Asymptotic behavior of coupled inclusions with variable exponents

This work concerns the study of asymptotic behavior of the solutions of a nonautonomous coupled inclusion system with variable exponents. We prove the existence of a pullback attractor and that the system of inclusions is asymptotically autonomous.


1.
Introduction. Nonlinear reaction-diffusion equations have been studied extensively in recent years and a special attention has been given to coupled reactiondiffusion equations from various fields of applied sciences arising from epidemics, biochemistry and engineering [18]. Reaction-diffusion systems are naturally applied in chemistry where the most common is the change in space and time of the concentration of one or more chemical substances. One interest in chemical kinetics is the construction of mathematical models that can describe the characteristics of a chemical reaction. Mathematical models for electrorheological fluids were considered in [19,20,21] and variable exponents do appear in the diffusion term (see also [7,9]). Reaction-diffusion systems can be perturbed by discontinuous nonlinear terms, which leads to study differential inclusions rather than differential equations, for example, evolution differential inclusion systems with positively sublinear upper semicontinuous multivalued reaction terms F and G (see [6]). This work concerns the coupled system of inclusions: − div(D 1 (t, ·)|∇u 1 | p(·)−2 ∇u 1 ) + |u 1 | p(·)−2 u 1 ∈ F (u 1 , u 2 ) t > τ ∂u 2 ∂t − div(D 2 (t, ·)|∇u 2 | q(·)−2 ∇u 2 ) + |u 2 | q(·)−2 u 2 ∈ G(u 1 , u 2 ) t > τ ∂u 1 ∂n (t, x) = ∂u 2 ∂n (t, x) = 0 in ∂Ω, t ≥ τ In this work we extend the results in [15] for a single inclusion to the case of a coupled inclusion system. We will prove that the strict generalized process (see Definition 2.7 in Section 2) defined by (S) possesses a pullback attractor. Moreover, we prove that the system (S) is in fact asymptotically autonomous. It makes use of a collection of ideas and results of some recent, distinct previous works [15,22,23,27] of the authors, which are applied here to a new problem to yield interesting new results. Regarding [13,14,15] where an equation and a single inclusion of this type of problems were considered, the coupled system can not be treated in the same way as the single case, the principal additional technical difficulty is to adjust the results considering two inclusions, in this sense, the main technical difficulty appears to prove dissipativity.
The paper is organized as follows. First, in Section 2 we provide some definitions and results on existence of global solutions and generalized processes. In Section 3 we prove the existence of the pullback attractor for the system (S). In Section 4 we say some words about forward attraction and in the last section we prove that the system (S) is asymptotically autonomous.
2. Preliminaries, existence of global solutions and generalized processes. Consider now the system (S) in the following abstract form where F and G are bounded, upper semicontinuous and positively sublinear multivalued maps (see Definitions 2.4, 2.3 and 2.5, respectively) and, for each t > τ, A(t) and B(t) are univalued maximal monotone operators of subdifferential type in a real separable Hilbert space H. Specifically, A(t) = ∂ϕ t and B(t) = ∂ψ t for nonnegative mappings ϕ t , ψ t with ∂ϕ t (0) = ∂ψ t (0) = 0, ∀ t ∈ R and the mappings ϕ t , ψ t satisfy: There is a set Z ⊂ (τ, T ] of zero measure such that φ t is a lower semicontinuous proper convex function from H into (−∞, ∞] with a nonempty effective domain 2) For any positive integer r there exist a constant K r > 0, an absolutely continuous function g r : [τ, T ] → R with g r ∈ L β (τ, T ) and a function of bounded variation where α is some fixed constant with 0 ≤ α ≤ 1 and Let us first review some concepts and results from the literature, which will be useful in the sequel. We refer the reader to [2,3,29] for more details about multivalued analysis theory.
2.1. Setvalued mappings. Let X be a real Banach space and M a Lebesgue measurable subset in R q , q ≥ 1. If G is a univalued map, the above definition is equivalent to the usual definition of a measurable function.
Definition 2.2. By a selection of E : M → P (X) we mean a function f : M → X such that f (y) ∈ E(y) a.e. y ∈ M , and we denote by SelE the set In order to obtain global solutions we impose the following suitable conditions on terms F and G.
hold for each f ∈ F (u, v) and each g ∈ G(u, v).

Generalized processes.
In order to study the asymptotic behavior of the solutions of the system (S) we will work with a multivalued process defined by a generalized process. We will review these concepts which had been considered in [22,23] and can be used in the study of infinite dimensional dynamical systems.
is called an exact (or strict) generalized process.
A family of sets K = {K(t) ⊂ X : t ∈ R} will be called a nonautonomous set. The family K is closed (compact, bounded) if K(t) is closed (compact, bounded) for all t ∈ R. The ω−limit set ω(t, E) consists of the pullback limits of all converging sequences {ξ n } n∈N where ξ n ∈ U G (t, s n )E, s n → −∞. Let A = {A(t)} t∈R be a family of subsets of X. We have the following concepts of invariance: pullback attracts all bounded subsets at τ , for each τ ∈ R. In this case, we say that the nonautonomous set A is pullback attracting.
A(t) pullback absorbs bounded sets at time t.

