that for any solution uof (2), the map t7→ ut∈E0is continuous. Remark 3. We note that if F:R×E0→Eis sequentially weakly continuous in bounded sets and the map t7→ ut∈E0is continuous, then t7→ F(t,ut)is weakly continuous, hence weakly measurable. If Eis separable, we obtain that t7→ F(t,ut)is strongly measurable. If we assume, moreover, that the map Fis bounded, then we have that F(·,u·)∈L1(τ,T;E). If F:R×E0→Eand t7→ ut∈E0are continuous, then the map t7→ F(t,ut)is continuous, hence strongly measurable. If we assume, moreover, that the map Fis bounded, then we have that F(·,u·)∈L1(τ,T;E). Then, we recall now some results ensuring the existence and uniqueness of solutions for problem (2), which were proved in [13]. Theorem 4. Assume that Eis reflexive and separable. Let f:R×E0→Ebe sequentially weakly continuous in bounded sets, and let Fbe a bounded map. Then, for each r>0, there exists a(r)>0such that if ψ∈E0and kψkE0≤r, problem (2) possesses at least one solution defined on [0,a(r)].Moreover, u(·)is a.e. differentiable and du dt =f(t,ut)for a.a. t∈0,a(r). If we assume additionally that f:R×E0→Eis continuous, then u∈C1[0,a]; Eand the separability of Eis not needed. Theorem 5. Assume the conditions of Theorem 4. If a solution u(·)of (2) has a maximal interval of existence [0,b)and there exists K>0such that ku(t)k≤K, for all t∈[0,b), then b= +∞, that is, u(·)is a globally defined solution. Let J:E→2E*be the duality map, i.e. J(y) = {ξ∈E*: (y,ξ) = kyk2=kξk2 *},∀y∈E. We state a result concerning uniqueness of solutions. Theorem 6. Assume the hypotheses of Theorem 4. Also, suppose that, for any M>0, there exists β(·,M)∈ L1 loc (R)such that β(t,M)≥0for a.a. t∈Rand the following inequality holds: (f(t,v)−f(t,w),j)≤β(t,M)kv−wk2 E0,(3) for all j∈J(v(0)−w(0)), all v,w∈E0with kvkE0,kukE0≤M, and a.a. t∈R. Then, for each r>0, there exists a(r)>0such that if ψ∈E0and kψkE0≤r, problem (2) has a unique solution defined on [0,a(r)]. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM 34 |Tomás Caraballo, Francisco Morillas, and José Valero 2.2 Lattice dynamical systems with delay: setting of the problem For a given τ∈R, consider the following first order lattice dynamical system with finite delay        dui dt −(ui−1−2ui+ui+1)+λui+fi(t,uit)= 0,t>τ,i∈Z, ui(s)=ψi(s−τ),∀s∈[τ−h,τ], (4) where λ∈R. We consider the separable Hilbert space `2={v=(vi)i∈Z:Pi∈Zv2 i<∞}with norm kvk=qPi∈Zv2 i and scalar product (w,v)=Pi∈Zwivi,and also the Banach space `∞={v=(vi)i∈Z: supi∈Z|vi|<∞}with norm kvk∞= supi∈Z|vi|. Further, we shall use the notation E=`2,E0=C[−h,0],`2,E1=C([−h,0],R), with the norms kukE0= maxs∈[−h,0] ku(s)k,kukE1= maxs∈[−h,0] |u(s)|. Also, put E∞=C[−h,0],`∞with norm kukE∞= maxs∈[−h,0] ku(s)k∞. We note that E0⊂E∞, as ku(t)−u(s)k∞= sup i∈Z |ui(t)−ui(s)|≤sX i∈Z |ui(t)−ui(s)|2=ku(t)−u(s)k,∀t,s∈[−h,0], and kukE∞= max s∈[−h,0] sup i∈Z |ui|≤max s∈[−h,0] sX i∈Z |ui|2=kukE0. We