Uniform Attractors for a Non-Autonomous Thermoviscoelastic Equation with Strong Damping

This paper considers the existence of uniform attractors for a non-autonomous thermoviscoelastic equation with strong damping in a bounded domain ( ) 1 n n Ω ⊆ ≥  by establishing the uniformly asymptotic compactness of the semi-process generated by the global solutions.


Introduction
In this paper we investigate the existence of uniform attractors for a nonlinear non-autonomous thermoviscoelastic equation with strong damping  where ( ) is a bounded domain with smooth boundary ∂Ω , u and θ are displacement and temperature difference, respectively.

( )
, u x t τ (the past history of u) is a given datum which has to be known for all t τ ≤ , the function g represents the kernel of a memory, ( ) ( ) , , , x t f f x t σ σ = = are non-autonomous terms, called symbols, and ρ is a real number such that Now let us recall the related results on nonlinear one-dimensional thermoviscoelasticity.Dafermos [1], Dafermos and Hsiao [2], proved the global existence of a classical solution to the thermoviscoelastic equations for a class of solid-like materials with the stress-free boundary conditions at one end of the rod.Hsiao and Jian [3], Hsiao and Luo [4] obtained the large-time behavior of smooth solutions only for a special class of solid-like materials.Ducomet [5] proved the asymptotic behavior for a non-monotone fluid in one-dimension: the positive temperature case.Watson [6] investigated the unique global solvability of classical solutions to a one-dimensional nonlinear thermoviscoelastic system with the boundary conditions of pinned endpoints held at the constant temperature and where the pressure is not monotone with respect to u and may be of polynomial growth.Racke and Zheng [7] proved the global existence and asymptotic behavior of weak solutions to a model in shape memory alloys with a stress-free boundary conditions at least at one end of the rod.Qin [8] [9] obtained the global existence, and asymptotic behavior of smooth solutions under more general constitutive assumptions, and more recently.Qin [10] has further improved these results and established the global existence, exponential stability and the existence of maximal attractors in ( ) As for the existence of global (maximal) attractors, we refer to [11] [12] [13].More recently, Qin and Lü [12] obtained the existence of (uniformly compact) global attractors for the models of viscoelasticity; Qin, Liu and Song [13] established the existence of global attractors for a nonlinear thermoviscoelastic system in shape memory alloys.
Our problem is derived from the form ( ) which has several modeling features.The aim of this paper is to extend the decay results in [14] for a viscoelastic system to those for the thermoviscoelastic system (1.1-1.2) and then to establish the existence of the uniform attractor for this thermoviscoelastic systems.In the case ( ) t f u is a constant, Equation (1.6) has been used to model extensional vibrations of thin rods (see Love [15], Chapter 20).In the case ( ) t f u is not a constant, Equation (1.6) can model materials whose density depends on the velocity t u .For instance, a thin rod which possesses a rigid surface and with an interior which can deforms slightly.We refer the reader to Fabrizio and Morro [16] for several other related models.
Let us recall some results concerning viscoelastic wave equations.In [17], the author concerned with the quasilinear viscoelastic equation he proved that the energy decays similarly with that of g.In [18], Wu considered the nonlinear viscoleastic wave equation with the same boundary and initial conditions as (1.7), the author proved that, for a class of kernels g which is singular at zero, the exponential decay rate of the 0, In the case 0 b = in (1.12), Messaoudi and Tatar [22] proved the exponential decay of global solutions to (1.12) without smallness of initial data, considering only the dissipation effect given by the memory.Considering nonlinear dissipation.
Recently, Araújo et al. [23] studied the following equation and proved the global existence, uniqueness and exponential stability, and the global attractor was also established, but they did not establish the uniform attractors for non-autonomous equation.Then, Qin et al. [24] established the existence of uniform attractors for a non-autonomous viscoelastic equation with a past history Moreover, we would like to mention some results in [25] [30] established the global existence of weak solutions and the uniform decay estimates for the energy by using the Faedo-Galerkin method and the perturbed energy method, respectively.To the best of our knowledge, there is no result on the existence of uniform attractors for non-autonomous thermoviscoelastic problem (1.1)- (1.4).Therefore in this paper, we shall establish the existence of uniform attractors for problem (1.1)-(1.4)by establishing uniformly asymptotic compactness of the semi-process generated by their global solutions.Noting that the symbol ( ) ( ) , , , x t f x t σ , which are dependent in t, so our estimates are more complicated than [23] [24] and we must use new methods to deal with the symbol ( ) ( ) , , , x t f x t σ as the change of time.Therefore we improved the results in [23] [24].For more results concerning attractors, we can refer to [31]- [37].
Motivated by [38] [39] [40], we shall add a new variable ( ) to the system which corresponds to the relative displacement history.Let us define and we can take as initial condition Thus, the original memory term can be written as and we get a new system with the boundary conditions 0 on , 0 on , and initial conditions , , .
The rest of our paper is organized as follows.In Section 2, we give some preparations for our consideration and our main result.The statements and the proofs of our main results will be given in Section 3 and Section 4, respectively.
For convenience, we denote the norm and scalar product in ( )

