PULLBACK DYNAMICS OF A NON-AUTONOMOUS MIXTURE PROBLEM IN ONE DIMENSIONAL SOLIDS WITH NONLINEAR DAMPING

. This paper is devoted to study the asymptotic behavior of a nonautonomous mixture problem in one dimensional solids with nonlinear damp- ing. We prove the existence of minimal pullback attractors with respect to a universe of tempered sets deﬁned by the sources terms. Moreover, we prove the upper-semicontinuity of pullback attractors with respect to non-autonomous perturbations.


1.
Introduction. The theory of mixtures of solids has been widely investigated in the last decades, see for instance the references [3,5,6,7] for a detailed presentation. In this paper, our interest is devoted to a special case of a theory of binary mixture of solids with nonlinear damping, sources terms and non-autonomous external forces. Qualitative properties of solutions to the problem defining this kind of material have been the scope of many investigations. In particular, several results concerning existence, uniqueness, continuous dependence and asymptotic stability can be found in the literature [1,2,14,20].
The main goal here is to prove the existence of minimal pullback D-attractors for the evolution process generated by the problem (1.4)-(1.5) with respect to a universe of tempered sets defined by the growth of sources terms f 1 (u, w), f 2 (u, w). We will also prove the upper-semicontinuity of pullback attractors as the non-autonomous perturbation tends to zero. In fact, we will prove that the family of pullback attractors associated to problem (1.4)-(1.5) with h j replaced by h j converges to the corresponding compact global attractor associated with the autonomous limit problem (1.1)-(1.2) when → 0.
This paper is organized as follows: In Section 2, we establish the existence and uniqueness of weak and strong solutions. This is presented in the Theorem 2.8. In Section 3, we recall the key definitions and results which concern the nonautonomous dynamical systems and pullback attractors. In Section 4, we prove that the evolution process generated by the problem (1.4)-(1.5) has a pullback Dabsorbing family and is pullback D-asymptotically compact and, consequently, the existence of minimal pullback attractors is established. Our main result is presented in the Theorem 4.6. In Section 5, we prove the upper-semicontinuity of global attractors as the non-autonomous perturbation tends to zero. More precisely, we prove that the family of pullback attractors associated to problem (1.4)-(1.5) with h j replaced by h j converges to the corresponding global attractor associated with the limit problem (1.1)-(1.2) as → 0. This is presented in the Theorem 5.1.

2.
Preliminaries and well-posedness. In this section, the existence and uniqueness of weak and strong solution of the problem (1.4)-(1.5) will be studied.
2.1. Assumptions. In this part, we present some notations and assumptions. We will use the following notations We recall the Poincaré's inequality Our study is given on the phase space It is a Hilbert space with the inner product: If z = (u, w, u , w ),z = (ũ,w,ũ ,w ) ∈ H, then we define • There exist p 1 and C > 0 such that, for j = 1, 2 Moreover, we assume that (iii) External forces: For the external forces, we assume that with σ 0 ∈ (0, σ 1 ], where σ 1 > 0 is a constant dependent only on the parameters of model specified later in Lemma 4.1. Remark 2.4. Observe that assumption (2.6) implies the monotonicity property, that is, and satisfies the following identity in the sense of distributions (2.14) If a weak solution satisfies further  We define, along a strong solution, the total energy by where E(t) is the linear energy given by Lemma 2.6. Let z = (u, w, u t , w t ) be a strong solution of (1.4)-(1.5), this way, the total energy satisfies (2.17) If in turn we multiply by w t the second equation in (1.4), we get d dt (2.18) Add (2.17) and (2.18) and then use (2.7) to obtain d dt It follows that d dt This proves that By (2.6) and Young's inequality we conclude that The proof is complete.
2.3. Semigroup formulation. In this part, we show that the system (1.4)-(1.5) is well-posed using nonlinear semigroup theory and monotone operators (see e.g. [10,4]). Let us write the problem (1.4)-(1.5) as a Cauchy problem The domain of A is given by (2.20) 2.4. Local and global solutions. This part, is dedicated to prove the existence of local and global solutions for the problem (1.4)-(1.5). We start by proving an auxiliary result which will be used in the sequel.
Then, there exist constants β 0 , C F > 0 such that