Strong solutions.
Consider the following initial value problem: (ii) u is absolutely continuous on any compact subset of (τ, T ); , v(t)) a.e. in (τ, T ), and such that (u, v) is a strong solution (see Definition 2.10) over (τ, T ) to the system (P 1 ) below: Let D(u(τ ), v(τ )) be the set of the solutions of (S2) with initial data (u τ , v τ ) and define G(τ ) : Theorem 2.13 ([27]). Under the conditions of Theorem 2.12, G is an exact generalized process.
Let Ω ⊂ R n , n ≥ 1, be a bounded smooth domain and write H := L 2 (Ω) and Y := W 1,p(·) (Ω) with p − > 2. Then Y ⊂ H ⊂ Y * with continuous and dense embeddings. We refer the reader to [7,8] and references therein to see properties of the Lebesgue and Sobolev spaces with variable exponents. In particular, with for u ∈ L p(·) (Ω) and p ∈ L ∞ + (Ω). Consider the operator A(t) defined in Y such that for each u ∈ Y is associated the following element of Y * , A(t)u : Y → R given by The authors proved in [13] that: is maximal monotone and A(t)(Y ) = Y * . • The realization of the operator A(t) in H = L 2 (Ω), i.e., • The operator A H (t) is the subdifferential ∂ϕ t p(·) of the convex, proper and lower semicontinuous map ϕ t p(·) : L 2 (Ω) → R ∪ {+∞} given by Using the following elementary assertion we can obtain estimates on the operator considering only two cases.
Lemma 3.2. Let (u 1 , u 2 ) be a solution of problem (S). Then there exist positive constants r 1 and T 1 > T 0 , which do not depend on the initial data, such that Proof. Take T 1 > T 0 . Since (u 1 , u 2 ) is a solution of (S) there exists a pair (f, g) Consider ϕ t p(·) as in (1). Using Assumption D (ii), and then we obtain Now by Lemma 3.1 and the fact that F and G are bounded, there exists a positive constant C 0 such that f (t) H ≤ C 0 for all t ≥ T 0 + τ . Then, by the definition of a subdifferential and the Uniform Gronwall Lemma (see [28]), there exists a positive constant C 1 such that ϕ t p(·) (u 1 (t)) ≤ C 1 for all t ≥ T 1 +τ. Consequently, there exists a positive constant K 1 such that u 1 (t) Y ≤ K 1 for all t ≥ T 1 + τ.
In a similar way, we conclude u 2 (t) Y ≤ K 2 for all t ≥ T 1 + τ for a positive constant K 2 . The assertion of the lemma then follows.
Let U G be the multivalued process defined by the generalized process G. We know from [23] that for all t ≥ s in R the map x → U G (t, s)x ∈ P (H × H) is closed, so we obtain from Theorem 18 in [4] the following result

