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 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 Blow-Up Phenomena for Porous Medium Equation with Nonlinear Flux on the Boundary

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Yan Hu ◽  
Jing Li ◽  
Liangwei Wang
Keyword(s):  
Porous Medium ◽  
Boundary Conditions ◽  
Blow Up ◽  
Neumann Boundary ◽  
Porous Medium Equation ◽  
Neumann Boundary Conditions ◽  
Medium Equation ◽  
Nonnegative Solutions

We investigate the blow-up phenomena for nonnegative solutions of porous medium equation with Neumann boundary conditions. We find that the absorption and the nonlinear flux on the boundary have some competitions in the blow-up phenomena.

scholarly journals LOWER BOUNDS FOR BLOW-UP TIME IN SOME NON-LINEAR PARABOLIC PROBLEMS UNDER NEUMANN BOUNDARY CONDITIONS

2011 ◽  
Vol 53 (3) ◽  
pp. 569-575 ◽  
Author(s):  
CRISTIAN ENACHE
Keyword(s):  
Boundary Conditions ◽  
Lower Bounds ◽  
Blow Up ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Neumann Boundary Conditions ◽  
Parabolic Problems ◽  
Non Linear ◽  
Inequality Technique ◽  
Initial Boundary Value

AbstractThis paper deals with some non-linear initial-boundary value problems under homogeneous Neumann boundary conditions, in which the solutions may blow up in finite time. Using a first-order differential inequality technique, lower bounds for blow-up time are determined.

Blow-up phenomena in parabolic problems with time-dependent coefficients under Neumann boundary conditions

2012 ◽  
Vol 142 (3) ◽  
pp. 625-631 ◽  
Author(s):  
L. E. Payne ◽  
G. A. Philippin
Keyword(s):  
Boundary Conditions ◽  
Global Solution ◽  
Lower Bounds ◽  
Blow Up ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Time Dependent ◽  
Parabolic Problems

This paper deals with the blow-up of solutions to a class of parabolic problems with time-dependent coefficients under homogeneous Neumann boundary conditions. For one set of problems in this class we show that no global solution can exist. For another we derive lower bounds for the time of blow-up when blow-up 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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