Stabilization of some coupled hyperbolic/parabolic equations

Abstract

First, we consider a coupled system consisting of the wave equation and the heat equation in a bounded domain. The coupling involves an operator parametrized by a real number $\mu$ lying in the interval [0,1]. We show that for $0\leq\mu<1$, the associated semigroup is not uniformly stable. Then we propose an explicit non-uniform decay rate. For $\mu=1$, the coupled system reduces to the thermoelasticity equations discussed by Lebeau and Zuazua [23], and subsequently by Albano and Tataru [1]; we show that in this case, the corresponding semigroup is exponentially stable but not analytic. Afterwards, we discuss some extensions of our results. Second, we consider partially clamped Kirchhoff thermoelastic plate without mechanical feedback controls, and we prove that the underlying semigroup is exponentially stable uniformly with respect to the rotational inertia. We use a constructive frequency domain method to prove the stabilization result, and we obtain an explicit decay rate by showing that the real part of the spectrum is uniformly bounded by a negative number that depends on the parameters of the system other than the rotational inertia; our approach is an alternative to the energy method applied by Avalos and Lasiecka [6].