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

The aim of the paper is to study the problem ⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪utt+dut−c2Δu=0μvtt−divΓ(σ∇Γv)+δvt+κv+ρut=0vt=∂νu∂νu=0u(0,x)=u0(x),ut(0,x)=u1(x)v(0,x)=v0(x),vt(0,x)=v1(x)inR×Ω,onR×Γ1,onR×Γ1,onR×Γ0,inΩ,onΓ1, 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, d is a suitably regular function in Ω 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 Ω is bounded, Γ1 is connected, r=2, ρ is constant and κ,δ,d≥0.