Research Article | Open Access

# Existence and Uniform Decay of Weak Solutions for Nonlinear Thermoelastic System with Memory

**Academic Editor:**Yeol Je Cho

#### Abstract

A nonlinear thermoelastic system with memory is considered, which is derived from a physical model with vibration in temperature environment. By some skillful and technical arguments, results of existence, uniqueness, and uniform decay on this generalized system are obtained.

#### 1. Introduction

In this work we consider the following initial boundary value problem: where , , , is a bounded domain in with boundary, , , is class function like , and , are positive constants: , is a known function and the function is positive and satisfies some conditions to be specified later.

However, (1.1) consists of a dynamical equation coupling a heat equation, which can be used to describe some physical process of thermoelastic material. Also, and represent the displacement and temperature, respectively, at position and time . The coupling of the heat equation in the model of vibrations presents important aspects because it represents better than the reality, that is, allowing to influence the vibrations in a more adequate way. appearing in the dynamical part of system (1.1) is a nonlinear perturbation of Moeover, Kirchhoff-Carrier's model which describes small vibrations of a stretched string (dimension *n* = 1) when tension is assumed to have only a vertical component at each point of the string. Many researchers have investigated several types of problems involving the Kirchhoff equation among which we can cite the work in [1, 2]. Clark and Lima [3] studied the local existence for of solutions to the mixed problem:

In this paper, we prove the global existence and uniqueness of weak solutions of (1.1) based on different definition of weak solution and estimate techniques from [3], we consider the Kirchhoff equation with the strong damping term and so-called “memory” term . Here we consider the memory effect in (1.1) because physically some materials could produce the viscosity of memory type [4]. Hence under appropriate assumptions on , , and , and making use of Galerkin's approximations and compactness argument, we establish global existence and uniqueness. Meanwhile, by some suitable estimate techniques, we deal with the memory term and another nonlinear term appearing in the mixed problem of viscoelastic wave equation. In order to obtain the exponential decay of the energy, we make use of the perturbed energy method, see Komornik and Zuazua [5].

The rest of this paper is organized as follows: In Section 2 we give out assumptions and state the main result. In Section 3 we exploit Faedo-Galerkin's approximation, priori estimates, and compactness arguments to obtain the existence of solutions of a penalty problem. In Section 4, uniqueness is proved. In Section 5, the exponential decay of solution is obtained by using the perturbed energy method.

#### 2. Assumptions and Main Results

Throughout this paper, we use the following notation:

Now we state the main hypotheses in this paper.

*(A.1) Assumption on Kernel h*

Let be a nonnegative and bounded function and suppose that there exist positive constants such that
Moreover, verifies .

*(A.2) Assumption on *

Let satisfies that
is given by the Sobolev embedding inequality for , in the general case, we denote .

*(A.3) Assumption on Initial Condition, f and g *

Assume that and . Next we define the energy with The main result is as follow.

Theorem 2.1. *If assumptions ()–() hold, then there exists a unique weak solution with , , , and such that
**
Furthermore, if , satisfy that and small enough, we have the following decay estimate:
**
where and are positive constants.*

#### 3. Existence of Solutions

*Proof of Theorem 2.1. *We use Galerkin's approximation. Let be a basis in which is orthonormal in and the subspace of generated by the first of . For each , we seek the approximate solution:
of the following Cauchy problem:
satisfying the initial conditions
According to the ODE theory, we can solve the system (3.2)-(3.3) by Picard's iteration. Hence, this system has unique solution on interval for each . The following estimates allow us to extend the solution to the closed interval .

In the following proof, we will use , , to denote various positive constants which may be different in different places and may be dependent on in some cases.*The First Estimate*

Taking in (3.2) and in (3.3), respectively, then adding the results and using assumption we have
Now integrating (3.5) over for we have
Moreover, from assumption (1), we have
where is arbitrary.

Hence letting small enough and using Gronwall's inequality we obtain the first estimate:

where is independent of .*The Second Estimate*

First we estimate the initial data in the -norm. Taking and in (3.2) we have
Hence, noticing the assumption on , and , we deduce
where is independent of .

Similarly, taking and in (3.3), we also deduce

where is independent of .

Differentiating (3.2) and (3.3), replacing by and respectively, and then adding the results, we get

From the first estimate and Young's inequality, we have
where is arbitrary.

