scholarly journals Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up

2019 ◽  
Vol 18 (3) ◽  
pp. 1351-1358 ◽  
Author(s):  
Yanbing Yang ◽  
◽  
Runzhang Xu ◽  
◽  
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Finite Time ◽  
Blow Up ◽  
Initial Energy

scholarly journals Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term

2019 ◽  
Vol 9 (1) ◽  
pp. 613-632 ◽  
Author(s):  
Wei Lian ◽  
Runzhang Xu
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Blow Up ◽  
Local Existence ◽  
High Energy ◽  
Initial Energy ◽  
Infinite Time ◽  
Logarithmic Term ◽  
Strong Damping

Abstract The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E(0) < d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) > 0 (ω = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, infinite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy infinite time blow up result is established.

scholarly journals General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Salah Boulaaras ◽  
Fares Kamache ◽  
Youcef Bouizem ◽  
Rafik Guefaifia
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Lyapunov Functional ◽  
Blow Up ◽  
Initial Energy ◽  
General Decay ◽  
Memory Term ◽  
Decay Of Solutions ◽  
With Memory

AbstractThe paper studies the global existence and general decay of solutions using Lyapunov functional for a nonlinear wave equation, taking into account the fractional derivative boundary condition and memory term. In addition, we establish the blow-up of solutions with nonpositive initial energy.

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.

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

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
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