Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms

This paper is concerned with long-time dynamics of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. We also establish the existence of a global attractor, and we study the fractal dimension and exponential attractors.


1.
Introduction. This discussion is devoted to a special case of a theory of binary mixture of solids with nonlinear damping and sources terms. It is important to note that the theory of mixtures of solids has been widely investigated in the last decades, see the references [3,6,7,8] for a detailed presentation. Qualitative properties of solutions to the problem defining this kind of material have been the scope of many investigations. Several results concerning existence, uniqueness, continuous dependence and asymptotic stability can be found in the literature [1,2,14,17].
The version of the theory of binary mixture of solids considered below bears the influence of nonlinear amplitude-modulated forcing terms that could either act as energy "sinks" with a estoring effect (e.g., as a nonlinear refinement on Hooke's law) or in the more interesting case as "sources" that contribute to the build-up of energy and potentially lead to a blow-up of solutions. In addition, to counterbalance the effects of potentially destabilizing strong sources, the system incorporates internal viscous (frictional) damping. Besides the Hadamard well-posedness of this problem, the influence of the source-damping interaction on the behavior of solutions is of the main interest in this work.
From the above, the problem we want to study can be stated as follows: Given L > 0 and T > 0, we consider a rod composed by a mixture of two interacting continuous with reference configuration [0, L]. Denoting by u(x, t), w(x, t) : [0, L] × [0, T ] → R the displacement of each constituent, the considered particles are suppose to occupy the same position at time t = 0, so that x = y. For i = 1, 2, we also introduce the mass densities ρ i > 0, the partial stresses T i and the external forces F i associated to u and w respectively. Indicating with P i the internal body forces, the motion equations are given by the differential system (1.1) We assume the constitutive equations of the partial stresses to be Finally, taking F 1 = −g 1 (u t ) and F 2 = −g 2 (w t ) and substituting (1.2) and (1.3) into system (1.1), we end up with ρ 1 u tt − a 11 u xx − a 12 w xx + g 1 (u t ) = f 1 (u, w), in (0, L) × (0, T ), ρ 2 w tt − a 12 u xx − a 22 w xx + g 2 (w t ) = f 2 (u, w), in (0, L) × (0, T ). (1.4) The two equations in (1.4) are supplemented with the boundary and initial conditions      u(0, t) = u(L, t) = w(0, t) = w(L, t) = 0, t > 0, (1.5) Note that in light of the assumptions on the matrix A we have the relation a 11 > 0 and a 11 a 22 − a 2 12 > 0. (1.6) The main goal here is to study long-time behavior of binary mixture of solids with nonlinear damping and sources terms, focusing on the interplay between nonlinear damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy. We will use nonlinear semigroups and the theory of monotone operators to obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we will prove that such unique solutions depend continuously on the initial data. We will also prove the existence of a global attractor and we will study the fractal dimension and exponential attractors. This is done by showing that the solution semigroup is gradient and quasi-stable in the sense of [9].
The paper is organized as follows: In the Section 2 we present the notations needed, we list the standing assumptions on the nonlinear terms and we summarize the main results. In the Section 3 we prove the existence of global solution. In Section 4 we study the global attractors. Finally, in Section 5 we study the fractal dimensional and exponential attractors.
2. Problem setting, assumptions and main results. In this section we will present the notations, assumptions and main results.
Assumption 2.1 (For existence of global attractors).
• g 1 , g 2 ∈ C 1 (R) are monotone increasing functions with g 1 (0) = g 2 (0) = 0. In addition, there exist positive constants α and β such that for all |s| 1, • There exists a constant C > 0 such that |∇f j (z)| C |u| p−1 + |w| p−1 + 1 , j = 1, 2, with p 1. (2.10) • There exist α 0 , β 0 > 0 such that Moreover, we assume that Observe that the assumption on the damping, implies that for any δ > 0 there exists a C δ > 0 such that In order to describe the results we introduce the definition of weak solution to the problem (1.4)-(1.5). • z = (u, w) satisfies the following identity in the sense of distributions for all . Also, we define the total energy by where E(t) is the linear energy defined by Main results. In this part, we present the main results of this paper.
