Nonexistence of type II blowup solution for a semilinear heat equation

Abstract

A solution u of a Cauchy problem for a semilinear heat equation { u t = Δ u + | u | p − 1 u in R N × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in R N is said to undergo type II blowup at t = T < ∞ if lim sup t → T ( T − t ) 1 / ( p − 1 ) | u ( t ) | ∞ = ∞ . Let p S and p JL be the exponents of Sobolev and of Joseph and Lundgren, respectively. We prove that when p S < p < p JL , 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 p S < p < p JL if the number of intersections with ± φ ∞ is at most finite, then the radial solution does not undergo type II blowup. We do not impose an assumption on the number of intersections with ± φ ∞ . For example, when a radial initial data u 0 is nonnegative and nonincreasing in r = | x | , the result in Matano and Merle (2009) [12] does not exclude type II blowup for p in the range, while our result does it.