Description of regional blow-up in a porous-medium equation

Abstract

We describe the (finite-time) blow-up phenomenon for a non-negative solution of a porous medium equation of the form $$ u_t = Delta u^m + u^m $$ in the entire space. Here $m>1$ and the initial condition is assumed compactly supported. Blow-up takes place exactly inside a finite number of balls with same radii and exhibiting the same self-similar profile.

Keywords

• Multiple-bump
• pattern formation
• mathematical biology
• singular perturbation.