Remarks on exponential stability for a coupled system of elasticity and thermoelasticity with second sound
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Journal of Evolution Equations
Remarks on exponential stability for a coupled system of elasticity and thermoelasticity with second sound Manuel Rissel
and Ya- Guang Wang
Abstract. We study the large time behavior of solutions to a linear transmission problem in one space dimension. The problem at hand models a thermoelastic material with second sound confined by a purely elastic one. We shall characterize all equilibrium states of the considered system and prove that every solution approaches one designated equilibrium state with an exponential rate as time goes to infinity. Hereto, we apply methods from the theory of strongly continuous semigroups. In particular, we obtain uniform resolvent bounds for the underlying generator. This removes the largeness assumption of elastic wave speeds imposed in Meng and Wang (Anal Appl (Singap) 13, 2015) for having an exponential energy decay rate when the problem only has the trivial equilibrium. In an appendix, we provide a similar exponential stability result for the case where heat conduction is modeled using Fourier’s law.
1. Introduction Many interesting applications give rise to transmission problems, which are systems of differential equations with discontinuous coefficients and transmission conditions imposed on some interfaces. In this note, we are concerned with a transmission problem for a coupled system of elasticity and thermoelasticity, as illustrated in Fig. 1. In this work, the heat conduction in thermoelasticity obeys the Cattaneo law, which transforms the classical thermoelastic system of hyperbolic-parabolic type into the thermoelastic system with second sound, a strictly hyperbolic one, cf. [19]. We aim to develop the linear semigroup theory to re-study the long time stability analysis of this problem initiated in [13] via the Lyapunov argument. In particular, by the means of uniform resolvent bounds, we shall show that, as time goes to infinity, every solution converges with an exponential rate to a stationary state of the system. While our motivation stems from the work [13], where only the model with Cattaneo’s law is considered, it seems to us that the same setting but with classical Fourier’s law for heat conduction has not been studied, in particular in the semigroup context, as well. For this reason, in the appendix, we also provide details on how to obtain exponential stability when Fourier’s law is employed and the thermoelastic part of the bar is modeled in the classical way. The one-dimensional systems investigated here serve as toy models for the curl-free Mathematics Subject Classification: 35B35, 35B40, 35M33, 47D06 Keywords: Transmission problem, Thermoelasticity, Elasticity, Second sound, Exponential stability.
J. Evol. Equ.
M. Rissel and Y.- G. Wang
E
TE
E
L1
0
L2
L3
Figure 1. Illustration of an elastic(E)-theormoelastic(TE)-elastic bar part of considerably more complicated vectorial cases and are also of interest from the perspective of control theory. The dynamics of the elastic-thermoelastic-elastic bar under consideration
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