Blow-up for the wave equation with nonlinear source and boundary damping terms

Alessio Fiscella; Enzo Vitillaro

doi:10.1017/S0308210515000165

Blow-up for the wave equation with nonlinear source and boundary damping terms

Part of: Partial differential equations Hyperbolic equations and systems

Published online by Cambridge University Press: 20 July 2015

Alessio Fiscella and

Enzo Vitillaro

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Alessio Fiscella: Affiliation:
Dipartimento di Matematica ‘Federigo Enriques’, Università di Milano, Via Cesare Saldini 50, 20133 Milano, Italy, (alessio.fiscella@unimi.it)
Enzo Vitillaro: Affiliation:
Dipartimento di Matematica ed Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy, (enzo.vitillaro@unipg.it)

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Abstract

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Abstract

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The paper deals with blow-up for the solutions of an evolution problem consisting in 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

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 and 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.