consider the following conditions: (H1)The operator f:R×E0→Egiven by the rule (f(t,v))i=fi(t,vi),i∈Z, is well defined and bounded. (H2)The maps fi:R×C([−h,0],R)→Rare continuous. We shall first prove the existence of solutions for problem (4). For this aim we shall rewrite it in an abstract form. We define the operator A:E→Eby (Av)i:= −vi−1+ 2vi−vi+1,i∈Z. Also, we define the operators B,B*:E→Eby (Bv)i:= vi+1 −vi,B*vi:= vi−1−vi. It is easy to check that A=B*B=BB*, B*w,v=(w,Bv). Then the operator F:R×E0→Eis defined by F(t,v)=−Av (0)−f(t,v)−λv (0) and (4) can be rewritten as        du dt =F(t,ut),t>τ, uτ=ψ,i.e. u(s)=ψ(s−τ),∀s∈[τ−h,τ]. (5) Lemma 7. Let (H1)-(H2) hold. Then the map f:R×E0→Eis sequentially weakly continuous in bounded sets. Also, the map A:E→Eis weakly continuous. Proof. Let tn→tin R, and vn→v∈E0,w,with vnE0≤M1for all n, and let w∈`2be arbitrary. For any ε>0we take K0(ε)>0such that P|i|≥K0|wi|2<ε. Since fis bounded, there exists M2>0such that Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM Attractors for non-autonomous retarded lattice dynamical systems |35 ftn,vn≤M2,kf(t,v)k≤M2,for all n. Also, as tn→tand vn i→viin C([−h,0],R), for all i, (H2) implies the existence of N(K0,ε)such that P|i|<K0fitn,vn i−fi(t,vi)2<ε2if n≥N.Hence, ftn,vn−f(t,v),w≤sX |i|<K0fitn,vn i−fi(t,vi)2kwk+kf(t,v)k+ftn,vnsX |i|≥K0 |wi|2 ≤εkwk+ 2M2ε. The result for the operator Acan be proved similarly. This completes the proof. Theorem 8. Let (H1)-(H2) hold. For each r>0there exists a(r)>0such that if ψ∈E0and kψkE0≤r, then problem (4) has at least one solution defined on [τ,τ+a(r)].Moreover, u(·)is a.e. differentiable and du dt =F(t,ut)for a.a. t∈τ,τ+a(r). Proof. Lemma 7 implies that the operator Fis sequentially weakly continuous in bounded sets. Since fis bounded, Fis also bounded. The result follows from Theorem 4. In order to obtain that the map fis continuous, we need an assumption which is stronger than (H1). (H3)The operator f:R×E0→Egiven by (f(t,v))i=fi(t,vi),i∈Z, is well defined, and for any (t,v)∈R×E0, we have X |i|≥K |fi(t,vi)|2≤CkvkE0 max s∈[−h,0] X |i|≥K v2 i(s)+bK(t) ,for all K∈Z+, where bK(t)→0+as K→∞uniformly in compact sets, and C(·)≥0is a continuous non-decreasing function. Remark 9. Condition (H3) implies that the map fis bounded. Lemma 10. Let (H2)-(H3) hold. Then, the map f:R×E0→Eis continuous. Proof. Let tn→tin R, and vn→vin E0. Then for any ε>0there exists K(ε)such that max s∈[−h,0] X |i|≥Kvn i(s)2<ε,max s∈[−h,0] X |i|≥K |vi(s)|2<ε. Then by (H3) one can choose K1(ε)≥K(ε)such that X |i|≥K1fitn,vn i2≤Rε,X |i|≥K1 |fi(t,vi)|2≤Rε, for some R>0.On the other hand, by (H2) we obtain the existence of N(ε,K)such that X |i|<K1fitn,vn i−fi(t,vi)2<εif n≥N. Thus, X i∈Zfitn,vn i−fi(t,vi)2≤X |i|<K1fitn,vn i−fi(t,vi)2+ 2 X |i|≥K1fitn,vn i2+ 2 X |i|≥K1 |fi(t,vi)|2 ≤ε+ 2Rε, if n≥N. Corollary 11. Underconditions(H2)-(H3),thesolutiongiveninTheorem 8 belongs to the space C1[τ,τ+a]; E. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM 36 |Tomás Caraballo, Francisco Morillas, and José Valero In order to obtain the uniqueness of solutions we need an additional Lipschitz assumption. (H4)For any M>0there exists β(t,M)≥0such that β(·,M)∈L1(R)and f(t,z)−f(t,v),z(0) −v(0)≥ −β(t,M)kz−vk2 E0, if kzkE0,kvkE0≤M,t∈R. Theorem 12. Assume (H1)-(H2) and (H4). Then the solution given in Theorem 8 is unique. Proof. Let z,v∈E0,kzkE0,kvkE0≤M,and w=z−v. It follows from (H4) and Aw(0),w(0)=Bw(0),Bw(0)≥ 0that F(t,z)−F(t,v),z(0) −v(0)=−Aw(0),w(0)−λkwkE0−f(t,z)−f(t,v),w(0) ≤β(t,M)kwkE0. Then the result follows from Theorem 6. We now aim to study the asymptotic behaviour of solutions for problem (4). In particular, we will show the existence of a non-autonomous attractor. When conditions (H1)-(H2), (H4) hold, if we assume that every solution is global (this is true if we obtain an estimate of the solutions by Theorem 5), then we can define the map U:Rd×E0→E0,R2 d={(t,τ)∈R2:t≥τ}by U(t,τ,ψ)=ut, where u(·)is the unique solution to (4) with uτ=ψ.Moreover, it is easy to prove, using (3) and Gronwall’s lemma, that the map ψ7→ U(t,τ,ψ)is continuous for any τ≤t.The map Uis a process, that is, U(τ,τ,ψ) = ψ and U(t,τ,ψ) = U(t,r,U(r,τ,ψ)) for all τ≤r≤tand ψ∈E0.(6) On the other hand, if we assume only (H1)-(H2) and that every solution is global, then we can define a multivalued semiflow by U:R2 d×E0→P(E0)(P(E0)is the set of all non-empty subsets of E0) by U(t,τ,ψ)={ut:u(·)is a solution of (4) with uτ=ψ}.(7) Since we do not have uniqueness of the Cauchy problem, this map is in general multivalued. In a similar way to the autonomous case [19, Lemma 13] one can prove that it is a multivalued process, that is: 1. U(τ,τ,·)=Id (the identity map); 2. U(t,τ,ψ)⊂U(t,r,U(r,τ,ψ)) for all ψ∈E0,τ≤r≤t. Moreover, it is strict, that is, U(t,τ,ψ)=U(t,r,U(r,τ,ψ)) for all ψ∈E0,τ≤r≤t. Now, we will recall the main results from the theory of pullback attractors. First, let us consider the case of a single-valued process [9], [10] (see also [18]). Let Xbe a complete metric space. Suppose that Dis a nonempty class of parameterized sets b D={D(t); t∈ R} ⊂ P(X),where P(X)denotes the family of all nonempty subsets of X. Definition 13. The process Uis said to be pullback D-asymptotically compact if for any t∈R, any b D∈D, any sequence τn→−∞,and any sequence yn∈U(t,τn,D(τn))}is relatively compact in X. Definition 14. It is said that b B∈Dis pullback D-absorbing for the process Uif for any t∈Rand any b D∈D, there exists a τ0(t,b D)≤tsuch that U(t,τ,D(τ)) ⊂B(t)for all τ≤τ0(t,b D). Definition 15. The family b A={A(t); t∈R} ⊂ P(X)is said to be a pullback D-attractor for U(·,·)if: Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM Attractors for non-autonomous retarded lattice dynamical systems |37 1. A(t)is compact for all t∈R, 2. b Ais pullback