Transmission Problem in Thermoelasticity

  • Margareth S. Alves1,

    Affiliated with

    • Jaime E. Muñoz Rivera2,

      Affiliated with

      • Mauricio Sepúlveda3Email author and

        Affiliated with

        • Octavio Vera Villagrán4

          Affiliated with

          Boundary Value Problems20112011:190548

          DOI: 10.1155/2011/190548

          Received: 24 November 2010

          Accepted: 17 February 2011

          Published: 9 March 2011

          Abstract

          We show that the energy to the thermoelastic transmission problem decays exponentially as time goes to infinity. We also prove the existence, uniqueness, and regularity of the solution to the system.

          1. Introduction

          In this paper we deal with the theory of thermoelasticity. We consider the following transmission problem between two thermoelastic materials:
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ1_HTML.gif
          (1.1)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ2_HTML.gif
          (1.2)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ3_HTML.gif
          (1.3)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ4_HTML.gif
          (1.4)

          We denote by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq1_HTML.gif a point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq2_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq3_HTML.gif ) while http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq4_HTML.gif stands for the time variable. The displacement in the thermoelasticity parts is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq6_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq8_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq9_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq10_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq12_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq13_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq14_HTML.gif is the variation of temperature between the actual state and a reference temperature, respectively. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq15_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq16_HTML.gif are the thermal conductivity. All the constants of the system are positive. Let us consider an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq17_HTML.gif -dimensional body which is configured in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq18_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq19_HTML.gif ).

          The thermoelastic parts are given by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq21_HTML.gif , respectively. The constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq22_HTML.gif are the coupling parameters depending on the material properties. The boundary of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq23_HTML.gif is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq24_HTML.gif and the boundary of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq25_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq26_HTML.gif . We will consider the boundaries http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq27_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq28_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq29_HTML.gif in the rest of this paper. The thermoelastic parts are given by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq31_HTML.gif , respectively, that is (see Figure 1),
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ5_HTML.gif
          (1.5)
          We consider for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq32_HTML.gif the operators
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ6_HTML.gif
          (1.6)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ7_HTML.gif
          (1.7)
          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq34_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq35_HTML.gif ) are the Lamé moduli satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq36_HTML.gif .
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Fig1_HTML.jpg
          Figure 1

          Domains http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq38_HTML.gif and boundaries of the transmission problem.

          The initial conditions are given by
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ8_HTML.gif
          (1.8)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ9_HTML.gif
          (1.9)
          The system is subject to the following boundary conditions:
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ10_HTML.gif
          (1.10)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ11_HTML.gif
          (1.11)
          and transmission conditions
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ12_HTML.gif
          (1.12)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ13_HTML.gif
          (1.13)

          The transmission conditions are imposed, that express the continuity of the medium and the equilibrium of the forces acting on it. The discontinuity of the coefficients of the equations corresponds to the fact that the medium consists of two physically different materials.

          Since the domain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq39_HTML.gif is composed of two different materials, its density is not necessarily a continuous function, and since the stress-strain relation changes from the thermoelastic parts, the corresponding model is not continuous. Taking in consideration this, the mathematical problem that deals with this type of situation is called a transmission problem. From a mathematical point of view, the transmission problem is described by a system of partial differential equations with discontinuous coefficients. The model (1.1)–(1.13) to consider is interesting because we deal with composite materials. From the economical and the strategic point of view, materials are mixed with others in order to get another more convenient material for industry (see [13] and references therein). Our purpose in this work is to investigate that the solution of the symmetrical transmission problem decays exponentially as time tends to infinity, no matter how small is the size of the thermoelastic parts. The transmission problem has been of interest to many authors, for instance, in the one-dimensional thermoelastic composite case, we can refer to the papers [47]. In the two-, three- or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq40_HTML.gif -dimensional, we refer the reader to the papers [8, 9] and references therein. The method used here is based on energy estimates applied to nonlinear problems, and the differential inequality is obtained by exploiting the symmetry of the solutions and applying techniques for the elastic wave equations, which solve the exponential stability produced by the boundary terms in the interface of the material. This methods allow us to find a Lyapunov functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq41_HTML.gif equivalent to the second-order energy for which we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ14_HTML.gif
          (1.14)

          In spite of the obvious importance of the subject in applications, there are relatively few mathematical results about general transmission problem for composite materials. For this reason we study this topic here.

