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 $$ \partial_{t}u-\Delta u^m+u^q=0 \quad \quad \hbox{in} \ (0,\infty)\times\real^N\, $$ 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 $q > 1$. 
Said Benachour (IECN), Razvan Gabriel Iagar (IMAR), Philippe Laurencot   (IMT)
math.AP