ASYMPTOTIC-BEHAVIOR NEAR FINITE-TIME EXTINCTION FOR THE FAST DIFFUSION EQUATION

We study the Cauchy problem for the fast diffusion equation u(t) = Del ta(u(m)), u greater than or equal to 0 in R-N x R+, when N greater tha n or equal to 3 and 0 < m < N-2/N. For a class of nonnegative radially symmetric N finite-mass solutions, which vanish identically at a give n time T > 0, we show that their asymptotic behaviour as t / T is desc ribed by a uniquely determined self-similar solution of the second kin d: u(r, t) = (T - t)(a)f(eta), where eta = r/(T - t)(beta) and alpha = 1-2 beta/1-m and r = \x\. Here beta is determined from a nonlinear e igenvalue problem involving an ordinary differential equation for the function f. Special attention is paid to the case when m N-2/N+2. Then beta = 0 and the function f can be found explicity. The proof is base d on a geometric Lyapunov-type argument and comparison arguments based on the intersection properties of the solution graphs.