          This paper is organized as follows. Before describing the main results, in Section 2, we briefly outline the notation and terminology to be used later on and we present some lemmas. In Section 3 we prove the existence and regularity of radially symmetric solutions to the transmission problem. In Section 4 we show the exponential decay of the solutions and we prove the main theorem.

          2. Preliminaries

          We will use the following standard notation. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq42_HTML.gif be a domain in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq43_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq45_HTML.gif are all real valued measurable functions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq46_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq47_HTML.gif is integrable for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq49_HTML.gif is finite for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq50_HTML.gif . The norm will be written as
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ15_HTML.gif
          (2.1)

          For a nonnegative integer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq52_HTML.gif , we denote by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq53_HTML.gif the Sobolev space of functions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq54_HTML.gif having all derivatives of order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq55_HTML.gif belonging to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq56_HTML.gif . The norm in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq57_HTML.gif is given by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq58_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq59_HTML.gif with norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq61_HTML.gif with norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq62_HTML.gif . We write http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq63_HTML.gif for the space of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq64_HTML.gif -valued functions which are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq65_HTML.gif -times continuously differentiable (resp. square integrable) in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq66_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq67_HTML.gif is an interval, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq68_HTML.gif is a Banach space, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq69_HTML.gif is a nonnegative integer. We denote by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq70_HTML.gif the set of orthogonal http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq71_HTML.gif real matrices and by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq72_HTML.gif the set of matrices in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq73_HTML.gif which have determinant 1.

          The following results are going to be used several times from now on. The proof can be found in [10].

          Lemma 2.1.

          Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq74_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq75_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq76_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq77_HTML.gif be arbitrary but fixed. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq78_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq79_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq80_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq81_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq82_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq83_HTML.gif satisfy
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ16_HTML.gif
          (2.2)
          Then the solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq86_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq87_HTML.gif of (1.1)–(1.13) has the form
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ17_HTML.gif
          (2.3)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ18_HTML.gif
          (2.4)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ19_HTML.gif
          (2.5)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ20_HTML.gif
          (2.6)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq88_HTML.gif , for some functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq89_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq91_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq92_HTML.gif .

          Lemma 2.2.

          One supposes that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq93_HTML.gif is a radially symmetric function satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq94_HTML.gif . Then there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq95_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ21_HTML.gif
          (2.7)
          Moreover one has the following estimate at the boundary:
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ22_HTML.gif
          (2.8)

          Remark 2.3.

          From (2.3) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ23_HTML.gif
          (2.9)
          The following straightforward calculations are going to be used several times from now on.
          1. (a)
            From (1.8) we obtain
            http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ24_HTML.gif
            (2.10)
             
          1. (b)
            Using (1.10) and (1.11) we have that
            http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ25_HTML.gif
            (2.11)
             
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ26_HTML.gif
          (2.12)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ27_HTML.gif
          (2.13)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ28_HTML.gif
          (2.14)
          1. (c)
            Using (1.6) we have that
            http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ29_HTML.gif
            (2.15)
             
          Thus, using (1.10) and (1.11) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ30_HTML.gif
          (2.16)
          Similarly, we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ31_HTML.gif
          (2.17)

          Throughout this paper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq96_HTML.gif is a generic constant, not necessarily the same at each occasion (it will change from line to line), which depends in an increasing way on the indicated quantities.

          3. Existence and Uniqueness

          In this section we establish the existence and uniqueness of solutions to the system (1.1)–(1.13). The proof is based using the standard Galerkin approximation and the elliptic regularity for transmission problem given in [11]. First of all, we define what we will understand for weak solution of the problem (1.1)–(1.13).

          We introduce the following spaces:
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ32_HTML.gif
          (3.1)

          for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq98_HTML.gif .

          Definition 3.1.

          One says that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq99_HTML.gif is a weak solution of (1.1)–(1.13) if
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ33_HTML.gif
          (3.2)
          satisfying the identities
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ34_HTML.gif
          (3.3)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ35_HTML.gif
          (3.4)
          for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq100_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq101_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq102_HTML.gif , and almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq103_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ36_HTML.gif
          (3.5)

          The existence of solutions to the system (1.1)–(1.13) is given in the following theorem.

