Blow-up and global solutions for nonlinear reaction–diffusion equations with Neumann boundary conditions

2008 ◽  
Vol 68 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Juntang Ding ◽  
Shengjia Li
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Global Solutions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Reaction

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 Phenomena for Nonlinear Reaction-Diffusion Equations under Nonlinear Boundary Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Juntang Ding
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Nonlinear Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Nonlinear Reaction

This paper deals with blow-up and global solutions of the following nonlinear reaction-diffusion equations under nonlinear boundary conditions:g(u)t=∇·au∇u+fu  in  Ω×0,T,  ∂u/∂n=bx,u,t  on  ∂Ω×(0,T),  u(x,0)=u0(x)>0,  in  Ω¯,whereΩ⊂RN  (N≥2)is a bounded domain with smooth boundary∂Ω. We obtain the conditions under which the solutions either exist globally or blow up in a finite time by constructing auxiliary functions and using maximum principles. Moreover, the upper estimates of the “blow-up time,” the “blow-up rate,” and the global solutions are also given.

Global Existence and Blow-Up in Finite Time of Solutions to One Kind of Reaction-Diffusion Educations with Neumann Boundary Conditions

2014 ◽  
Vol 971-973 ◽  
pp. 1017-1020
Author(s):  
Jun Zhou Shao ◽  
Ji Jun Xu
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Blowing Up ◽  
Processing Techniques

This paper deals with the properties of one kind of reaction-diffusion equations with Neumann boundary conditions based on the comparison principles. The relations of parameter and the situation of the coupled about equations are used to construct the global existent super-solutions and the blowing-up sub-solutions, and then we obtain the conditions of the global existence and blow-up in finite time solutions with the processing techniques of inequality.

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