Backward selfsimilar solutions of supercritical parabolic equations

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