Continuation of blowup solutions of nonlinear heat equations in several space dimensions

Abstract

The possible continuation of solutions of the nonlinear heat equation in R N R+ ut = u m + u p with m> 0 ;p > 1 ; after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m +p 2 we find a phenomenon of nontrivial continuation where the regionfx : u(x;t )= 1g is bounded and propagates with finite speed. This we call incomplete blowup. For N 3 and p>m ( N +2 )= (N 2) we find solutions that blow up at finite t = T and then become bounded again for t>T . Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation ut = u m u p ;m > 0 :