Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity

The Cauchy problem is considered, where τ>0, p>m 1 and u0(x) (>0) is a bounded continuous radially symmetric function in RN. We choose p in some open interval (ps,pp) with ps=m(N+2)/[N−2]+ such that a peaking solution (incomplete blow-up solution) of (P) exists. Denote the solution of (P) by uτ. We show that if u0(x) is nonincreasing in large r=x and decays slowly: u0(x)=O(x−α) as x→∞ (2/(p−m)<α), then uτ is classified into one of the next three types according to the value τ as follows: There exists τ1 (0,∞) such that (I) uτ blows up completely in finite time if τ>τ1, (II) uτ blows up incompletely in finite time and as t→∞ if τ=τ1, (III) uτ does not blow up in finite time and as t→∞ if 0<τ<τ1.