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

The aim of the paper is to study the problem utt−c2Δu=0 in R×Ω, μvtt−divΓ(σ∇Γv)+δvt+κv+ρut=0 on R×Γ1, vt=∂νu on R×Γ1,∂νu=0 on R×Γ0, u(0,x)=u0(x) and ut(0,x)=u1(x) in Ω, v(0,x)=v0(x) and vt(0,x)=v1(x) on Γ1, where Ω is a open domain of RN with uniformly Cr boundary (N≥2, r≥1), Γ=∂Ω, (Γ0,Γ1) is a relatively open partition of Γ with Γ0 (but not Γ1) possibly empty. Here divΓ and ∇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 Γ1 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=∞, ρ is constant, κ,δ≥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≥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 ∫Ωut=c2∫Γ1v.