The self-similar solutions to a fast diffusion equation

A quasilinear equation Δu^α-x·▽u/2+f(u)=0 is studied, where f(u)=-μu+u^β, μ > 0, 0<α. <1, β>1 and x εRⁿ. The equation arises from the study of blow-up self-similar solutions of the heat equation ψ_t=Δψ^α+ψ^β. We prove the existence and non-existence of ground state for various combination of μ, α and β. In particular, we prove that when β/α < ∞ for n=1,2 or β/α < (n + 2) /(n - 2) for n ≥ 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where β/α > (n + 2)/(n - 2) there exists an infinite number of non-constant positive radial self-similar solutions. © 1994 Birkhäuser Verlag