Nonexistence of type II blowup solution for a semilinear heat equation

AbstractA solution u of a Cauchy problem for a semilinear heat equation{ut=Δu+|u|p−1uin RN×(0,T),u(x,0)=u0(x)in RN is said to undergo type II blowup at t=T<∞ iflim supt→T(T−t)1/(p−1)|u(t)|∞=∞. Let pS and pJL be the exponents of Sobolev and of Joseph and Lundgren, respectively. We prove that when pS<p<pJL, a radial solution u does not exhibit type II blowup if u does not blow up at infinity. Let φ∞ be the positive singular stationary solution with radial symmetry. It was shown in Matano and Merle (2009) [12] that for pS<p