Open Access

Transmission Problem in Thermoelasticity

  • Margareth S. Alves1,
  • Jaime E. Muñoz Rivera2,
  • Mauricio Sepúlveda3Email author and
  • Octavio Vera Villagrán4
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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ4_HTML.gif
(1.4)

We denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq1_HTML.gif a point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq2_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq3_HTML.gif ) while https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq6_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq8_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq10_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq12_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq13_HTML.gif , and https://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. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq15_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq17_HTML.gif -dimensional body which is configured in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq18_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq19_HTML.gif ).

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

Domains https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq37_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ8_HTML.gif
(1.8)
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ10_HTML.gif
(1.10)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ11_HTML.gif
(1.11)
and transmission conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ12_HTML.gif
(1.12)
https://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 https://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 https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq42_HTML.gif be a domain in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq43_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq44_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq46_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq47_HTML.gif is integrable for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq49_HTML.gif is finite for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq50_HTML.gif . The norm will be written as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ15_HTML.gif
(2.1)

For a nonnegative integer https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq52_HTML.gif , we denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq53_HTML.gif the Sobolev space of functions in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq54_HTML.gif having all derivatives of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq55_HTML.gif belonging to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq56_HTML.gif . The norm in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq57_HTML.gif is given by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq58_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq59_HTML.gif with norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq61_HTML.gif with norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq62_HTML.gif . We write https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq63_HTML.gif for the space of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq64_HTML.gif -valued functions which are https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq66_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq67_HTML.gif is an interval, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq68_HTML.gif is a Banach space, and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq70_HTML.gif the set of orthogonal https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq71_HTML.gif real matrices and by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq72_HTML.gif the set of matrices in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq74_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq75_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq76_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq77_HTML.gif be arbitrary but fixed. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq78_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq79_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq80_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq81_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq82_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq83_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ16_HTML.gif
(2.2)
Then the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq85_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq86_HTML.gif , and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ17_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ18_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ19_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ20_HTML.gif
(2.6)

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

Lemma 2.2.

One supposes that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq93_HTML.gif is a radially symmetric function satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq94_HTML.gif . Then there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq95_HTML.gif such that
https://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:
https://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
https://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
    https://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
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ25_HTML.gif
    (2.11)
     
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ26_HTML.gif
(2.12)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ27_HTML.gif
(2.13)
https://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
    https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ30_HTML.gif
(2.16)
Similarly, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ31_HTML.gif
(2.17)

Throughout this paper https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ32_HTML.gif
(3.1)

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

Definition 3.1.

One says that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ33_HTML.gif
(3.2)
satisfying the identities
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ34_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ35_HTML.gif
(3.4)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq100_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq101_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq102_HTML.gif , and almost every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq103_HTML.gif such that
https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ38_HTML.gif
(3.7)
Moreover, if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ39_HTML.gif
(3.8)
verifying the boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ40_HTML.gif
(3.9)
and the transmission conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ41_HTML.gif
(3.10)
then the solution satisfies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq105_HTML.gif , denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq106_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq107_HTML.gif the projections on the subspaces
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ43_HTML.gif
(3.12)
of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq109_HTML.gif , respectively. Let us write
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ44_HTML.gif
(3.13)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq110_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq111_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ45_HTML.gif
(3.14)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ46_HTML.gif
(3.15)
with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ47_HTML.gif
(3.16)

for almost all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq112_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq113_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq114_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq115_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq117_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq120_HTML.gif .

Energy Estimates

Multiplying (3.14) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq121_HTML.gif , summing up over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq122_HTML.gif , and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq123_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ48_HTML.gif
(3.17)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ49_HTML.gif
(3.18)
Multiplying (3.15) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq124_HTML.gif , summing up over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq125_HTML.gif , and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq126_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ50_HTML.gif
(3.19)
where
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ52_HTML.gif
(3.21)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ53_HTML.gif
(3.22)
Integrating over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq127_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq128_HTML.gif , we have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ54_HTML.gif
(3.23)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ55_HTML.gif
(3.24)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ56_HTML.gif
(3.25)
In particular,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ57_HTML.gif
(3.26)
and it follows that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq130_HTML.gif , and we denote
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ59_HTML.gif
(3.28)
Taking
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ60_HTML.gif
(3.29)
we can see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq131_HTML.gif satisfies (1.1)–(1.4). Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq132_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq134_HTML.gif satisfies
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ62_HTML.gif
(3.31)
Thus https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq136_HTML.gif , (1.2) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq137_HTML.gif , (1.3) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq138_HTML.gif , and (1.4) by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq140_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ63_HTML.gif
(3.32)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq141_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq142_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq143_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq145_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ64_HTML.gif
(3.33)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ65_HTML.gif
(3.34)
Therefore, we find that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq147_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq148_HTML.gif -regular ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq149_HTML.gif ) if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ67_HTML.gif
(3.36)
where the values of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq151_HTML.gif are given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ68_HTML.gif
(3.37)
verifying the boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ69_HTML.gif
(3.38)
and the transmission conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ70_HTML.gif
(3.39)
for https://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 https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ72_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ73_HTML.gif
(4.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq155_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ74_HTML.gif
(4.3)

