International Journal of Differential EquationsVolume 2011 (2011), Article ID 403264, 10 pagesdoi:10.1155/2011/403264

Research Article

Exponential Decay to Thermoelastic Systems over Noncylindrical Domains

Luci Harue Fatori and Michelle Klaiber

Department of Mathematics, State University of Londrina, Campus Universitario, 86051-990 Londrina, PR, Brazil

Received 13 May 2011; Accepted 4 July 2011

Academic Editor: Alberto Cabada

Copyright © 2011 Luci Harue Fatori and Michelle Klaiber. 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.

Abstract.

This paper is concerned with linear thermoelastic systems defined in domains with moving boundary. The uniform rate of decay of the energy associated is proved.

1. Introduction

In the study of asymptotic behavior for thermoelastic systems, a pioneering work is the one by Dafermos [1] concerned with the classical linear thermoelasticity for inhomogeneous and anisotropic materials, where the existence of a unique global solution and asymptotic stability of the system were proved. The existence of solution and asymptotic behavior to thermoelastic systems has been investigated extensively in the literature. For example, Muñoz Rivera [2] showed that the energy of the linear thermoelastic system (on cylindrical domain) decays to zero exponentially as . In [3], Burns et al. proved the energy decay for a linear thermoelastic bar. The asymptotic behaviour of a semigroup of the thermoelasticity was established in [4]. Concerning nonlinear thermoelasticity we can cite [5–7].

In the last two decades, several well-known evolution partial differential equations were extended to domains with moving boundary, which is also called noncylindrical problems. See, for instance, [8–10] and the references therein. In this work we studied the linear thermoelastic system in a noncylindrical domain with Dirichlet boundary conditions. This problem was early considered by Caldas et al. [11], which concluded that the energy associated to the system decreases inversely proportional to the growth of the functions that describes the noncylindrical domain. However they did not establish a rate of decay. The goal in the present work is to provide a uniform rate of decay for this noncylindrical problem.

Let us consider noncylindrical domains of the form with lateral boundary where is a given function. Then our problem is with initial conditions and boundary conditions where , , and are positive real constants.

The function and the constants , and satisfy the following conditions.

() and

There exists a positive constant such that

Problem (1.3)–(1.6) is slightly different from the one of [11] with respect to condition (1.6). Indeed, they assumed that , for all . Because of this mixed boundary condition in (1.6), we are able to construct a suitable Liapunov functional to derive decay rates of the energy. This is sufficient to provide a uniform rate of decay for this noncylindrical problem.

The existence and uniqueness of global solutions are derived by the arguments of [11] step by step, that is, to prove that the result of existence and uniqueness is based on transforming the system (1.3)–(1.6) into another initial boundary-value problem defined over a cylindrical domain whose sections are not time-dependent. This is done using a suitable change of variable. Then to show the existence and uniqueness for this equivalent system using Galerkin Methods and the existence result on noncylindrical domains will follows using the inverse of the transformation.

Therefore, we have the following result.

Theorem 1.1. Let and be the intervals , , and , respectively. Then, given and , there exist unique functions satisfying the following conditions: which are solutions of (1.3)–(1.6) in .

2. Energy Decay

In [11] the authors proved that the energy associated with (1.3)–(1.6) decays at the rate with ; that is, the energy is decreasing inverserly proportional to the increase of sections of . We make a slightly difference from the one of [11] with respect to the hypotheses about ; we are able to construct a suitable Liapunov functional to derive decay rates of the energy. This is done with the thermal dissipation only. More specifically, in this section we prove that the energy associated with (1.3)–(1.6) decays exponentially. Instead considering an auxiliary problem, we work directly on the original problem (1.3)-(1.4) in its noncylindrical domain.

In order to decay rates of the energy let us suppose the following hypotheses.

There exist positive constants and such that

There exists a positive constant such that

Let us introduce the energy functional

Our main result is the following.

Theorem 2.1. Under the hypotheses (H1)–(H4), there exist positive constants and such that

The proof of Theorem 2.1 is given by using multipliers techniques. The notations and function spaces used here are standard and can be found, for instance, in the book by Lions [8].

Lemma 2.2. Let be solution of (1.3)–(1.5) given by Theorem 1.1; then one obtains where .

Proof. From hypothesis it follows that Multiplying (1.3) by , integrating in the variable , and from (2.6) we obtain Now, applying integration by parts and using (2.6) it follows that Thus, from inequalities (2.7) and (2.8) we have Multiplying (1.4) by and integrating in the variable and using (2.6) we obtain Multiplying (2.10) by and summing with (2.9) it follows that Thus, following the hypothesis (H3), which concludes the demonstration.

To estimate the term of the energy we use the following lemma.

Lemma 2.3. With the same hypothesis of Lemma 2.2, one gets where is Poincare's constant.

Proof. From the outline condition follows that Replacing in the derivative above we get Applying Cauchy-Schwartz's inequality, Young's inequality, and Poincare's inequality in (2.15) we have Therefore our conclusion follows.

To estimate the term of the energy we introduce the function . By these conditions we have the following lemma.

Lemma 2.4. With the same hypothesis of Lemma 2.2, there are positive constants and such that where and .

Proof. Calculate the derivative From (1.3) and recording that , we get As and we obtain Now, integrating (1.4) from to , multiplying by , and after integrating from to , it follows that Replacing in and from (2.6) we get Estimating some terms of (2.22) we obtain Applying Poincare's inequality in the first term of the previous inequality, using the hypothesis (H3), and grouping the common terms, we obtain From hypothesis (H4) we have where and are positive constants. This concludes the demonstration of the lemma.

Now we use the above auxiliary lemmas to conclude the proof of Theorem 2.1.

Proof of Theorem 2.1. Consider the functional From Lemmas 2.3 and 2.4 we obtain Finally we introduce the functional where will be chosen later.From Lemma 2.2 and from (2.27) it follows that Taking sufficiently large we find that there is a positive constant such that Observe that and are equivalents, that is, there exists positive constant satisfying Therefore, Now, from equivalence (2.31) it follows that where and . The proof is now complete.

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