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

Abstract

The aim of the paper is to study the problem $$u_{tt}+d{u}_t-c^2\Delta u=0\text{in R×Ω,}$$ $$\mu v_{tt}-{\text{div}}_{\Gamma }(\sigma {\nabla }_{\Gamma }v)+\delta v_t+\kappa v+\rho u_t=0\text{on R×Γ1,}$$ $$v_t={\partial }_{\nu }u\text{on R×Γ1,}$$ $${\partial }_{\nu }u=0\text{on R×Γ0,}$$ $$u(0,x)=u_0(x),u_t(0,x)=u_1(x)\text{in Ω,}$$ $$v(0,x)=v_0(x),v_t(0,x)=v_1(x)\text{on Γ1,}$$ 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, $d$ is a suitably regular function in $\Omega$ 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 $\Omega$ is bounded, ${\Gamma }_1$ is connected, $r=2$, $\rho$ is constant and $\kappa ,\delta ,d\ge 0$.

The paper also extends some results in arXiv:2105.09219

References in corpus (1)

• General enegy decay for a viscoelastic equation of Kirchhoff type with acoustic boundary conditions