          Theorem 3.2.

          One considers the following initial data satisfying
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ37_HTML.gif
          (3.6)
          Then there exists only one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq104_HTML.gif of the system (1.1)–(1.13) satisfying
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ38_HTML.gif
          (3.7)
          Moreover, if
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ39_HTML.gif
          (3.8)
          verifying the boundary conditions
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ40_HTML.gif
          (3.9)
          and the transmission conditions
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ41_HTML.gif
          (3.10)
          then the solution satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ42_HTML.gif
          (3.11)

          Proof.

          The existence of solutions follows using the standard Galerking approximation.

          Faedo-Galerkin Scheme

          Given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq105_HTML.gif , denote by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq107_HTML.gif the projections on the subspaces
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ43_HTML.gif
          (3.12)
          of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq109_HTML.gif , respectively. Let us write
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ44_HTML.gif
          (3.13)
          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq110_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq111_HTML.gif satisfy
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ45_HTML.gif
          (3.14)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ46_HTML.gif
          (3.15)
          with
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ47_HTML.gif
          (3.16)

          for almost all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq112_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq113_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq114_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq115_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq116_HTML.gif are the zero vectors in the respective spaces. Recasting exactly the classical Faedo-Galerkin scheme, we get a system of ordinary differential equations in the variables http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq118_HTML.gif . According to the standard existence theory for ordinary differential equations there exists a continuous solution of this system, on some interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq119_HTML.gif . The a priori estimates that follow imply that in fact http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq120_HTML.gif .

          Energy Estimates

          Multiplying (3.14) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq121_HTML.gif , summing up over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq122_HTML.gif , and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq123_HTML.gif we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ48_HTML.gif
          (3.17)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ49_HTML.gif
          (3.18)
          Multiplying (3.15) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq124_HTML.gif , summing up over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq125_HTML.gif , and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq126_HTML.gif we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ50_HTML.gif
          (3.19)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ51_HTML.gif
          (3.20)
          Adding (3.17) with (3.19) we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ52_HTML.gif
          (3.21)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ53_HTML.gif
          (3.22)
          Integrating over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq127_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq128_HTML.gif , we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ54_HTML.gif
          (3.23)
          Thus,
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ55_HTML.gif
          (3.24)
          Hence,
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ56_HTML.gif
          (3.25)
          In particular,
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ57_HTML.gif
          (3.26)
          and it follows that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ58_HTML.gif
          (3.27)

          The system (1.1)–(1.4) is a linear system, and hence the rest of the proof of the existence of weak solution is a standard matter.

          The uniqueness follows using the elliptic regularity for the elliptic transmission problem (see [11]).We suppose that there exist two solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq130_HTML.gif , and we denote
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ59_HTML.gif
          (3.28)
          Taking
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ60_HTML.gif
          (3.29)
          we can see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq131_HTML.gif satisfies (1.1)–(1.4). Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq133_HTML.gif are weak solutions of the system we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq134_HTML.gif satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ61_HTML.gif
          (3.30)
          Using the elliptic regularity for the elliptic transmission problem we conclude that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ62_HTML.gif
          (3.31)
          Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq135_HTML.gif satisfies (1.1)–(1.4) in the strong sense. Multiplying (1.1) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq136_HTML.gif , (1.2) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq137_HTML.gif , (1.3) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq138_HTML.gif , and (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq139_HTML.gif and performing similar calculations as above we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq140_HTML.gif , where
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ63_HTML.gif
          (3.32)

          which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq141_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq142_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq143_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq144_HTML.gif . The uniqueness follows.

          To obtain more regularity, we differentiate the approximate system (1.1)–(1.4); then multiplying the resulting system by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq146_HTML.gif and performing similar calculations as in (3.23) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ64_HTML.gif
          (3.33)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ65_HTML.gif
          (3.34)
          Therefore, we find that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ66_HTML.gif
          (3.35)

          Finally, our conclusion will follow by using the regularity result for the elliptic transmission problem (see [11]).

          Remark 3.3.

          To obtain higher regularity we introduce the following definition.

          Definition 3.4.

