The wave equation with acoustic boundary conditions on non-locally reacting surfaces

Abstract

The aim of the paper is to study the problem $u_{tt}-c^2\Delta u=0$ in $\mathbb{R}\times\Omega$, $\mu v_{tt}- \text{div}_\Gamma (\sigma \nabla_\Gamma v)+\delta v_t+\kappa v+\rho u_t =0$ on $\mathbb{R}\times \Gamma_1$, $v_t =\partial_\nu u$ on $\mathbb{R}\times \Gamma_1$,$\partial_\nu u=0$ on $\mathbb{R}\times \Gamma_0$, $u(0,x)=u_0(x)$ and $u_t(0,x)=u_1(x)$ in $\Omega$, $v(0,x)=v_0(x)$ and $v_t(0,x)=v_1(x)$ 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 while $c$ is a positive constant. This problem have been proposed long time ago by Beale and Rosencrans, when $N=3$, $\sigma=0$, $r=\infty$, $\rho$ is constant, $\kappa,\delta\ge 0$, to model acoustic wave propagation with locally reacting boundary. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we give precise qualitative results for solutions when $\Omega$ is bounded and $r=2$, $\rho$ is constant, $\kappa,\delta\ge 0$. These results motivate a detailed discussion of the derivation of the problem in Theoretical Acoustics and the consequent proposal of adding to the model the integral condition $\int_\Omega u_t=c^2\int_{\Gamma_1}v$.