Large time behavior for the fast diffusion equation with critical absorption

Said Benachour (1) , Razvan Gabriel Iagar (2) , Philippe Laurencot (3)

1 IECL - Institut Élie Cartan de Lorraine
2 IMAR - "Simion Stoilow" Institute of Mathematics
3 IMT - Institut de Mathématiques de Toulouse UMR5219

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 $q > 1$.

Mots clés: critical absorption; large time behavior; gradient estimates; fast diffusion; lower bound

Domaines: Equations aux dérivées partielles [math.AP]

Said Benachour, Razvan Gabriel Iagar, Philippe Laurencot. Large time behavior for the fast diffusion equation with critical absorption. Journal of Differential Equations, 2016, 260, pp.8000-8024. ⟨10.1016/j.jde.2016.02.008⟩. ⟨hal-01061516⟩