          One will say that the initial data http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq147_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq148_HTML.gif -regular ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq149_HTML.gif ) if
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ67_HTML.gif
          (3.36)
          where the values of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq151_HTML.gif are given by
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ68_HTML.gif
          (3.37)
          verifying the boundary conditions
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ69_HTML.gif
          (3.38)
          and the transmission conditions
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ70_HTML.gif
          (3.39)
          for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq152_HTML.gif . Using the above notation we say that if the initial data is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq153_HTML.gif -regular, then we have that the solution satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ71_HTML.gif
          (3.40)

          Using the same arguments as in Theorem 3.2, the result follows.

          4. Exponential Stability

          In this section we prove the exponential stability. The great difficulty here is to deal with the boundary terms in the interface of the material. This difficulty is solved using an observability result of the elastic wave equations together with the fact that the solution is radially symmetric.

          Lemma 4.1.

          Let one suppose that the initial data http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq154_HTML.gif is 3-regular; then the corresponding solution of the system (1.1)–(1.13) satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ72_HTML.gif
          (4.1)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ73_HTML.gif
          (4.2)
          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq155_HTML.gif with
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ74_HTML.gif
          (4.3)

          and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq156_HTML.gif .

          Proof.

          Multiplying (1.1) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq157_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq158_HTML.gif , and using (2.16) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ75_HTML.gif
          (4.4)
          Multiplying (1.2) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq159_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq160_HTML.gif , and using (2.17) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ76_HTML.gif
          (4.5)
          Multiplying (1.3) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq161_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq162_HTML.gif , and using (2.11) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ77_HTML.gif
          (4.6)
          Multiplying (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq163_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq164_HTML.gif , using (2.11), and performing similar calculations as above we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ78_HTML.gif
          (4.7)
          Adding up (4.4), (4.5), (4.6), and (4.7) and using (1.12) and (1.13) we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ79_HTML.gif
          (4.8)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ80_HTML.gif
          (4.9)
          Thus
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ81_HTML.gif
          (4.10)

          In a similar way we obtain (4.2).

          Lemma 4.2.

          Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.11) satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ82_HTML.gif
          (4.11)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ83_HTML.gif
          (4.12)
          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq165_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq166_HTML.gif are positive constants and
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ84_HTML.gif
          (4.13)

          Proof.

          Multiplying (1.1) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq167_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq168_HTML.gif , and using (1.10) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ85_HTML.gif
          (4.14)
          Then
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ86_HTML.gif
          (4.15)
          Hence
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ87_HTML.gif
          (4.16)
          Thus
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ88_HTML.gif
          (4.17)
          Hence
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ89_HTML.gif
          (4.18)
          Therefore
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ90_HTML.gif
          (4.19)
          Similarly, multiplying (1.2) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq169_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq170_HTML.gif , and performing similar calculations as above we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ91_HTML.gif
          (4.20)
          Multiplying (1.3) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq171_HTML.gif and integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq172_HTML.gif we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ92_HTML.gif
          (4.21)
          Hence
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ93_HTML.gif
          (4.22)
          Then
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ94_HTML.gif
          (4.23)
          Using (1.10) and (2.9) and performing similar calculations as above we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ95_HTML.gif
          (4.24)
          Replacing (1.1) in the above equation we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ96_HTML.gif
          (4.25)
          On the other hand
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ97_HTML.gif
          (4.26)
          Therefore
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ98_HTML.gif
          (4.27)
          Multiplying (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq173_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq174_HTML.gif , and performing similar calculations as above we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ99_HTML.gif
          (4.28)
          Adding (4.19) with (4.27) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ100_HTML.gif
          (4.29)
          Adding (4.20) with (4.28) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ101_HTML.gif
          (4.30)
          Moreover, by Lemma 2.2, there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq175_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq176_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ102_HTML.gif
          (4.31)
          Therefore we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ103_HTML.gif
          (4.32)
          Similarly
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ104_HTML.gif
          (4.33)

          The result follows.

          Lemma 4.3.

          Under the same hypotheses of Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ105_HTML.gif
          (4.34)
          with
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ106_HTML.gif
          (4.35)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq177_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq178_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq179_HTML.gif are positive constants.

          Proof.

