Multiple blow-up for a porous medium equation with reaction

The present paper is concerned with the Cauchy problem {partial derivative(t)u = Delta u(m) + u(p) in R-N x (0, infinity), u(x, 0) = u(0)(x) >= 0 in R-N, with p, m > 1. A solution u with bounded initial data is said to blow up at a finite time T if lim sup(t NE arrow T) parallel to u(t)parallel to(L)infinity(R-N) = infinity. For N >= 3 we obtain, in a certain range of values of p, weak solutions which blow up at several times and become bounded in intervals between these blow-up times. We also prove a result of a more technical nature: proper solutions are weak solutions up to the complete blow-up time.