Large time behavior for the fast diffusion equation with critical absorption

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption partial derivative(t)u - Delta u(m) + u(q) = 0 in (0, infinity) x R-N, with 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. (C) 2016 Elsevier Inc. All rights reserved.