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 ${\int }_{\Omega }{u}_t=c^2{\int }_{{\Gamma }_1}v$.

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