D-attracting, i.e., lim τ→−∞ dist(U(t,τ,D(τ)),A(t)) = 0, for all b D∈D, and all t∈R, 3. b Ais invariant, i.e., U(t,τ,A(τ)) = A(t),for − ∞ <τ≤t<+∞. We have the following result. Theorem 16. Suppose that the map ψ7→ U(t,τ,ψ)is continuous for any τ≤tand that the process Uis pullback D-asymptotically compact. Let b B∈Dbe a family of pullback D-absorbing sets for U(·,·). Then, the family b A={A(t); t∈R} ⊂ P(X)defined by A(t) = Λ(b B,t),t∈R,where Λ(b D,t) = \ s≤t [ τ≤s U(t,τ,D(τ))!,for each b D∈D, is a pullback D-attractor for U(·,·)which satisfies in addition that A(t) = [ b D∈D Λ(b D,t),for t∈R. Furthemore, b Ais minimal in the sense that if b C={C(t); t∈R} ⊂ P(X)is a family of closed sets such that limτ→−∞ dist(U(t,τ,B(τ)),C(t)) = 0,then A(t)⊂C(t). The family Dis said to be inclusion-closed if b D∈Dand ∅=B(t)⊂D(t),for all t∈R, implies b B∈D.If the family is inclusion-closed and the absorbing set b B∈Dsatisfies that the sets B(t)are closed, then A(t)⊂B(t) implies that the attractor b Abelongs to D. Let us consider now the case of a multivalued process. The following result is proved in [7] (see also [6] for a more general non-autonomous and random framework). The definitions of pullback D-asymptotically compactness, pullback D-absorbing family and pullback D-attraction are the same as in the single-valued case. For fixed τ≤tthe mapping U(t,τ,·)is said to be upper-semicontinuous if for any x0∈Xand for every neighborhood Nin Xof the set U(t,τ,x0),there exists δ>0such that U(t,τ,y)⊂Nwhenever dX(x0,y)<δ. Definition 17. A family b A=A(t) : t∈R⊂P(X)is said to be a global pullback D-attractor for the MNDS U if A(t)is compact for any t∈R,b Ais pullback D-attracting, and b Ais negatively invariant, i.e., A(t)⊂U(t,τ,A(τ)), for any (t,τ)∈R2 d. b Ais said to be a strict global pullback D-attractor if the invariance property in the third item is strict, i.e., A(t) = U(t,τ,A(τ)), for (t,τ)∈R2 d. Theorem 18. Assume that the map ψ7→ U(t,τ,ψ)is upper-semicontinuous and possesses closed values. Let b B=B(t) : t∈R∈Dbe pullback D-absorbing and such that Uis asymptotically compact with respect to b B. Then, the set b Agiven by A(t) := Λb B,t=\ s≤t[ τ≤s U(t,τ,B(τ)) t∈R,(8) is a pullback D-attractor for the MNDS U. Moreover, suppose that Dis inclusion-closed and that B(t)is closed in Xfor any t∈R. Then the family b A defined by (8) belongs to D, and is the unique pullback D-attractor with this property. In addition, in this case, if Uis a strict MNDS, then b Ais strictly invariant. