scholarly journals Blow-up Rate Estimates and Blow-up Set for a System of Two Heat Equations with Coupled Nonlinear Neumann Boundary Conditions

2020 ◽  
pp. 147-152
Author(s):  
Maan A. Rasheed ◽  
Miroslav Chlebik
Keyword(s):  
Boundary Conditions ◽  
Exponential Type ◽  
Parabolic System ◽  
Blow Up ◽  
Neumann Boundary ◽  
Upper Bounds ◽  
Neumann Boundary Conditions ◽  
Heat Equations ◽  
Blow Up Rate ◽  
Nonlinear Neumann Boundary Conditions

This paper deals with the blow-up properties of positive solutions to a parabolic system of two heat equations, defined on a ball in  associated with coupled Neumann boundary conditions of exponential type. The upper bounds of blow-up rate estimates are derived. Moreover, it is proved that the blow-up in this problem can only occur on the boundary.

scholarly journals Some Results on Blow-up Phenomenon for Nonlinear Porous Medium Equations with Weighted Source

2020 ◽  
Vol 13 (3) ◽  
pp. 645-662
Author(s):  
Huafei Di ◽  
Lin Chen ◽  
Zefang Song
Keyword(s):  
Porous Medium ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Blow Up ◽  
Neumann Boundary ◽  
Upper Bounds ◽  
Neumann Boundary Conditions ◽  
Integral Inequalities ◽  
Porous Medium Equations ◽  
Blow Up Rate

This paper deals with the blow-up phenomena for a type of nonlinear porous medium equations with weighted source ut −4um = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions. Based on the auxiliary functions and differential-integral inequalities, the blow-up criterions which ensure that u cannot exist all time are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived respectively. Moreover, we use three different methods to determine the lower bounds for blow-up time and blow-up rate estimates if blow-up does occurs.

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.

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