Forward attraction.
Pullback attractors contain all of the bounded entire solutions of the nonautonomous dynamical system [11,12]. Simple counterexamples show that a pullback attractor need not be attracting in the forward sense [11]. However, since the pullback absorbing set D above is also forward absorbing (the absorption time is independent of the initial time τ ), the forward omega limit sets ω f (τ, D) of the multivalued process starting at time τ are nonempty and compact subsets of the compact set D. Moreover, it follows by the positive invariance of the D and the two-parameter semi-group property that they are increasing in time. The forward limiting dynamics thus tends to the nonempty compact subset ω ∞ f (D) = ∪ τ ≥0 ω f (τ, D) ⊂ D, which was called the forward attracting set in [16]. (It is related to the Vishik uniform attractor, when that exists, but can be smaller since the attraction here need not be uniform in the initial time).
As shown in Proposition 8 of [16] (in the context of single valued difference equations, but a similar proof holds here) the forward attracting set ω ∞ f (D) is asymptotically positively invariant with respect to the set valued process U G (t, τ ), i.e., if for any monotone decreasing sequence ε p → 0 as p → ∞ there exists a monotone increasing sequence T p → ∞ as p → ∞ such that for each τ ≥ T p Simple counterexamples show that the set ω ∞ f (D) need not be invariant or even positive invariant, although it may be in special cases depending on the nature of the time varying terms in the system. For asymptotically autonomous systems ω ∞ f (D) is contained in the global attractor A ∞ for the multivalued semigroup G associated with the limiting autonomous system.
Moreover, it is possible to compare the global attractor A ∞ with the limit-set A(∞) defined by A(∞) := t∈R ∪ r≥t A(r) and which can be characterized by This kind of comparison was done in [26] for the multivalued context.  26]). Suppose the pullback attractor A is forward compact, i.e., ∪ r≥t A(r) is precompact for each t ∈ R. Moreover, suppose that for each solution u of problem (8) there exists a solution v of problem (9) such that u(t + τ ) → v(t) in X as τ → +∞ for each t ≥ 0 whenever ψ τ ∈A(τ ) and ψ τ → ψ 0 in X as τ → +∞.
To obtain the equality A ∞ = A(∞) we need to assume stronger conditions as in the next result. (b) lim t→+∞ sup x∈A∞ dist(G(t)x, U G (t, 0)x) = 0. 5. Asymptotic upper semicontinuity. In this section we establish the asymptotic upper semicontinuity of the elements of the pullback attractor. Specifically, we prove that the system (S) is asymptotically autonomous. 5.1. Theoretical results. In this subsection motivated by problem (S), we study the asymptotic behavior of an abstract nonautonomous multivalued problem in a Hilbert space H of the form compared with that of an autonomous multivalued problem of the form (v 1 (0), v 2 (0)) = (ψ 1,0 , ψ 2,0 )) =: ψ 0 , where A(t), B(t), A ∞ and B ∞ are univalued operators in H × H and F , G : Under appropriate relationships between the operators A(t), A ∞ and B(t), B ∞ , the autonomous problem (9) is the asymptotic autonomous version of the nonautonomous problem (8) . In particular, we establish the convergence in the Hausdorff semi-distance of the component subsets of the pullback attractor of the nonautonomous problem (8) to the global autonomous attractor of the autonomous problem (9).
Some definitions on multivalued semigroups are recalled here, see for example [5,17,24] for more details.
It is called strict (or exact) if G(t 1 + t 2 , x) = G(t 1 , G(t 2 , x)), for all x ∈ X and t 1 , Definition 5.2. Let G be a multivalued semigroup on X. The set A ⊂ X attracts the subset B of X if lim t→∞ dist H (G(t, B), A) = 0. The set M is said to be a global B-attractor for G if M attracts any nonempty bounded subset B ⊂ X.
Suppose that the multivalued evolution process {U (t, τ ) : t ≥ τ } in H × H associated with problem (8) has a pullback attractor A = {A(t) : t ∈ R} and that the multivalued semigroup G : R + × H × H → P (H × H) associated with problem (9) has a global autonomous B-attractor A ∞ in the Hilbert space H × H. The following result will be used later to establish the convergence in the Hausdorff semi-distance of the component subsets A(t) of the pullback attractor A to A ∞ as t → ∞. Theorem 5.3. Suppose that C := ∪ τ ≥0 A(τ ) is a compact subset of H × H. In addition, suppose that for each solution u of problem (8) there exists a solution v of problem (9) with initial values ψ τ and ψ 0 , respectively, such that u(t + τ ) → v(t) in H × H as τ → +∞ for each t ≥ 0 whenever ψ τ ∈A(τ ) and ψ τ → ψ 0 in H as τ → +∞. Then Proof. Suppose that this is not true. Then there would exist an 0 > 0 and a real sequence {τ n } n∈N with τ n +∞ such that dist H×H (A(τ n ), A ∞ ) ≥ 3 0 for all n ∈ N. Since the sets A(τ n ) are compact, there exist a n ∈ A(τ n ) such that dist H×H (a n , A ∞ ) = dist H×H (A(τ n ), A ∞ ) ≥ 3 0 , for each n ∈ N. By the attraction property of the multivalued semigroup, we have dist H×H (G(τ n0 , C), A ∞ ) ≤ 0 for n 0 > 0 large enough. Moreover, by the negative invariance of the pullback attractor there exist b n ∈ A (τ n − τ n0 ) ⊂ C for n > n 0 such that a n ∈ U (τ n , τ n − τ n0 ) b n for each n > n 0 . Since C is compact, there is a convergent subsequence b n → b ∈ C. Since a n ∈ U (τ n , τ n − τ n0 ) b n there exists a solution u n = (u 1n , u 2n ) of du 2n dt (t) + B(t)u 2n (t) ∈ G(u 1n (t), u 2n (t)) u n (τ n − τ n0 ) = b n , such that a n = u n (τ n ). Writing τ n = τ n0 + (τ n − τ n0 ) and using the hypotheses with t = τ n0 and τ = τ n − τ n0 → +∞ (as n → +∞), there exists a solution v n of for n large enough. Hence, dist H×H (a n , A ∞ ) = dist H×H (u n (τ n ), A ∞ ) ≤ u n (τ n ) − v n (τ n0 ) H×H + dist H×H (v n (τ n0 ) , A ∞ ) ≤ u n (τ n ) − v n (τ n0 ) H×H + dist H×H (G (τ n0 , C) , A ∞ ) which contradicts (10).
The next result is very useful for checking that the hypothesis of asymptotic continuity of the nonautonomous flow in the preceeding theorem for problems like (8) holds. In order to obtain the result we suppose that the operators A(t) and A ∞ satisfy the following assumption.