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

Proof.

Multiplying (1.1) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq157_HTML.gif , integrating in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ75_HTML.gif
(4.4)
Multiplying (1.2) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq159_HTML.gif , integrating in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ76_HTML.gif
(4.5)
Multiplying (1.3) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq161_HTML.gif , integrating in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ77_HTML.gif
(4.6)
Multiplying (1.4) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq163_HTML.gif , integrating in https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ79_HTML.gif
(4.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ80_HTML.gif
(4.9)
Thus
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ82_HTML.gif
(4.11)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ83_HTML.gif
(4.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq165_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq166_HTML.gif are positive constants and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq167_HTML.gif , integrating in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ85_HTML.gif
(4.14)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ86_HTML.gif
(4.15)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ87_HTML.gif
(4.16)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ88_HTML.gif
(4.17)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ89_HTML.gif
(4.18)
Therefore
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq169_HTML.gif , integrating in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ91_HTML.gif
(4.20)
Multiplying (1.3) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq171_HTML.gif and integrating in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq172_HTML.gif we have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ92_HTML.gif
(4.21)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ93_HTML.gif
(4.22)
Then
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ96_HTML.gif
(4.25)
On the other hand
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ97_HTML.gif
(4.26)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ98_HTML.gif
(4.27)
Multiplying (1.4) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq173_HTML.gif , integrating in https://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
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq175_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq176_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ102_HTML.gif
(4.31)
Therefore we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ103_HTML.gif
(4.32)
Similarly
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ105_HTML.gif
(4.34)
with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ106_HTML.gif
(4.35)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq177_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq178_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq180_HTML.gif , integrating in https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ108_HTML.gif
(4.37)
Multiplying (1.2) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq182_HTML.gif , integrating in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ109_HTML.gif
(4.38)
Multiplying (1.3) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq184_HTML.gif and integrating in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq185_HTML.gif , we have that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ111_HTML.gif
(4.40)
Multiplying (1.4) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq186_HTML.gif , integrating in https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ113_HTML.gif
(4.42)
with
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ116_HTML.gif
(4.45)
Similarly
https://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
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ119_HTML.gif
(4.48)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ120_HTML.gif
(4.49)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq188_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq189_HTML.gif is a ball with center https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq190_HTML.gif and radius https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ121_HTML.gif
(4.50)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ122_HTML.gif
(4.51)

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

Proof.

Using Lemma A.1, taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq197_HTML.gif as above, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq199_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq200_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ123_HTML.gif
(4.52)
Applying the hypothesis on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq201_HTML.gif and since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ124_HTML.gif
(4.53)
we have that
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ127_HTML.gif
(4.56)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq202_HTML.gif ; there exists a positive constant https://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
https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq204_HTML.gif -variable we have that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq205_HTML.gif and integrating in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq206_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ131_HTML.gif
(4.60)
Hence
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq207_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq208_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq209_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq210_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ133_HTML.gif
(4.62)
Multiplying (4.61) by https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ135_HTML.gif
(4.64)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq212_HTML.gif and https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ137_HTML.gif
(4.66)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq214_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq215_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq217_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq218_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ138_HTML.gif
(4.67)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ139_HTML.gif
(4.68)
Hence, taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq220_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq221_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ140_HTML.gif
(4.69)
where we have used
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ142_HTML.gif
(4.71)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ143_HTML.gif
(4.72)

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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq223_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq226_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq227_HTML.gif such that
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ147_HTML.gif
(4.76)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq228_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq229_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_Equ149_HTML.gif
(4.78)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq230_HTML.gif . Note that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F190548/MediaObjects/13661_2010_Article_27_IEq231_HTML.gif large enough we have that
https://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

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© 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.