Large time behavior for the fast diffusion equation with critical absorption

Abstract

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorptionwith m c := (N − 2) + /N < m < 1 and q = m + 2/N . Given an initial condition u 0 decaying arbitrarily fast at infinity, we show that the asymptotic behavior of the corresponding solution u is given by a Barenblatt profile with a logarithmic scaling, thereby extending a previous result requiring a specific algebraic lower bound on u 0 . A by-product of our analysis is the derivation of sharp gradient estimates and a universal lower bound, which have their own interest and hold true for general exponents q > 1.