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

Part of: Hyperbolic equations and systems Partial differential equations

Published online by Cambridge University Press:  20 July 2015

Alessio Fiscella  and
Enzo Vitillaro
Alessio Fiscella
Affiliation:
Dipartimento di Matematica ‘Federigo Enriques’, Università di Milano, Via Cesare Saldini 50, 20133 Milano, Italy, ([email protected])
Enzo Vitillaro
Affiliation:
Dipartimento di Matematica ed Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy, ([email protected])
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Abstract

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.

Keywords

wave equationboundary dampingblow-upsource

MSC classification

Primary: 35L05: Wave equation 35L20: Initial-boundary value problems for second-order hyperbolic equations
Secondary: 35A01: Existence problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 4 , August 2015 , pp. 759 - 778
Copyright
Copyright © Royal Society of Edinburgh 2015 

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