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM 38 |Tomás Caraballo, Francisco Morillas, and José Valero 3A lattice system with sublinear non-autonomous retarded terms We shall consider a function f:R×E0→Egiven by the rule (f(t,v))i=fi(t,vi)and fi(t,vi)=F0,i(t,vi(0)) +F1,it,vi−ρ(t)+ 0 Z −h bi(t,s,vi(s)) ds, where ρ(·)∈C1(R)and ρ(t)∈[0,h]for all t∈R, that is, putting v=ut=u(t+·), problem (4) can be rewritten as        dui dt −(ui−1−2ui+ui+1)+λui+F0,i(t,ui(t)) +F1,it,uit−ρ(t) +R0 −hbi(t,s,ui(t+s)) ds = 0,t>τ,i∈Z, ui(s)=ψi(s−τ),∀s∈[τ−h,τ]. (9) We consider the following conditions: (C1) λ>0. (C2) F0,i:R2→Rare continuous and satisfy that F0,i(x)x≥ −C0,i(t),C0∈C(R;`1)and t Z −∞ kC0(s)k`1eδsds <∞,for all t∈Rand δ>0. (C3) F0,i(t,x)≤H(|x|)|x|+C1,i(t),for all x∈R, where C1∈C(R;`2), and H(·)≥0is a continuous and non-decreasing function. (C4) F1,i:R2→Rare continuous and satisfy that F1,i(t,x)≤K1|x|+C2,i(t), for all x∈R,where C2∈ C(R;`2),K1>0and t Z −∞ kC2(s)k2 `2eδsds <∞,for all t∈Rand δ>0. (C5) |bi(t,s,x)|≤m0,i(t,s)+m1,i(s)|x|,for all x∈Rand a.a. s∈(−h,0), where biare Caratheodory in the sense that it is measurable in sand continuous in (t,x). Also, m0,i(t,·),m1,i(·)∈L1(−h,0),m0,i(t,s),m1,i(s)≥0and defining M0,i(t) = R0 −hm0,i(t,s)ds and M1,i=R0 −hm1,i(s)ds we assume that M1:= qPi∈ZM2 1,i<∞,M0(t) := qPi∈ZM2 0,i(t)<∞, M0∈C(R;R+)and t Z −∞ (M0(s))2eδsds <∞,for all t∈Rand δ>0. (C6) ρ∈C1R,[0,h]and ρ′(t)≤ρ*<1. Let us check conditions (H1)-(H3). First, in order to obtain (H1) we prove that fis well defined and bounded. We note that |fi(t,vi)|≤F0,i(t,vi(0))+F1,it,vi−ρ(t)+ 0 Z −h |bi(t,s,vi(s))|ds.(10) For the first term we have by (C3) that F0,i(t,vi(0))2≤2H2(|vi(0)|)|vi(0)|2+C2 1,i(t)(11) ≤2χ(kvkE0)|vi(0)|2+ 2C2 1,i(t), where χ(kvkE0) = maxi∈ZH2(|vi(0)|), which exists because H(·)is non-decreasing and v∈E0.Then, X i∈ZF0,i(t,vi(0))2≤2χ(kvkE0)kvk2 E0+ 2 kC1(t)k2. (12) Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM Attractors for non-autonomous retarded lattice dynamical systems |39 As for the second term we obtain thanks to (C4) that X i∈ZF1,it,vi−ρ(t)2≤2K2 1X i∈Zvi−ρ(t)2+ 2 kC2(t)k2 ≤2K2 1kvk2 E0+ 2 kC2(t)k2. (13) Now, for the term with the integral delay, taking into account (C5), we proceed as follows: 0 Z −h |bi(t,s,vi(s))|ds ≤ 0 Z −hm0,i(s)+m1,i(s)|vi(s)|ds ≤M0,i(t) + kvkE∞M1,i. Then X i∈Z  0 Z −h |bi(t,s,vi(s))|ds  2 ≤2X i∈Z M2 0,i+ 2 kvk2 E∞X i∈Z M2 1,i ≤2M2 0(t) + 2 kvk2 E0M2 1. (14) Using (12)-(14) in (10) we obtain that fis well defined and bounded. Now, we check (H2), i.e., that the maps fi:R×C([−h,0],R)→Rare continuous. We consider tn∈ R,vnn∈N⊂C([−h,0],R)and t0∈R,v0∈C([−h,0],R)such that tn→t0,vn→v0in C([−h,0],R). Now, we have fitn,vn−fit0,v0≤F0,itn,vn(0)−F0,it0,v0(0) +F1,itn,vn−ρ(tn)−F1,it0,v0−ρ(t0) + 0 Z −h bitn,s,vn(s)ds − 0 Z −h bit0,s,v0(s)ds . From (C2) and (C4), F0,iand F1,iare continuous functions. Also, from (C5) and Lebesgue’s theorem, the last term converges to 0. Thus, the continuity of fifollows. To check (H3) we observe that X |i|≥K  0 Z −h |bi(t,s,vi(s))|ds  2 ≤2X |i|≥K  0 Z −h m0,i(s)ds  2 + 2 X |i|≥K  0 Z −h m1,i(s)|vi(s)|ds  2 ≤2X |i|≥K M2 0,i(t) + 2 kvk2 E0X |i|≥K M2 1,i. Also, by (10), (11) and (C4) we have X |i|≥K |fi(t,vi)|2≤R χ(kvkE0)X |i|≥K |vi(0)|2+X |i|≥K C2 1,i(t) + K2 1X |i|≥Kvi−ρ(t)2 +X |i|≥K C2 