The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces

Abstract

The aim of the paper is to study the problem $$u_{tt}+du_t-c^2\Delta u=0 \qquad \text{in $\mathbb{R}\times\Omega$,}$$ $$\mu v_{tt}- \text{div}_\Gamma (\sigma \nabla_\Gamma v)+\delta v_t+\kappa v+\rho u_t =0\qquad \text{on $\mathbb{R}\times \Gamma_1$,}$$ $$v_t =\partial_\nu u\qquad \text{on $\mathbb{R}\times \Gamma_1$,}$$ $$\partial_\nu u=0 \text{on $\mathbb{R}\times \Gamma_0$,}$$ $$u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x)\quad \text{in $\Omega$,}$$ $$v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) \quad \text{on $\Gamma_1$,}$$ where $\Omega$ is a open domain of $\mathbb{R}^N$ with uniformly $C^r$ boundary ($N\ge 2$, $r\ge 1$), $\Gamma=\partial\Omega$, $(\Gamma_0,\Gamma_1)$ is a relatively open partition of $\Gamma$ with $\Gamma_0$ (but not $\Gamma_1$) possibly empty. Here $\text{div}_\Gamma$ and $\nabla_\Gamma$ denote the Riemannian divergence and gradient operators on $\Gamma$, $\nu$ is the outward normal to $\Omega$, the coefficients $\mu,\sigma,\delta, \kappa, \rho$ are suitably regular functions on $\Gamma_1$ with $\rho,\sigma$ and $\mu$ uniformly positive, $d$ is a suitably regular function in $\Omega$ and $c$ is a positive constant. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when $\Omega$ is bounded, $\Gamma_1$ is connected, $r=2$, $\rho$ is constant and $\kappa,\delta,d\ge 0$.