THE SELF-SIMILAR SOLUTIONS TO A FAST DIFFUSION EQUATION

A quasilinear equation DELTAu(alpha) - x . DELTAu/2 + f(u) = 0 is studied, where f(u) = - muu + u(beta), mu > 0, 0 < alpha < 1, beta > 1 and x element-of R(n). The equation arises from the study of blow-up self-similar solutions of the heat equation psi(t) = DELTApsi(alpha) + psi(beta). We prove the existence and non-existence of ground state for various combination of mu, alpha and beta. In particular, we prove that when beta/alpha < infinity for n = 1,2 or beta/alpha < (n + 2)/(n - 2) for n greater-than-or-equal-to 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where beta/alpha > (n + 2)/(n - 2) there exists an infinite number of non-constant positive radial self-similar solutions.