Blow up for the wave equation with a nonlinear dissipation of cubic convolution type in

2004 ◽  
Vol 148 (3) ◽  
pp. 759-771 ◽  
Author(s):  
Nasser-eddine Tatar
Keyword(s):  
Wave Equation ◽  
Blow Up ◽  
Convolution Type ◽  
Nonlinear Dissipation

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

2015 ◽  
Vol 145 (4) ◽  
pp. 759-778 ◽  
Author(s):  
Alessio Fiscella ◽  
Enzo Vitillaro
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Initial Data ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Neumann Boundary ◽  
Nonlinear Dissipation ◽  
Typical Problem ◽  
Nonlinear Source ◽  
Boundary Damping

The paper deals with blow-up for the solutions of an evolution problem consisting in a semilinear wave equation posed in a boundedC1,1open subset of ℝn, supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied iswhere∂Ω=Γ0∪Γ1,Γ0∩Γ1= ∅,σ(Γ0) > 0, 2 <p≤ 2(n− 1)/(n− 2) (whenn≥ 3),m> 1,α∈L∞(Γ1),α≥ 0 andβ≥ 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.

scholarly journals Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation

2012 ◽  
Vol 5 (4) ◽  
pp. 777-829 ◽  
Author(s):  
Matthieu Hillairet ◽  
Pierre Raphaël
Keyword(s):  
Wave Equation ◽  
Blow Up ◽  
Type Ii

scholarly journals Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation

2019 ◽  
Vol 19 (4) ◽  
pp. 639-675
Author(s):  
Thierry Cazenave ◽  
Yvan Martel ◽  
Lifeng Zhao
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Blow Up ◽  
Finite Energy ◽  
Energy Solution ◽  
Compact Set ◽  
Existence Of A Solution ◽  
Blowing Up ◽  
Finite Energy Solution

AbstractWe prove that any sufficiently differentiable space-like hypersurface of {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation {\partial_{tt}u-\Delta u=|u|^{p-1}u} on {{\mathbb{R}}\times{\mathbb{R}}^{N}}, for any {1\leq N\leq 4} and {1<p\leq\frac{N+2}{N-2}}. We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at {t=0} for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at {t=0}. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with {H^{2}\times H^{1}} solutions for the transformed problem.

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