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

We consider the blowup rate of solutions for a semilinear heat equationut=Δu+|u|p−1u,x∈Ω⊂RN,t>0, with critical power nonlinearity p=(N+2)/(N−2)p=(N+2)/(N−2) and N⩾3N⩾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=3N=3.