Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity

The Cauchy problem {u(t) - Deltau(m) = u(p) in R-N x (0, T), u(x, 0) = tauu(0) (x) in R-N, (P) is considered, where tau > 0, p > m greater than or equal to I and u(0) (x) (> 0) is a bounded continuous radially symmetric function in R-N. We choose p in some open interval (p(s),p(p)) with p(s) = m(N + 2)1[N - 2](+) such that a peaking solution (incomplete blow-up solution) of (P) exists. Denote the solution of (P) by u(tau). We show that if u(0)(x) is nonincreasing in large r = \x\ and decays slowly: u(0)(x) = O(\x\(-alpha)) as \x\ --> infinity (2/(p - m)<alpha), then u(tau) is classified into one of the next three,types according to the value tau as follows: There exists tau(1) is an element of (0, infinity) such that (1) u(tau) blows up completely in finite time if tau > tau(l), (11) u(tau) blows up incompletely in finite time and \\u(tau)(t)\\(Linfinity)(R-N) = O(t(-1/p-1)) as t --> infinity if tau = tau(1), (111) u(tau) does not blow up in finite time and I \\u(tau)(t)\\(Linfinity)(R-N) = O(t(-1/p-1)) as t --> infinity if 0 < tau < tau(1). (C) 2002 Elsevier Science (USA). All rights reserved.