Explicit blow-up time for complete porous medium problems

VIGLIALORO, GIUSEPPE

2015-01-01

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Abstract

Mainly, this paper deals with the blow-up phenomena of solutions to porous medium problems, prescribed by reaction-diffusion equations, defined in a bounded domain of Rn, with n = {2, 3}. We distinguish two situations, depending on the presence or not of a gradient nonlinearity in the reaction term of the corresponding equation. Specifically, important heoretical and general results regarding estimates of blow-up time t∗ to solutions of such problems are summarized. More exactly, by means of an energy function explicit lower bounds for t∗, if blow-up occurs, are derived in the case n = 3. On the other hand, first a general resolution method is proposed and, then, it is applied in the two dimensional case for some examples to both compute the explicit blow-up times of such unbounded solutions and to analyze and discuss some of their properties.