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 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 Solution of a Porous Medium Equation with Nonlocal Boundary Conditions

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Porous Medium ◽  
Boundary Conditions ◽  
Blow Up ◽  
Dimensional Space ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Porous Medium Equation ◽  
Nonlocal Boundary Conditions ◽  
Medium Equation ◽  
Inequality Technique

In this paper, we devote to studying the blow-up phenomena for a porous medium equation under nonlocal boundary conditions. Based on auxiliary functions and differential inequality technique, we derive the sufficient conditions to guarantee the existence of blow-up solutions under different measures and obtain an upper bound for blow-up time. Moreover, we demonstrate the lower bounds for blow-up time under some appropriate measures in R3 and in the higher-dimensional space Rnn≥3, respectively. At last, two examples are given to illustrate the applications of main results.

scholarly journals Roles of Weight Functions to a Nonlocal Porous Medium Equation with Inner Absorption and Nonlocal Boundary Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zhong Bo Fang ◽  
Jianyun Zhang ◽  
Su-Cheol Yi
Keyword(s):  
Boundary Condition ◽  
Porous Medium ◽  
Blow Up ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Porous Medium Equation ◽  
Weight Functions ◽  
Medium Equation ◽  
Nonnegative Solutions ◽  
Inner Absorption

This work is concerned with an initial boundary value problem for a nonlocal porous medium equation with inner absorption and weighted nonlocal boundary condition. We obtain the roles of weight function on whether determining the blowup of nonnegative solutions or not and establish the precise blow-up rate estimates under some suitable condition.

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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