Stabilization of some coupled hyperbolic/parabolic equations
Louis Tebou · Discrete and Continuous Dynamical Systems - B · 2010
First, we consider a coupledsystem consisting of the wave equation and the heat equation in abounded domain. The coupling involves an operator parametrized bya 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 equationsdiscussed by Lebeau and Zuazua [23], and subsequently by Albano andTataru [1]; we show that in this case, the corresponding semigroup isexponentiallystable but not analytic. Afterwards, we discuss some extensions of our results.Second, we consider partially clamped Kirchhoff thermoelastic plate withoutmechanical feedback controls, and we prove that the underlying semigroup isexponentially stable uniformly with respect to the rotational inertia. We use aconstructive frequency domain method to prove the stabilization result, and weobtain an explicit decay rate by showing that the real part of the spectrum isuniformly bounded by a negative number that depends on the parameters of thesystem other than the rotational inertia; our approach is an alternative to theenergy method applied by Avalos and Lasiecka [6].