• The weak solution U ∈ C([0, ∞); H) depend continuously on initial data U 0 in the phase space H. In particular, such solutions are unique. • If U 0 ∈ D(A) then the weak solution is strong.
Theorem 2.5 (Existence of global attractors). Suppose that Assumption 2.1 hold. Then the dynamical system (H, S t ) generated by the equation (1.4) is dissipative and asymptotically smooth, and hence, it has a compact global attractor A. Moreover, the global attractor A is characterized by (2.19) where N is the set of stationary point of S t and M u (N ) is the unstable manifold of N .
To obtain the next result, we will need to replace the assumption (2.8) by Remark 2.6. Observe that assumption (2.20) implies the monotonicity property where m j > 0, j = 1, 2. • The complete trajectories (z(·), z t (·)) in A satisfy for some constant C > 0.
• The dynamical system (H, S t ) has a generalized exponential attractor A exp with finite fractal dimension in a extended space H −η , defined as interpolation of for any η ∈ (0, 1].
3. Existence global. In this section, we obtain the result on the existence of global solution stated in Theorem 2.4. We start by proving two lemmas which will be used in sequence.
By (2.10) and the Mean Value Theorem, we have for j = 1, 2, All terms on the right-hand side of (3.1) are estimated analogously. Using the Hölder's inequality, Sobolev embedding H 1 0 (0, L) → L q (0, L) for all 1 q ∞, assumptions p 1 and u 1,2 C(R), we obtain Therefore, by (3.2) and (2.5) we see that Using the above estimate and definition of F it is easy to see that there exists a constant L(R) > 0 such that The proof is complete.
Then • There exists a positive constant C 0 such that • The following identity holds Proof. Using Poincaré's inequality, (2.11) and (2.4) we obtain Consequently, We need to verify that the operator A is maximal monotone. We recall that [18] we write the operator A as Step 1: A is monotone. By straightforward computation, for all U,Û ∈ D(A) we get where we have used the monotonicity of g 1 and g 2 . Therefore A is monotone.
Step 1: A is maximal monotone. We need to prove that the range of A + I is all of H. LetV = (v, v 1 ) ∈ H. We will prove that there exists U = (z, z 1 ) ∈ D(A) such that (A + I)U =V , that is, Observe that (3.7) is equivalent to Since v ∈ V , then the right hand side of (3.8) belongs to V . Thus, we define the operator S : V → V by So we need to prove that S is surjective. In view of [5, Corollary 1.2], we only need to prove that S is maximal monotone and coercive. We split S as two operators C, G : V → V : Step 3: C is maximal monotone and coercive. In view of [5, Theorem 1.3], we only need to prove that C is monotone and hemicontinuous. Let z = (u, w) ∈ V andẑ = (û,ŵ) ∈ V . We have This proves that C is monotone. Next, we will show that C is hemicontinuous, that is, w-lim 10) which implies that C is hemicontinuous.
Step 3: G is maximal monotone and coercive. Since g 1 and g 2 are monotones it is easy to see that G is monotone. We now prove the hemicontinuity. Let z = (u, w), z 1 = (u , w 1 ) ∈ V andẑ = (û,ŵ) ∈ V . Firstly, we observe that (3.11) Since g 1 is continuous we have g 1 (u+λu 1 )û → g 1 (u)û pointwise as λ → 0. Moreover, by the assumption on the damping |g 1 (s)| β 1 (|s|+1) for all s. Therefore, if |λ| 1, we have that By Lebesgue's dominated convergence theorem we obtain lim λ→0 L 0 g 1 (u + λu 1 )û dx = This proves that G is hemicontinuous. Now, since C and G are both maximal monotone and D(C) = D(G) = V , by [5, Theorem 1.5], we conclude that S = C + G is maximal monotone. Moreover, since g 1 , g 2 are monotones and C0 = G0 = 0, then from (3.9) it follows that, for all z ∈ V , Therefore S is coercive. Consequently the maximal monotonicity of A is proved.
