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

Abstract

The aim of the paper is to study the problem utt+dut-c2Δu=0inR×Ω,μvtt-divΓ(σ∇Γv)+δvt+κv+ρut=0onR×Γ1,vt=∂νuonR×Γ1,∂νu=0onR×Γ0,u(0,x)=u0(x),ut(0,x)=u1(x)inΩ,v(0,x)=v0(x),vt(0,x)=v1(x)onΓ1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_{tt}+du_t-c^2\\Delta u=0 \\qquad &{}\ ext {in}\\, {\\mathbb {R}}\ imes \\Omega ,\\\\ \\mu v_{tt}- \ extrm{div}_\\Gamma (\\sigma \ abla _\\Gamma v)+\\delta v_t+\\kappa v+\\rho u_t =0\\qquad &{}\ ext {on}\\, {\\mathbb {R}}\ imes \\Gamma _1,\\\\ v_t =\\partial _\ u u\\qquad &{}\ ext {on}\\, {\\mathbb {R}}\ imes \\Gamma _1,\\\\ \\partial _\ u u=0 &{}\ ext {on}\\, {\\mathbb {R}}\ imes \\Gamma _0,\\\\ u(0,x)=u_0(x),\\quad u_t(0,x)=u_1(x) &{} \ ext {in}\\, \\Omega ,\\\\ v(0,x)=v_0(x),\\quad v_t(0,x)=v_1(x) &{} \ ext {on}\\, \\Gamma _1, \\end{array}\\right. } \\end{aligned}$$\\end{document}where Omega is a open domain of {mathbb {R}}^N with uniformly C^r boundary (Nge 2, rge 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 textrm{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 ,dge 0.