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

Huafei Di, Lin Chen, Zefang Song

Abstract

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.

Keywords

Porous medium equation, Upper and lower bounds, Blow-up rate, Weighted source

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