21)
and Proof. By (2.10) and (2.5) it follows that Using (2.9), we obtain (2.21) with The assertion (2.22) follows from the assumption (2.8) and the fact that If z τ ∈ D(A), then the solution is strong. Moreover, the weak solutions depend continuously on the initial data z τ in the phase space H.
Proof. The proof of the present theorem is done by using the equivalent Cauchy problem (2.19). The existence of solution for the autonomous case (F(t, z) = F(z)) was proven in [21]. Here we will present the main ideas of the proof.
Step 1: Local solutions. Note that A is maximal monotone, and that for each Step 2: Global solutions. By Lemma 2.6 we have It follows that Using (2.21) we obtain for some constant C > 0 independent of t. Hence, by (2.24) we conclude that which implies that t max = ∞.
Step 3: Continuous dependence. Let z 1 = (u 1 , w 1 , u 1 t , w 1 t ) and Then (u, w, u t , w t ) is the solution of with Dirichlet boundary conditions and initial condition Multiplying the first equation in (2.25) by u t and second by w t and then integrating over [0, L] we obtain (2.26) From (2.8), Hölder's inequality and embedding H 1 In a similar way we obtain that It follows from (2.27), (2.28) and (2.5) that  By (2.13) we conclude that (2.30) Substituting the estimates (2.29) and (2.30) in (2.26), we get that Applying Gronwall's lemma to (2.31) we conclude that , the continuous dependence follows from (2.32). The proof of Theorem 2.8 is complete.
3. Abstract results on the theory of pullback attractors. To describe the next results, we need some notations, definitions, and results (see, for instance [19,15,8,13,9] and references therein) which will be used throughout the following sections. We begin with precise definitions of the notions of a evolution process in metric space (X, d).
A process U is said to be closed if for any sequence x n → x in X and U (t, τ )x n → y in X, then U (t, τ )x = y. In addition, U is said to be continuous if the mapping U (t, τ ) : X → X is continuous for each t τ fixed.
Remark 3.2. It is clear that every continuous process is closed.
Throughout this paper, we denote by D a family of parameterised subsets in X, that is, is a non-empty subset of X for all t ∈ R. We say that a universe D is inclusion closed if given D ∈ D and C such that C(t) ⊂ D(t) for all t ∈ R, then C ∈ D.
Definition 3.4. A family B of non-empty sets is pullback D-absorbing for the process U if for any t ∈ R and any D ∈ D, there exists a τ 0 ( B, t) t such that Observe that in the above definition the set B does not necessarily belong to the class D.
Definition 3.5. Given a family D, an evolution process U is said to be pullback D-asymptotically compact if for any t ∈ R and any sequences τ n → −∞ and x n ∈ D(τ n ) the set {U (t, τ n )x n } n∈N is precompact in X. If a process is pullback Dasymptotically compact for any D ∈ D, then we say it is pullback D-asymptotically compact. Now, we recall a criterion, which is useful for verifying the pullback D-asymptotic compactness of evolutions process generated by non-autonomous hyperbolic equations (see [11,12] for autonomous system and [23,24,16] for non-autonomous system). Theorem 3.7. Let U be a evolution process on a Banach space X. Assume that U possesses a pullback D-absorbing family B 0 and for any t ∈ R and > 0 there exists τ t and a contractive function Ψ : Then the process is pullback D-asymptotically compact.
In the sequel we introduce the concept of pullback D-attractor. (iv) A is minimal in the sense that if C is a family of closed sets which is pullback D-attracting, then A(t) ⊂ C(t) for all t ∈ R.
The following result ensures the existence of a minimal pullback attractor (see [13,Theorem 3.11]). Theorem 3.9. Let U be a closed evolution process in a metric space X. Consider a universe D in X and suppose that U admits a pullback D-absorbing family B 0 and that U is pullback B 0 -asymptotically compact. Then, the family A D = {A(t)} t∈R defined by where denotes the pullback omega-limit is the minimal pullback D-attractor for the process U . If B 0 ∈ D, then Moreover, if B 0 (t) is closed for all t ∈ R and the universe D is inclusion closed, then the pullback attractor A D ∈ D.
To define a suitable universe in H for our purposes, we first establish the following stability inequality.
Lemma 4.1. Suppose that Assumption 2.3 holds. Then, for any z ∈ H there exist constants σ 1 > 0 and C 1 , C 2 , C 3 > 0 such that Proof. For each > 0, we shall define the perturbed energy by where E(t) is defined in (2.15) and Firstly, let us prove that there exists 0 > 0 such that Indeed, by Young's inequality, (2.5) and (2.21) we have that Choosing 0 = β 0 min 1, then the inequality (4.2) holds. Now, we claim that there exist constants C 4 , C 5 > 0 independent of t such that d dt From definition of N and (1.4) we have (4.5) Integrating by parts over [0, L] and using (2.7), we obtain
Thanks to Lemma 4.1, we can define a suitable tempered universe D in H.
is pullback D-absorbing.
Proof. Firstly we observe that (2.12) implies that (4.18) is well-defined. Now, let D ∈ D and t ∈ R. Since σ 0 σ 1 , we conclude Therefore, by (4.1) we have for all z τ ∈ D(τ ). Then, by the tempered condition of D it follows that Then, there exists τ 0 ( D, t) < t such that This proves that B 0 is a pullback D-absorbing family. The proof is complete.