2
L Ω by ⋅ and ( ) C denotes a general positive constant, which may be different in different estimates.
Similar to Theorem 2.1, we have the following existence and uniqueness result.
Theorem 2.3.Assume that 1 G E ∈ and Σ is defined by (2.8), then the family of processes

The Well-Posedness
The global existence of solutions is the same as in [23] [30] [40], so we omit the details here.Next we prove the uniqueness of solutions.
We consider two symbols with Dirichlet boundary conditions and initial conditions ( ) ( ) ( ) The corresponding energy for (3.1)-(3.3) is defined ( ) It is easy to see that , 0 To simplify notations, let us say that the norm of the initial data is bounded by some 0 R > .Then given T τ > we use RT C to denote several positive constants which depend on R and T. By Young's inequality and the interpolation inequalities, we derive ( ) ( ) which, together with (3.6)-(3.9),yields for some Integrating (3.10) from τ to t and using Hölder's inequality, we have Noting that ( ) Applying Gronwall's inequality, we see that which, together with (3.13), gives for all .
This shows that solutions of (1.17)-(1.21)depend continuously on the initial data.We complete the proof of Theorem 2.1.

Uniform Attractors
In this section, we shall establish the existence of uniform attractors for system (1.17)-(1.21).To this end, we shall introduce some basic conceptions and basic lemmas.For more results concerning uniform attractors, we can refer to [31] [36] Let X be a Banach space, and Σ be a parameter set.The operators

{ }
T s be the translation semigroup on Σ , we say that a family of processes In the following, as usual, (w.r.t) will represent "with respect to".
Definition 4.2.The family of semi-processes , dist ⋅ ⋅ stands for the usual Hausdorff semidistance between two sets in X.In particular, a closed uniformly attracting set  A Σ is said to be the uniform (w.r.t Ĝ ∈ Σ ) attractor of the family of the semi-process if it is contained in any closed uniformly attracting set (minimality property).Definition 4.4.Let X be a Banach space and B be a bounded subset of , X Σ be a symbol (or parameter) space.We call a function ( ) We denote the set of all contractive functions on B B × by ( ) be a family of semi-processes satisfying the translation identities (4.3) and (4.4) on Banach space X and has a bounded uniformly (w.r.t Ĝ ∈ Σ ) absorbing set 0 B X ⊆ .Moreover, assuming that for any 0 ε > , there exist ( ) ,0 ,0 , ; , , , , .
Proof.This lemma is a version for semi-processes of a result by Khanmamedov [45].A proof can be found in Sun et al. [43], Theorem 4.2.
Next, we will divide into two subsections to prove Theorem 2.3.

Uniformly (w.r.t. G ∈ Σ ) Absorbing Set in H
In this subsection we shall establish the family of processes Now we define ( ) From (1.17), integration by parts and Young's inequality, we derive for any ( ) hereinafter we use λ to represent the Poincaré constant.
From the expression of ( ) and the embedding theorem , we have for any which, together with (4.20) and Poincaré's inequality, gives Now we take ( ) Hence from (4.21)-( 4.22), it follow We define the functional It follows from (1.17) that .
From Young's inequality, Hölder's inequality and Poincaré's inequality, we derive for any ( ) and for any where M and ε are positive constants.
Then it follows from (4.10), (4.23), (4.34) and (2.2) that Now we claim that there exist two constants 1 2 , 0 For any t τ ≥ , we take ε so small that For fixed ε , we choose δ small enough and M so large that Then there exist a constant which, together with (4.37), gives  1 e Now for any bounded set 0 ⊆  H , for any ( ) , is a uniform absorbing ball for any 1 G E ∈ .The proof is now complete.

>.
are constants, they proved the energy decay for the viscoelastic equation with nonlinear damping.Then Park and Park [20] established the general decay for the viscoelastic problem with nonlinear weak damping ( ) boundary condition, where 0 ρ > is a constant.In [14], Cavalcanti et al. studied the following equation with Dirichlet boundary conditions They established a global existence result for 0 γ ≥ and an exponential decay of energy for 0 γ > , and studied the interaction within the t tt u u ρ and the memory term g u * ∆ .Messaoudi and Tatar [21] established, for small initial data, the global existence and uniform stability of solutions to the equation s inequality, Young's inequality, Poinceré's inequality and Theorem 4.1, we get