BLOW–UP FOR THE WAVE EQUATION WITH NONLINEAR SOURCE AND BOUNDARY DAMPING TERMS

Abstract. The paper deals with blow–up for the solutions of an evolution problem consisting on a semilinear wave equation posed in a bounded C1,1 open subset of Rn, supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied is utt −∆u = |u|p−2u in [0,∞) × Ω, u = 0 on (0,∞) × Γ0, ∂νu = −α(x) (|ut|m−2ut + β|ut|µ−2ut) on (0,∞) × Γ1, u(0, x) = u0(x), ut(0, x) = u1(x) in Ω, where ∂Ω = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅, σ(Γ0)> 0, 2 < p ≤ 2(n − 1)/(n − 2) (when n ≥ 3), m> 1, α ∈ L∞(Γ1), α ≥ 0, β ≥ 0. The initial data are posed in the energy space. The aim of the paper is to improve previous blow–up results concerning the problem. 1