On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity

This paper is concerned with a Cauchy problem [GRAPHICS] where p > p(*) = (N + 2)/(N - 2), lambda > 0 and phi is a nonnegative radially symmetric function in C-1 (R-N) with compact support. Denote the solution of (P) by u(lambda). Let p* = infinity if 3 less than or equal to N less than or equal to 10 and p* = 1 + 6/(N - 10) if N greater than or equal to 11. We show that if p(*) < p < p*, then there is lambda(phi) > 0 such that: (i) If lambda < λ(φ), then u(λ) exists globally in time in the classical sense an, u(λ)(t) converges to zero locally uniformly in R-N as t --> infinity. (ii) If lambda = lambda(phi), then u(lambda) blows up incompletely in finite time. (iii) If lambda > lambda(phi), then u(lambda) blows up completely in finite time.