The self-similar solutions to a fast diffusion equation

A quasilinear equation u-x·u/2+f(u)=0 is studied, wheref(u)=–u+u, > 0, 0<. <1, >1 andx Rⁿ. The equation arises from the study of blow-up self-similar solutions of the heat equation _t=+. We prove the existence and non-existence of ground state for various combination of , and . In particular, we prove that when / < forn=1,2 or / < (n + 2) /(n – 2) forn 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where / > (n + 2)/(n – 2) there exists an infinite number of non-constant positive radial self-similar solutions.