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.