Vanishing time behavior of solutions to the fast diffusion equation

Abstract

Let n >= 3, 0 < m < (n-2) (n) and T > 0. We construct positive solu-tions to the fast diffusion equation u(t )= Delta u(m) in R-n x (0, T), which vanish at time T. By introducing a scaling parameter 0 inspired by [DKS], we study the second-order asymptotics of the self-similar solutions associated with 0 at spatial infinity. We also investigate the asymptotic behavior of the solutions to the fast diffusion equation near the vanishing time T, provided that the initial value of the solution is close to the initial value of some self-similar solution and satisfies some proper decay condition at infinity. Depending on the range of the parameter 0, we prove that the rescaled solution converges either to a self-similar profile or to zero as t NE arrow T. The former implies asymptotic stabilization towards a self-similar solution, and the latter is a new vanishing phenomenon even for the case n >= 3 and m = (n-2 )(n+2) which corresponds to the Yamabe flow on R-n with metric g = u (4 )(n+2 dx)2.