Blowup and life span of solutions for a semilinear parabolic equation

This paper is concerned with the Cauchy problem [GRAPHICS] where p > 1. Let Omega be a set in R-N given by Omega = {(r,omega) is an element of R+ x SN-1; r > R, d(omega,omega 0) < cr(-mu)} for some R > 0, c > 0, omega(0) is an element of SN-1, and 0 less than or equal to mu < 1, where d(.,.) denotes the standard distance on SN-1. It is shown that if u(0) decays like [x](-alpha) as [x] --> infinity in Omega with 0 < alpha < 2(1-mu)/(p-1), then the solution blows up in finite time regardless of the behavior of u(0) outside Omega. Moreover the life span of such a solution with u(0) = lambda phi is estimated from above for small lambda > 0 in terms of p, alpha, and mu. The optimality of these results is also studied.