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 double-struck Rn, supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied is utt - Δu = |u|p-2u in [0, ∞) × Omega;, 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 Omega;, where ∂Omega; = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = 0, σ(Γ0) > 0, 2 [removed] 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.
Fiscella, A., Vitillaro, E. (2015). Blow-up for the wave equation with nonlinear source and boundary damping terms. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SECTION A. MATHEMATICS, 145(4), 759-778 [10.1017/S0308210515000165].
Blow-up for the wave equation with nonlinear source and boundary damping terms
Fiscella A
;
2015
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 double-struck Rn, supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied is utt - Δu = |u|p-2u in [0, ∞) × Omega;, 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 Omega;, where ∂Omega; = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = 0, σ(Γ0) > 0, 2 [removed] 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.
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