          Multiplying (1.1) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq180_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq181_HTML.gif , using (2.9), and performing straightforward calculations we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ107_HTML.gif
          (4.36)
          Using (1.10) we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ108_HTML.gif
          (4.37)
          Multiplying (1.2) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq182_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq183_HTML.gif , and performing similar calculations as above we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ109_HTML.gif
          (4.38)
          Multiplying (1.3) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq184_HTML.gif and integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq185_HTML.gif , we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ110_HTML.gif
          (4.39)
          Performing similar calculations as above we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ111_HTML.gif
          (4.40)
          Multiplying (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq186_HTML.gif , integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq187_HTML.gif , and performing similar calculation as above we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ112_HTML.gif
          (4.41)
          Adding (4.37), (4.38), (4.40), and (4.41), using (1.13), and performing straightforward calculations we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ113_HTML.gif
          (4.42)
          with
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ114_HTML.gif
          (4.43)
          Using the Cauchy inequality we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ115_HTML.gif
          (4.44)
          and, from trace and interpolation inequalities, we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ116_HTML.gif
          (4.45)
          Similarly
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ117_HTML.gif
          (4.46)
          Replacing in the above equation we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ118_HTML.gif
          (4.47)

          The result follows.

          We introduce the following integrals:
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ119_HTML.gif
          (4.48)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ120_HTML.gif
          (4.49)

          with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq188_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq189_HTML.gif is a ball with center http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq190_HTML.gif and radius http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq191_HTML.gif .

          Lemma 4.4.

          Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ121_HTML.gif
          (4.50)
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ122_HTML.gif
          (4.51)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq192_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq193_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq194_HTML.gif are positive constants and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq195_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq196_HTML.gif .

          Proof.

          Using Lemma A.1, taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq197_HTML.gif as above, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq198_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq199_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq200_HTML.gif , we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ123_HTML.gif
          (4.52)
          Applying the hypothesis on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq201_HTML.gif and since
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ124_HTML.gif
          (4.53)
          we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ125_HTML.gif
          (4.54)
          Using (2.8) and the Cauchy-Schwartz inequality in the last term and performing straightforward calculations we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ126_HTML.gif
          (4.55)
          Finally, considering (1.1) and applying the trace theorem we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ127_HTML.gif
          (4.56)

          with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq202_HTML.gif ; there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq203_HTML.gif which proves (4.51).

          We now introduce the integrals
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ128_HTML.gif
          (4.57)

          Lemma 4.5.

          With the same hypotheses as in Lemma 4.1, the following equality holds:
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ129_HTML.gif
          (4.58)

          Proof.

          Differentiating (1.2) in the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq204_HTML.gif -variable we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ130_HTML.gif
          (4.59)
          Multiplying the above equation by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq205_HTML.gif and integrating in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq206_HTML.gif we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ131_HTML.gif
          (4.60)
          Hence
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ132_HTML.gif
          (4.61)
          On the other hand, using Lemma A.1 for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq207_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq208_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq209_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq210_HTML.gif we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ133_HTML.gif
          (4.62)
          Multiplying (4.61) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq211_HTML.gif and adding with (4.62) we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ134_HTML.gif
          (4.63)

          The result follows.

          We introduce the integral
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ135_HTML.gif
          (4.64)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq212_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq213_HTML.gif are positive constants.

          Lemma 4.6.

          Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ136_HTML.gif
          (4.65)

          Proof.

          From (4.11), (4.12), and (4.34), using the Cauchy-Schwartz inequality and performing straightforward calculations we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ137_HTML.gif
          (4.66)
          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq214_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq215_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq216_HTML.gif are positive constants. By Lemma 2.2, there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq217_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq218_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ138_HTML.gif
          (4.67)
          Then
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ139_HTML.gif
          (4.68)
          Hence, taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq219_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq220_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq221_HTML.gif we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ140_HTML.gif
          (4.69)
          where we have used
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ141_HTML.gif
          (4.70)
          Using (1.10), we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ142_HTML.gif
          (4.71)
          Thus
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ143_HTML.gif
          (4.72)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq222_HTML.gif .

          We define the functional
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ144_HTML.gif
          (4.73)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq223_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq224_HTML.gif are positive constants.

          Theorem 4.7.