2,i(t)+X |i|≥K M2 0,i(t) + kvk2 E0X |i|≥K M2 1,i  ≤CkvkE0 max s∈[−h,0] X |i|≥K v2 i(s)+bK(t) , where bK→0+as K→∞uniformly in compact sets, and C(·)≥0is a continuous non-decreasing function. Thus, (H3) holds. Then Theorem 8 and Corollary 11 imply that for any ψ∈E0there exists, at least, one solution u(·)∈ C1[τ,α),Ein a maximal interval [τ,α). In order to obtain that every solution is globally defined we need to prove some estimates. This will be done in the next section. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM 46 |Tomás Caraballo, Francisco Morillas, and José Valero Taking ϵ=λand using condition (16), we have eηt X i∈Z ρK,i|ui(t)|2≤eητ X i∈Z ρK,i|ui(τ)|2+2 ηeηt −eητC(t2) K + t Zτ eηs    2ρ1 2 KC2(s) 2 λ+Pi∈ZρK,iM2 0,i(s) ˆ ϵ+ 2 kρKC0(s)k`1  ds +2K2 1 λ eηh η1−ρ*ρ1 2 Kψ 2 E0eητ −eη(τ−h)+ 2M1 t Zτ eηs ρ1 2 Kus 2 E0 ds.(40) Let θ∈[−h,0]. We replace tby t+θin (40), and use that ρ1 2 Ku(t+θ)=ρ1 2 Kψ(t+θ)≤ρ1 2 KψE0 if t+θ<τ; multiplying by e−η(t+θ)we obtain X i∈Z ρK,i|ui(t+θ)|2≤eητe−η(t+θ)X i∈Z ρK,i|ui(τ)|2 +e−η(t+θ) t+θ Zτ eηs    2ρ1 2 KC2(s) 2 λ+ρ1 2 KM0(s) 2 ˆ ϵ+ 2 kρKC0(s)k`1  ds +2 ηeη(t+θ)−eητC(t2) Ke−η(t+θ) +2K2 1 λ eηhe−η(t+θ) η1−ρ*ρ1 2 Kψ 2 E0eητ −eη(τ−h)+ 2M1e−η(t+θ) t+θ Zτ eηs ρ1 2 Kus 2 E0 ds, and eηt ρ1 2 Kut 2 E0 ≤ eηh +2K2 1e2ηh λη 1−ρ*!ρ1 2 Kψ 2 E0 eητ +eηh t Zτ eηs    2ρ1 2 KC2(s) 2 λ+Pi∈ZρK,iM2 0,i(s) ˆ ϵ+ 2 kρKC0(s)k`1  ds +2 η C(t2) Keηt + 2M1eηh t Zτ eηs ρ1 2 Kus 2 E0 ds. We can rewrite this expression as eηt ρ1 2 Kut 2 E0 ≤2 η C(t2) Keηt +ˆ Cρ1 2 Kψ 2 E0 eητ + t Zτ eηsβρK(s)ds +L t Zτ eηs ρ1 2 Kus 2 E0 ds, (41) where we have used the notation ˆ C:= eηh +2K2 1 λη 1−ρ*e2ηh, βρK(t):= eηh    2ρ1 2 KC2(t) 2 λ+Pi∈ZρK,iM2 0,i(s) ˆ ϵ+ 2 kρKC0(t)k`1  , L:= 2M1eηh. Now, proceeding in a similar way to (32) and using η−L>0(see Remark 20) we obtain ρ1 2 Kut 2 E0 ≤2e(L−η)te(η−L)τˆ Cρ1 2 Kψ 2 E0 +2C(t2) K(η−L)+e−ηt t Zτ eηsβρK(s)ds +e(L−η)t t Zτ e(η−L)sβρK(s)ds. (42) Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM Attractors for non-autonomous retarded lattice dynamical systems |47 Now, it is convenient to keep in mind the definition of βρk, and its dependence on C0,M0and C2. It follows that βρK(s)→0as K→∞for any s. Hence, Lebesgue’s Dominated Convergence Theorem implies that t Z −∞ eδsβρK(s)ds →0,as K→∞for any t∈[t1,t2],δ>0. Thus, there exist Tϵ,t1,t2,b Bη≤t1,Kϵ,t1,t2,b Bη≥1such that max s∈[−h,0] sX |i|≥2K (ui(t+s))2≤max s∈[−h,0] sX i∈Z ρK,i(ui(t+s))2 =ρ1 2 KutE0 ≤ϵ,if τ≤T,t∈[t1,t2]. 3.3 Existence of the pullback attractor: general case We know that under the assumptions of Proposition 19, the map Ugiven by (7) is a strict multivalued process. For any initial data ψ∈E0we denote Dτ(ψ)=u(·)is a global solution of (9) with initial data uτ=ψ. We will prove that the map ψ7→ U(t,τ,ψ)is upper-semicontinuous and has closed values, and also that Uis asymptotically compact with respect to the pullback Dη-absorbing family b Bηdefined in Corollary 22. First, we obtain an auxiliary lemma. Lemma 26. We assume the conditions of Lemma 23. Let ψn→ψin E0. Then: 1. For arbitrary ϵ>0,τ≤Tthere exists K(ϵ,τ,T)such that for any un(·)∈Dτψn, max s∈[−h,0] sX |i|≥2Kun i(t+s)2≤ϵ,∀t∈[τ,T].