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

Abstract

The aim of the paper is to study the problem $u_{tt}-c^2Δu=0$ in $\mathbb{R}\timesΩ$, $μv_{tt}- \text{div}_Γ(σ\nabla_Γv)+δv_t+κv+ρu_t =0$ on $\mathbb{R}\times Γ_1$, $v_t =\partial_νu$ on $\mathbb{R}\times Γ_1$,$\partial_νu=0$ on $\mathbb{R}\times Γ_0$, $u(0,x)=u_0(x)$ and $u_t(0,x)=u_1(x)$ in $Ω$, $v(0,x)=v_0(x)$ and $v_t(0,x)=v_1(x)$ on $Γ_1$, where $Ω$ is a open domain of $\mathbb{R}^N$ with uniformly $C^r$ boundary ($N\ge 2$, $r\ge 1$), $Γ=\partialΩ$, $(Γ_0,Γ_1)$ is a relatively open partition of $Γ$ with $Γ_0$ (but not $Γ_1$) possibly empty. Here $\text{div}_Γ$ and $\nabla_Γ$ denote the Riemannian divergence and gradient operators on $Γ$, $ν$ is the outward normal to $Ω$, the coefficients $μ,σ,δ, κ, ρ$ are suitably regular functions on $Γ_1$ with $ρ,σ$ and $μ$ uniformly positive while $c$ is a positive constant. This problem have been proposed long time ago by Beale and Rosencrans, when $N=3$, $σ=0$, $r=\infty$, $ρ$ is constant, $κ,δ\ge 0$, 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 $Ω$ is bounded and $r=2$, $ρ$ is constant, $κ,δ\ge 0$. 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 $\int_Ωu_t=c^2\int_{Γ_1}v$.