4.2.
Pullback D-asymptotic compactness. In this part, we prove that the evolution process generated by the problem (1.4)-(1.5) is pullback D-asymptotically compact. We first prove a stabilization inequality.
Lemma 4.4. Suppose that Assumption 2.3 holds. Let B 0 the pullback D-absorbing family given in Lemma 4.3 and let U (t, τ )z j = (u j , w j , u j t , w j t ) be the weak solutions of (1.4)-(1.5) with initial condition z j ∈ B 0 (τ ), j = 1, 2. Then, there exists a constant σ 2 > σ 1 , and a constant C τ,t > 0 depending on τ t such that where u = u 1 − u 2 and w = w 1 − w 2 .

4.3.
Existence of pullback D-attractors. As a consequence of the previous results, we obtain the following theorem, which is the main result of this section.
Theorem 4.6. Suppose that Assumption 2.3 holds. Then the evolution process associated to the problem (1.4)-(1.5) possesses a minimal pullback D-attractor, where D is defined in (4.17). If in addition, σ 0 < 2σ1 p+1 , then above pullback D-attractor belongs to D.
Proof. Lemmas 4.3 and 4.5 indicate that the evolution process generated by the problem (1.4)-(1.5) has a pullback D-absorbing family and it is pullback D-asymptotically compact. Then, Theorem 3.9 implies the existence of the minimal pullback D-attractor.
Now, we will prove that the pullback attractor belongs to universe D under assumption σ 0 < 2σ1 p+1 . It is clear that D is inclusion closed. Then we must show that B 0 ∈ D, that is, lim By (4.18) we see that (4.47) Since the integral involved in (4.47) does not increase as τ decreases and σ 0 < 2σ1 p+1 , we conclude that lim Hence, (4.46) holds and B 0 ∈ D. Then from Theorem 3.9 the pullback attractor belongs to D. The proof is complete.
Multiplying the first equation in (5.5) by u t and second by w t and then integrating over [0, L] we obtain (5.6) By (2.6) we see that  Also, we observe that