Research Article | Open Access
Existence and Uniform Decay of Weak Solutions for Nonlinear Thermoelastic System with Memory
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.
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  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 , 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 . 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 .
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 ) 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.
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
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.
This work is supported by NSFC of Yunnan Province (07Y40422, 2007A196M) and the National Natural Science Foundation of China under Grant 10471072.
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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.