Step 4: Global solution: Since A is maximal monotone and F locally Lipschitz, cf. Lemma 3.1, then by [9, Theorem 7.2] for all U 0 ∈ D(A) there exists a t max ∞ and a unique strong solution U for (2.6) defined on the interval [0, t max ). Moreover, if U 0 ∈ H then (2.6) has a unique weak solution U ∈ C([0, t max ); H) and such solutions satisfy lim sup t→t − max U (t) H = ∞, provided t max < ∞. A standard approximation argument shows that the energy identity (2.18) hold. To prove the existence global, we need to prove that t max = ∞. Let U a strong solution on [0, t max ). By (3.3) it follows that Using density argument, we obtain that (3.15) is also satisfied for weak solutions. We conclude that t max = ∞.
Step 5: Continuous dependence: The solution of (2.6), denoted by U (t), can be expressed by the following variation of constants formula Since, by Lemma 3.1 F is locally Lipschitz continuous, we conclude that there exists a constant C T > 0 such that  Hence, z t (t) = 0, for all t 0, a.e. in (0, L), which implies that z(t) = z 0 , for all t 0.
Therefore, S t U = U = (z 0 , 0) for all t 0. The proof is complete.
In next, we will prove that the dynamical system (H, S t ) has a bounded absorbing set. The proof is inspired by the argument of [16,Lemma 4.6], also see [9, Lemma 3.2]. Integrating (4.4) from 0 to T we get Using the last estimate in (4.5) we see that  We need to estimate all terms on the right-hand side of (4.6). Estimate for the term: − (z t (T ), z(T )) H − (z t (0), z(0)) H . Using Hölder's inequality we have (4.7) Using Lemma 3.2 in (4.7) we get Estimate for the term: We introduce the sets: By (2.8) we have g 1 (s)s α 1 |s| 2 for |s| 1, thus (4.10) Similarly, Then, by (4.10) and (4.11) we obtain that Estimate for the term: By Hölder's and Young's inequalities we have (4.13) Since g 1 is continuous and increasing we see that Z1 g 1 (u t ) 2 dxdt max{g 1 (−1) 2 , g 1 (1) 2 }LT. (4.14) By (2.8), we have |g 1 (s)s| β 1 |s| 2 for |s| 1. Therefore Substituting the estimates (4.14) and (4.15) in (4.13) we have Similarly, Combining (4.16) and (4.17) we obtain We choose = C0 4 , then by Lemma 3.2 we conclude that there exists a constant C 3 > 0 such that We apply the estimates (4.8), (4.12) and (4.18) in (4.6) to get Using the fact that E(T ) E(t) and energy identity (2.18) we have Hence, taking T > 0 sufficiently large such that T > 2C 2 , we obatin By iterating the estimate on intervals [mT, (m + 1)T ], m = 1, 2, . . . , we have Using the fact that γ T < 1 and the argument presented in [16, p.p. 2485-2486] it follows that there exists constants γ, α > 0 such that Combining the last estimate and Lemma 3.2 we obtain Therefore, the closed ball B = B(0, R 0 ) in H of center zero and radius R 0 , where is a bounded absorbing set. The proof is complete.

Proof of Theorem 2.5 (Existence of global attractors).
In this part, we conclude the proof of Theorem 2.5. We start by citing a criterion of asymptotic smoothness by [10, Theorem 7.1.11] which will be used in sequence. for every sequence {y n } from B. Then (X, S t ) is an asymptotically smooth dynamical system.
Next, we will show that the dynamical system (H, S t ) is asymptotically smooth. The argument of the next lemma is similar to the proof of Lemma 4.2 in [16].
Since we have shown that (H, S t ) is a dissipative asymptotically smooth dynamical system, we can apply the result in [ [4,13,20,15]). Thus the proof of Theorem 2.5 is complete. 5. Fractal dimension, regularity and exponential attractors. This section is devoted to prove the Theorem 2.7. 5.1. Quasistability. In this part, we will establish the quasi-stability of dynamical system (H, where S t U j = (z j , z j t ) = (u j , w j , u j t , w j t ), j = 1, 2 weak solutions of (1.4)-(1.5) with initial condition U 1 , U 2 ∈ B and z = (u, w) = z 1 − z 2 .