          Let us suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq225_HTML.gif is a strong solution of the system (1.1)–(1.13). Then there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq226_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq227_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ145_HTML.gif
          (4.74)

          Proof.

          We will assume that the initial data is 3-regular. The conclusion will follow by standard density arguments. Using Lemmas 4.3 and 4.5 and considering boundary conditions, we find that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ146_HTML.gif
          (4.75)
          From (4.1), (4.2), and (4.75) we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ147_HTML.gif
          (4.76)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ148_HTML.gif
          (4.77)
          Using the Cauchy inequality, we see that there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq228_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq229_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ149_HTML.gif
          (4.78)
          Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq230_HTML.gif . Note that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq231_HTML.gif large enough we have that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ150_HTML.gif
          (4.79)

          From the above two inequalities our conclusion follows.

          Declarations

          Acknowledgments

          This work was done while the third author was visiting the Federal University of Viçosa. Viçosa, MG, Brazil and the National Laboratory for Scientific Computation (LNCC/MCT). This research was partially supported by PROSUL Project. Additionally, it has been supported by Fondecyt project no. 1110540, FONDAP and BASAL projects CMM, Universidad de Chile, and CI2MA, Universidad de Concepción.

          Authors’ Affiliations

          (1)
          Departamento de Matemática, Universidade Federal de Viçosa (UFV)
          (2)
          National Laboratory for Scientific Computation
          (3)
          CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción
          (4)
          Departamento de Matemática, Universidad del Bío-Bío

          References

          1. Balmès E, Germès S: Tools for viscoelastic damping treatment design: application to an automotive floor panel. Proceedings of the 28th International Seminar on Modal Analysis (ISMA '02), 2002, Leuven, Belgium
          2. Oh K: Theoretical and experimental study of modal interactions in metallic and lamined com- posite plates, Ph.D. thesis. Virginia Polytechnic Institute and State Unversity, Blacksburg, Va, USA; 1994.
          3. Rao MD: Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. Journal of Sound and Vibration 2003, 262(3):457-474. 10.1016/S0022-460X(03)00106-8View Article
          4. Alves MS, Raposo CA, Muñoz Rivera JE, Sepúlveda M, Villagrán OV: Uniform stabilization for the transmission problem of the Timoshenko system with memory. Journal of Mathematical Analysis and Applications 2010, 369(1):323-345. 10.1016/j.jmaa.2010.02.045View ArticleMathSciNetMATH
          5. Fatori LH, Lueders E, Muñoz Rivera JE: Transmission problem for hyperbolic thermoelastic systems. Journal of Thermal Stresses 2003, 26(7):739-763. 10.1080/713855994View ArticleMathSciNet
          6. Marzocchi A, Muñoz Rivera JE, Naso MG: Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity. Mathematical Methods in the Applied Sciences 2002, 25(11):955-980. 10.1002/mma.323View ArticleMathSciNetMATH
          7. Muñoz Rivera JE, Portillo Oquendo H: The transmission problem for thermoelastic beams. Journal of Thermal Stresses 2001, 24(12):1137-1158. 10.1080/014957301753251665View ArticleMathSciNet
          8. Lebeau G, Zuazua E: Decay rates for the three-dimensional linear system of thermoelasticity. Archive for Rational Mechanics and Analysis 1999, 148(3):179-231. 10.1007/s002050050160View ArticleMathSciNetMATH
          9. Muñoz Rivera JE, Naso MG: About asymptotic behavior for a transmission problem in hyperbolic thermoelasticity. Acta Applicandae Mathematicae 2007, 99(1):1-27. 10.1007/s10440-007-9152-8View ArticleMathSciNetMATH
          10. Marzocchi A, Muñoz Rivera JE, Naso MG: Transmission problem in thermoelasticity with symmetry. IMA Journal of Applied Mathematics 2003, 68(1):23-46. 10.1093/imamat/68.1.23View ArticleMathSciNetMATH
          11. Ladyzhenskaya OA, Ural'tseva NN: Linear and Quasilinear Elliptic Equations. Academic Press, New York, NY, USA; 1968:xviii+495.MATH

          Copyright

          © Margareth S. Alves et al. 2011

          This article is published under license to BioMed Central Ltd. 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.