Asymptotic analysis to blow-up points for the porous medium equation with a weighted non-local source

Abstract

This article deals with the porous medium equation with a more complicated source term,

subject to the homogeneous Dirichlet condition, where is a ball with radius R, m > 1 and the non-negative constants satisfying . We investigate how the three factors (the non-local source , the local source and the weight function a(x)) influence the asymptotic behaviour of the solutions. It is proved that (i) when p < 1, the non-local source plays a dominating role, i.e. the blow-up set of the system is the whole domain B _{R , a} , where . (ii) When p > m, this system presents single blow-up patterns. In other words, the local term dominates the non-local term in the blow-up profile. Moreover, the blow-up rate estimate is established with more precise coefficients determined.

Keywords:

• asymptotic analysis
• porous medium equation
• non-linear non-local source
• blow-up set
• global blow-up
• single blow-up
• weight source

AMS Subject Classifications::

• 35B40
• 35K55

Acknowledgements

This work is supported in part by NNSF of China (10771226), in part by Natural Science Foundation Project of CQ CSTC (2007BB0124) and in part by Natural Science Foundation Project of China SWU, SWU208029.