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The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces

The aim of the paper is to study the problem $$u_{tt}+du_t-c^2Δu=0 \qquad \text{in $\mathbb{R}\timesΩ$,}$$ $$μv_{tt}- \text{div}_Γ(σ\nabla_Γv)+δv_t+κv+ρu_t =0\qquad \text{on $\mathbb{R}\times Γ_1$,}$$ $$v_t =\partial_νu\qquad \text{on $\mathbb{R}\times Γ_1$,}$$ $$\partial_νu=0 \text{on $\mathbb{R}\times Γ_0$,}$$ $$u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x)\quad \text{in $Ω$,}$$ $$v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) \quad \text{on $Γ_1$,}$$ 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, $d$ 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, $Γ_1$ is connected, $r=2$, $ρ$ is constant and $κ,δ,d\ge 0$.