scholarly journals Uniform Blow-Up Rates and Asymptotic Estimates of Solutions for Diffusion Systems with Nonlocal Sources

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
Zhoujin Cui ◽  
Zuodong Yang
Keyword(s):  
Blow Up ◽  
Nonnegative Solution ◽  
Local Existence ◽  
Diffusion Equations ◽  
Asymptotic Estimates ◽  
Nonlocal Sources ◽  
Estimates Of Solutions ◽  
Diffusion Systems ◽  
Blow Up Rate ◽  
Nonlocal Reaction

This paper investigates the local existence of the nonnegative solution and the finite time blow-up of solutions and boundary layer profiles of diffusion equations with nonlocal reaction sources; we also study the global existence and that the rate of blow-up is uniform in all compact subsets of the domain, the blow-up rate of|u(t)|∞is precisely determined.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

scholarly journals Blow-Up and Global Existence for a Degenerate Parabolic System with Nonlocal Sources

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Ling Zhengqiu ◽  
Wang Zejia
Keyword(s):  
Global Existence ◽  
Critical Exponent ◽  
Parabolic System ◽  
Blow Up ◽  
Nonnegative Solution ◽  
Degenerate Parabolic System ◽  
Nonlocal Sources ◽  
Nonnegative Solutions ◽  
Large Ball ◽  
Degenerate Parabolic

This paper investigates the blow-up and global existence of nonnegative solutions for a class of nonlocal degenerate parabolic system. By using the super- and subsolution techniques, the critical exponent of the system is determined. That is, ifPc=p1q1−(m−p2)(n−q2)<0, then every nonnegative solution is global, whereas ifPc>0, there are solutions that blowup and others that are global according to the size of initial valuesu0(x)andv0(x). WhenPc=0, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.

scholarly journals Blow-Up Rate Estimates for a System of Reaction-Diffusion Equations with Gradient Terms

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Maan A. Rasheed ◽  
Hassan Abd Salman Al-Dujaly ◽  
Talat Jassim Aldhlki
Keyword(s):  
Blow Up ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Dirichlet Problems ◽  
Blow Up Rate

This paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diffusion equations with gradient terms. The main goal is to study the influence of the gradient terms on the blow-up profile. Namely, under some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.

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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