Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball.

1996 ◽  
Vol 1996 (472) ◽  
pp. 17-52 ◽  
Keyword(s):  
Positive Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotic Periodicity

Uniqueness, multiplicity and stability for positive solutions of a pair of reaction–diffusion equations

1996 ◽  
Vol 126 (4) ◽  
pp. 777-809 ◽  
Author(s):  
Yihong Du
Keyword(s):  
Diffusion Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Reaction Diffusion ◽  
Single Equation ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Complete Understanding ◽  
Satisfactory Description

We study the number and stability of the positive solutions of a reaction–diffusion equation pair. When certain parameters in the equations are large, the equation pair can be viewed as singular or regular perturbations of some single (or essentially single) equation problems, for which the number and stability of their solutions can be well understood. With the help of these simpler equations, we are able to obtain a rather complete understanding of the number and stability of the positive solutions for the equation pair for the cases that certain parameters are large. In particular, we obtain a fairly satisfactory description of the positive solution set of the equation pair.

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.

Sign in / Sign up

Export Citation Format

Share Document