(43) 2. Let un(·)∈Dτψn. Then there exists u(·)∈Dτ(ψ)and a subsequence unkof unsuch that unk→uin C([τ,T],E)for all T>τ.(44) Proof. It follows from ψn→ψin E0the existence of K1(ϵ)>0such that X i∈Z ρK,iψn i(s)2<ϵ,∀n,s∈[−h,0] X i∈Z ρK,iψ0 i(s) 2<ϵ,∀s∈[−h,0], if K≥K1.Now, from (42) we obtain the existence of K(ϵ,τ,T)≥K1such that ρ1 2 Kun t 2 E0 ≤2e(L−η)te(η−L)τˆ Cρ1 2 Kψn 2 E0 +2C(T) K(η−L)+e−ηt t Zτ eηsβρK(s)ds +e(L−η)t t Zτ e(η−L)sβρK(s)ds ≤ϵ, Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM 48 |Tomás Caraballo, Francisco Morillas, and José Valero for all t∈[τ,T], where we have used βρK(s)→0as K→∞for any sand the Lebesgue Dominated Convergence Theorem. Therefore, max s∈[−h,0] sX |i|≥2Kun i(t+s)2≤max s∈[−h,0] sX i∈Z ρK,iun i(t+s)2 =ρ1 2 Kun tE0 ≤ϵ, which proves (43). Next, from Proposition 19 we have that un tis bounded in E0. Then, using (43) one can prove in a standard way (see [13, p.71] for the details) that un(t)is precompact in Efor any t∈[τ,T]. After that, following the same lines as in [13, p.71], we can obtain the existence of u(·)∈Dτ(ψ)and a subsequence such that un(·)→u(·)in C([τ,T],E)for all T>τ. As a direct consequence we have the following result. The proof is rather similar to that in [13, p.72]. Corollary 27. Assume the conditions of Lemma 23. Then, the multivalued map ψ7→ G(t,τ,ψ)possesses closed graph and is upper semicontinous. Moreover, it has compact values. Lemma 28. AssumetheconditionsofLemma23. Then,themultivalued process UispullbackDη-asymptotically compact. In particular, it is pullback asymptotically compact with respect to the pullback Dη-absorbing family b Bη. Proof. We consider ξn=un t∈U(t,τn,ψn), where un(·)∈Dτnψn,ψn∈D(τn), and b D={D(t)} ∈ Dη. In view of Corollary 22, for nlarge enough we have un t∈Bη(t). Hence, un t(s)≤C,∀s∈[−h,0], for some C>0. For fixed s∈[−h,0]we can find a subsequence (denoted again as un) such that un(t+s)→ωsin Ew. Using a similar argument as in [13, p.71] (with the help of Lemma 25) we obtain that un(tn+s)→ωsin E. Therefore, un t(s)is a precompact sequence for any s∈[−h,0]. In order to apply the Ascoli-Arzelà theorem, we need to obtain the equicontinuity property. Using Proposition 19, the boundedness of the sequence ψn2 E0e(η−L)τn, the fact that the operator Fis bounded and the integral representation of solution we can obtain that un(t+s2)−un(t+s1)≤ t+s2 Z t+s1Fr,un rdr ≤K(s2−s1),if −h≤s1<s2≤0. Then, the Ascoli-Arzelà theorem implies that ξnis relatively compact in E0.Since by Lemma 23 we have that b Bη∈Dη,Uis pullback asymptotically compact with respect to this family as well. The existence of the pullback attractor follows now from Proposition 19, Lemma 28, Corollaries 22, 27 and Theorem 18. Theorem 29. Assume the conditions of Lemma 23. Then, the multivalued process Upossesses a unique pullback Dη-attractor b A, which belongs to Dη. Moreover, it is strictly invariant. