Determination of the blow-up rate for the semilinear wave equation

2003 ◽  
Vol 125 (5) ◽  
pp. 1147-1164 ◽  
Author(s):  
Frank Merle ◽  
Hatem Zaag
Keyword(s):  
Wave Equation ◽  
Blow Up ◽  
Semilinear Wave Equation ◽  
Blow Up Rate

scholarly journals LYAPUNOV FUNCTIONAL AND BLOW-UP RESULTS FOR A CLASS OF PERTURBATIONS OF SEMILINEAR WAVE EQUATIONS IN THE CRITICAL CASE

2012 ◽  
Vol 09 (02) ◽  
pp. 195-221 ◽  
Author(s):  
MOHAMED ALI HAMZA ◽  
HATEM ZAAG
Keyword(s):  
Wave Equation ◽  
Singular Solution ◽  
Lyapunov Functional ◽  
Wave Equations ◽  
Critical Case ◽  
Critical Power ◽  
Blow Up ◽  
Semilinear Wave Equation ◽  
Nirenberg Inequality ◽  
Blow Up Rate

We study a class of perturbations for the semilinear wave equation with critical power nonlinearity (in the conformal transform sense). Working in the framework of similarity variables, we introduce a Lyapunov functional for this problem. Using a two-step argument based on interpolation and a critical Gagliardo–Nirenberg inequality, we establish that the blow-up rate of any singular solution is given by the solution of the nonperturbed associated ODE, specifically u″ = up.

scholarly journals Blow-up for discrete reaction-diffusion equations on networks

2015 ◽  
Vol 9 (1) ◽  
pp. 103-119 ◽  
Author(s):  
Soon-Yeong Chung ◽  
Jae-Hwang Lee
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Blow Up Rate

In this paper, we discuss the conditions under which blow-up occurs for the solutions of reaction-diffusion equations on networks. The analysis of this class of problems includes the existence of blow-up in finite time and the determination of the blow-up time and the corresponding blow-up rate. In addition, when the solution blows up, we give estimates for the blow-up time and also provide the blow-up rate. Finally, we show some numerical illustrations which describe the main results.

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