Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity

Abstract

We consider the blowup rate of solutions for a semilinear heat equation u t = Δ u + | u | p − 1 u , x ∈ Ω ⊂ R N , t > 0 , with critical power nonlinearity p = ( N + 2 ) / ( N − 2 ) and N ⩾ 3 . First we investigate the profiles of backward self-similar solutions by making use of the variational method, and then, by employing the intersection comparison argument with a particular self-similar solution, we derive the criteria of the blowup rate of solutions, assuming the positivity of solutions in backward space–time parabola. In particular, we show the existence of the so-called type II blowup solutions for the Cauchy–Dirichlet problems on suitable shrinking domains in the case N = 3 .