CRITICAL EXPONENT FOR THE BIPOLAR BLOWUP IN A SEMILINEAR PARABOLIC EQUATION

The blowup of solutions of the initial-boundary value problem [GRAPHIC S] is studied, where Omega is a cone in R-N. We say that a solution u exhibits bipolar blowup sup/x is an element of Omega u(x, t) --> infin ity and inf/x is an element of Omega u(x, t) --> -infinity as t up arr ow T for some T < infinity. It is shown that there exists a critical e xponent p > 1 for the bipolar blowup in the following sense. If 1 < p less than or equal to p, then there exist arbitrarily small initial data such that the solution exhibits the bipolar blowup, whereas if p > p, then the bipolar blowup does not occur for any sufficiently smal l initial data. The value of p is expressed in terms of the dimension N and the second Dirichlet eigenvalue of the Laplace-Beltrami operato r on Omega boolean AND SN-1. In the case of Omega = R, we define k-pol ar blowup, and determine critical exponents for the k-polar blowup for k = 1, 2, 3,.... (C) 1998 Academic Press.