1. Large time behavior for the fast diffusion equation with critical absorption
- Author
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Said Benachour, Philippe Laurençot, Razvan Gabriel Iagar, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Zero order absorption ,35B33, 35B40, 35B45, 35K67 ,Diffusion equation ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,gradient estimates ,01 natural sciences ,Upper and lower bounds ,large time behavior ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,fast diffusion ,FOS: Mathematics ,Initial value problem ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Absorption (logic) ,0101 mathematics ,lower bound ,critical absorption ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
International audience; 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 $q > 1$.
- Published
- 2016
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