Global existence for the wave equation with nonlinear boundary damping and source terms

Abstract

The paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is u tt− Δu=0 in (0,∞)×Ω, u=0 on [0,∞)×Γ 0, ∂u ∂ν =−|u t| m−2u t+|u| p−2u on [0,∞)×Γ 1, u(0,x)=u 0(x), u t(0,x)=u 1(x) on Ω, where Ω⊂ R n (n⩾1) is a regular and bounded domain, ∂Ω=Γ 0∪Γ 1 , m>1, 2⩽ p< r, where r=2( n−1)/( n−2) when n⩾3, r=∞ when n=1,2, and the initial data are in the energy space. We prove local existence of the solutions in the energy space when m> r/( r+1− p) or n=1,2, and global existence when p⩽ m or the initial data are inside the potential well associated to the stationary problem.