Noticing , assumption and the first estimate, we have

and by assumption (1), we have
Therefore, combining (3.14)–(3.16), (3.10), (3.11) and integrating (3.12) over we have
Moreover, consider that

Hence, from (3.17), (3.18), the first estimate, letting small enough and using Gronwall's inequality, we get the second estimate:

where is independent of .*The Third Estimate*

Taking in (3.3), we have
Hence we easily get , and is independent of .*The Fourth Estimate*

Let be two natural numbers and consider , . From the system (3.2), we have

Taking in (3.21), we have

Noticing that
hence, using assumption (2.2) and integrating (3.22) over , we get
Notice that
where is arbitrary:
Moreover, by mean value theorem and assumption we have
Therefore, by (3.25)–(3.27), letting small enough, by the first estimate, and using the Gronwall's lemma of integral form (see [6]) in (3.24) we obtain that
*Passage to the Limit*

From above estimates, we deduce that there exist functions and subsequences of , which we still denote by , satisfying
Moreover, according to the compactness of Aubin-Lions, we have
Hence combing (3.31) and the fourth estimate (3.28), we deduce that

Thus we can pass the limit in system (3.2)-(3.3). Let , we prove that is a weak solution of the system (1.1).

#### 4. Uniqueness of the Solution

The proof of uniqueness of solution is similar to the fourth estimate, but for integrity, we still give the detailed proof.

Let and be two solutions of couple system (1.1) under the conditions of Theorem 2.1, then we have verifying

Taking in (4.1) and in (4.2), respectively, and adding the results, we have Noticing that hence, using assumption (2.2) and integrating (4.3) over , we get Notice that where is arbitrary: Moreover, by mean value theorem and assumption , we have Therefore, by (4.6)–(4.8), Cauchy inequality, Young's inequality, and using Gronwall's lemma in (4.5), we get Thus, we have proved the uniqueness consequence.

#### 5. Existence of Solutions

In this section, we follow the additional assumptions appeared in Theorem 2.1. We introduce the energy where we define

*Remark 5.1. *Taking in (2.6) and in (2.7), respectively, then adding the results we have
Noticing
and combining the assumptions on appeared in Theorem 2.1, we deduce
where we denote . Thus, we have the energy is uniformly bounded (by *e*(0) and is decreasing in .

*Remark 5.2. *Furthermore, from the assumption , we have

For every , we define the perturbed energy by setting

Lemma 5.3. *There exists such that
*

*Proof. *From (5.7), we obtain
hence we have
where .

Lemma 5.4. *There exists and such that for *

*Proof. *By using the problem (1.1), we obtain
Notice that
where is arbitrary.

Hence, from (5.12)–(5.15), we have

Therefore, from (5.5) and (5.16), we get
Taking and small enough, we have . Moreover if we denote
and choosing we obtain
for some constant .

*Proof of Decay*

Let us define and consider . From Lemma 5.3, we have
and so
From (5.21), we get
Hence from (5.22) and Lemma 5.4, we obtain
that is,
Integrating last inequality over [0,*t*], we get
From (5.21) and (5.25), we have

Hence, from (5.6) and (5.26), we obtain

that is,
where and .

Therefore, we have proved the exponential decay of solution.

#### Acknowledgment

This work is supported by NSFC of Yunnan Province (07Y40422, 2007A196M) and the National Natural Science Foundation of China under Grant 10471072.

#### References

- M. P. Matos and D. C. Pereira, “On a hyperbolic equation with strong damping,”
*Funkcialaj Ekvacioj*, vol. 34, no. 2, pp. 303–311, 1991. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. A. Medeiros and M. M. Miranda, “On a nonlinear wave equation with damping,”
*Revista Matemática de la Universidad Complutense de Madrid*, vol. 3, no. 2-3, pp. 213–231, 1990. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. R. Clark and O. A. Lima, “On a mixed problem for a coupled nonlinear system,”
*Electronic Journal of Differential Equations*, no. 6, pp. 1–11, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet - G. Duvaut and J.-L. Lions,
*Inequalities in Mechanics and Physics*, Grundlehren der Mathematischen Wissenschaften, 21, Springer, Berlin, Germany, 1976. View at: MathSciNet - V. Komornik and E. Zuazua, “A direct method for the boundary stabilization of the wave equation,”
*Journal de Mathématiques Pures et Appliquées*, vol. 69, no. 1, pp. 33–54, 1990. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. E. Showalter,
*Monotone Operators in Banach Space and Nonlinear Partial Differential Equations*, vol. 49 of*Mathematical Surveys and Monographs*, American Mathematical Society, Providence, RI, USA, 1996.

#### Copyright

Copyright © 2009 Liu Haihong and Su Ning. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.