Explicit blow-up time for two porous medium problems with different reaction terms

This paper deals with the blow-up phenomena of classical solutions to porous medium problems, defined in a bounded domain of ℝn, with n ≥ 1. We distinguish two situations: in the first case, no gradient nonlinearity is present in the reaction term contrarily to the other case. Specifically, some theoretical and general results concerning the mathematical model, existence analysis and estimates of the blow-up time t∗ of unbounded solutions to these problems are summarized and discussed. More exactly, for both problems, explicit lower bounds of t∗ if blow-up occurs are derived in the case n = 3 and in terms of an auxiliary function. On the other hand, in order to compute the real blow-up times of such blowing-up solutions and discuss their properties, a general resolution method is proposed and used in some two-dimensional examples.