A further blow-up analysis for a localized porous medium equation

This paper deals with the porous medium equation with a localized sourcevτ=Δvm+avp1vq1(x0,τ),x∈Ω,τ>0subject to homogeneous Dirichlet condition. We investigate the influence of the localized source and local term on blow-up properties for this system. It is proved that: (i) when p1 ⩽ 1, the localized source plays a dominating role, i.e. the system has global blow-up and the uniformly blow-up profile is obtained. (ii) When p1 > 1, we obtain the blow-up rate estimates under some appropriate hypotheses on initial datum. Moreover, if p1 > m, this system presents single blow-up pattern. In other words, the local term dominates the localized term in the blow-up profile. This extends and generalizes a recent work of Chen and Xie [Y. Chen, C. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. and Comput., 159 (2004) 79-93], which only considered the blow-up profile in the special case p1 = 0.