$L_p$ theory for the linear thermoelastic plate equations in bounded and exterior domains

Abstract

The paper is concerned with linear thermoelastic plate equations in a domain $\Omega$: $$ u_{tt} + \Delta^2u+ \Delta \theta = 0 \ \text{ and }\ \theta_t - \Delta\theta - \Delta u_t = 0 \quad\text{in $\Omega\times(0, \infty)$}, $$ subject to the Dirichlet boundary condition $u|_\Gamma = D_\nu u|_\Gamma = \theta|_\Gamma = 0$ and initial condition $(u, u_t, \theta)|_{t=0} = (u_0, v_0, \theta_0) \in W^2_{p, D}(\Omega)\times L_p\times L_p$. Here, $\Omega$ is a bounded or exterior domain in ${{\mathbb R}}^n$ ($n\geq2$). We assume that the boundary $\Gamma$ of $\Omega$ is a $C^4$ hypersurface and we define $W^2_{p, D}$ by the formula $W^2_{p, D} = \{ u \in W^2_p : u|_\Gamma = D_\nu u|_\Gamma = 0\}$. We show that, for any $p \in (1, \infty)$, the associated semigroup $\{T(t)\}_{t\geq0}$ is analytic. Moreover, if $\Omega$ is bounded, then $\{T(t)\}_{t\geq0}$ is exponentially stable.