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 absorption $${\partial }_{t}u-\Delta u^m+u^q=0in(0,\infty )\times {\Re }^N$$ 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 $q>1$.