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM Attractors for non-autonomous retarded lattice dynamical systems |49 3.4 Existence of the pullback attractor: case of uniqueness We can prove uniqueness of the solution of the Cauchy problem (9) if we assume the following extra assumption: (C6) For any x,y∈Rand s∈[−h,0] we have F0,i(t,x)−F0,i(t,y)≤k0(t)C3(|x|,|y|)|x−y|, F1,i(t,x)−F1,i(t,y)≤k1(t)C4(|x|,|y|)|x−y|, |bi(t,s,x)−bi(t,s,y)|≤k2(t)k3(s)C5(|x|,|y|)|x−y|, where Cj(·,·)≥0are continuous and non-decreasing functions in both variables and ki(·)∈L2 loc (R) for i= 0,1,2,k3(·)∈L2(−h,0). Lemma 30. If (C6) holds, the map f:R×E0→Esatisfies the local Lipschitz assumption (H4). Proof. Let v,w∈E0be such that kvkE0,kwkE0≤M. On the one hand, we have that X i∈ZF0,i(t,vi(0)) −F0,i(t,wi(0))2≤k2 0(t)max i∈Z(C3(|vi(0)|,|wi(0)|))2X i∈Z |vi(0)−wi(0)|2 ≤k2 0(t)χ2 3kvkE0,kwkE0kv−wk2 E0, X i∈ZF1,i(vi(−h1)) −F1,i(wi(−h1))2≤k2 1(t)max i∈Z(C4(|vi(−h1)|,|wi(−h1)|))2X i∈Z |vi(−h1)−wi(−h1)|2 ≤k2 1(t)χ2 4kvkE0,kwkE0kv−wk2 E0, where χjkvkE0,kwkE0= maxi∈Z,s∈[−h,0](Cj(|vi(s)|,|wi(s)|)). On the other hand, X i∈Z  0 Z −h |bi(t,s,vi(s)) −bi(t,s,wi(s))|ds  2 ≤k2 2(t)max i∈Z,s∈[−h,0](C5(|vi(s)|,|wi(s)|))2X i∈Z  0 Z −h k3(s)|vi(s)−wi(s)|ds  2 ≤k2 2(t)χ2 5kvkE0,kwkE0X i∈Z 0 Z −h k2 3(s)ds 0 Z −h |vi(s)−wi(s)|2ds =k2 2(t)χ2 5kvkE0,kwkE00 Z −h k2 3(s)ds 0 Z −hX i∈Z |vi(s)−wi(s)|2ds ≤k2 2(t)χ2 5kvkE0,kwkE0h 0 Z −h k2 3(s)ds kv−wk2 E0, where χ5kvkE0,kwkE0= maxi∈Z,s∈[−h,0](C5(|vi(s)|,|wi(s)|)). The fact that the sum and the integral can be exchanged follows easily using Lebesgue’s theorem. Thus, there exist K(M),β(·)∈L1 loc (R)such that kf(t,v)−f(t,w)k2≤β(t)K(M)kv−wk2 E0, which proves the result. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 9/29/15 11:23 AM 50 |Tomás Caraballo, Francisco Morillas, and José Valero Then, if we assume conditions (C1)-(C6) and (15)-(16), Theorems 5, 8, 12, Corollary 11 and Proposition 19 imply that for any ψ∈E0there exists a unique global solution u(·)∈C1[τ,∞),Ewith u(τ)=ψ. Hence, as shown in Section 2.2, we can define the process Uby putting U(t,τ,ψ)=ut,where u(·)is the unique solution to (9) with ψ=u0.Moreover, this map is continuous with respect to the initial data ψ. We obtain now the existence of a pullback attractor. Theorem 31. Assume conditions (C1)-(C6) and (15)-(16), (34). Then, the process Upossesses a pullback Dηattractor b A, which belongs to Dη. Proof. Proposition 19, Lemma 28, Corollary 22 and Theorem 16 imply the existence of the pullback Dηattractor b A. Since the sets Bη(t)of the absorbing family are closed and Bη∈Dη, we obtain that b A∈Dη. 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