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摘要
This paper deals with the porous medium equation with a localized source v(tau) = Delta v(m) + av(p1)v(q1) (x(0), tau), x is an element of Omega, tau > 0 subject 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 p(1) <= 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 p(1) > 1, we obtain the blow-up rate estimates under some appropriate hypotheses on initial datum. Moreover, if p(1) > 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 p(1) = 0.
