scholarly journals Blow up of the Solutions of Nonlinear Wave Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-51
Author(s):  
Svetlin Georgiev Georgiev
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Blow Up

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.

General decay and blow up of solution for a nonlinear wave equation with a fractional boundary damping

2020 ◽  
Vol 43 (12) ◽  
pp. 7175-7193 ◽  
Author(s):  
Radhouane Aounallah ◽  
Salah Boulaaras ◽  
Abderrahmane Zarai ◽  
Bahri Cherif
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Blow Up ◽  
General Decay ◽  
Boundary Damping

scholarly journals Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

2011 ◽  
Vol 74 (18) ◽  
pp. 6933-6949 ◽  
Author(s):  
Le Xuan Truong ◽  
Le Thi Phuong Ngoc ◽  
Alain Pham Ngoc Dinh ◽  
Nguyen Thanh Long
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Nonlinear Wave ◽  
Exponential Decay ◽  
Nonlinear Wave Equation ◽  
Blow Up ◽  
Decay Estimates ◽  
Point Type
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