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

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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.