A probabilistic approach to blow-up of a semilinear heat equation

Abstract

We introduce a probabilistic approach to the study of blow-up of positive solutions to a class of semilinear heat equations. This then gives a representation of the coefficients in the power series expansion of the solutions. In a special case, this approach leads to a path-valued Markov process which can also be understood via the theory of Dawson-Watanabe superprocesses. We demonstrate the utility of the approach by proving a result on ‘complete blow-up’ of solutions.