# Large time behavior for the fast diffusion equation with critical absorption

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1 EDP - Equations aux dérivées partielles IECL - Institut Élie Cartan de Lorraine 2 IMAR -Simion Stoilow- Institute of Mathematics 3 IMT - Institut de Mathématiques de Toulouse UMR5219

Abstract : We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption$$\partial {t}u-\Delta u^m+u^q=0 \quad \quad \hbox{in} \0,\infty\times eal^N\ ,$$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 $q > 1$.

Keywords : lower bound large time behavior gradient estimates fast diffusion critical absorption

Autor: Said Benachour - Razvan Iagar - Philippe Laurencot -

Fuente: https://hal.archives-ouvertes.fr/

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