On the blow-up by comparison

Abstract

It is known that it is difficult to estimate the blow-up time of positive solutions of nonlinear reaction-diffusion equations u(t) = …u + f(x,u,..u). The main difficulty lies in the fact that as the solution becomes unbounded, known methods such as the maximum principle method are inapplicable. To remedy this situation, we shall transform the equation so that the blow-up set is equivalent to the set of zeros of a solution of a new problem. Then by looking at the steady state, we shall be able to decide whether we have a blow-up or not for the original equation.