Blowup in diffusion equations: A survey

This paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems.