A potential well theory for the wave equation with nonlinear source and boundary damping terms

Abstract

The paper deals with local existence, blow-up and global existence for the solutions of a wave equation with an internal nonlinear source and a nonlinear boundary damping. The typical problem studied is {uu - Δu = |u|p-2u in [0, ∞) × Ω, u = 0 on [0, ∞) × Γ0, ∂u/∂ν =-α(x)|ut|m-2ut on [0, ∞) × Γ1, u(0,x) = uo(x), ut(0, x) = u1(x) on Ω, where Ω ⊂ ℝn (n ≥ 1) is a regular and bounded domain, ∂Ω = Γ0 ∪ Γ1, λn-1(Γ0) > 0, 2 1, α ∈ L∞(Γ1), α ≥ 0, and the initial data are in the energy space. The results proved extend the potential well theory, which is well known when the nonlinear damping acts in the interior of Ω, to this problem.