Blow-up for a porous medium equation with a localized source

In this paper we investigate the blow-up properties of the positive solutions to a localized porous medium equation v(tau) = Deltav(m) + av(p1) v(q1) (x(0), tau) subject to homogeneous Dirichlet condition and positive initial datum v(0)(x). Under appropriate hypotheses, we establish the local existence and obtain that in the case of p(1) + q(1) < m or p(1) + q(1) = m and a is sufficiently small, there exists a global solution of the above problem; in the case of p(1) + q(1) > m, the solution of the above problem blows up for large initial datum while it admits a global solution for small initial datum. Moreover, for the special case p, = 0, q(1) > m and a is large, under an additional hypothesis on the initial datum, we can also obtain the asymptotic behavior of the blow-up solution. (C) 2003 Elsevier Inc. All rights reserved.