Backward selfsimilar solutions of supercritical parabolic equations

We consider the exponential reaction–diffusion equation in space-dimension n∈(2,10)n∈(2,10). We show that for any integer k≥2k≥2 there is a backward selfsimilar solution which crosses the singular steady state kk-times. The same holds for the power nonlinearity if the exponent is supercritical in the Sobolev sense and subcritical in the Joseph–Lundgren sense.