Comptes Rendus
Probability Theory/Ordinary Differential Equations
Asymptotic behavior for doubly degenerate parabolic equations
Comptes Rendus. Mathématique, Volume 337 (2003) no. 5, pp. 331-336.

We use mass transportation inequalities to study the asymptotic behavior for a class of doubly degenerate parabolic equations of the form

ρ t= div ρc * F'(ρ)+V in (0,)×Ω, and ρ(t=0)=ρ 0 in {0}×Ω,(1)
where Ω is n , or a bounded domain of n in which case ρc * [(F'(ρ)+V)]·ν=0 on (0,)×Ω. We investigate the case where the potential V is uniformly c-convex, and the degenerate case where V=0. In both cases, we establish an exponential decay in relative entropy and in the c-Wasserstein distance of solutions – or self-similar solutions – of (1) to equilibrium, and we give the explicit rates of convergence. In particular, we generalize to all p>1, the HWI inequalities obtained by Otto and Villani (J. Funct. Anal. 173 (2) (2000) 361–400) when p=2. This class of PDEs includes the Fokker–Planck, the porous medium, fast diffusion and the parabolic p-Laplacian equations.

Nous utilisons des inégalités de transport de masse pour étudier le comportement asymptotique des équations paraboliques doublement dégénérées de la forme (1), où Ω est soit n , ou un domaine borné de n auquel cas ρc * [(F'(ρ)+V)]·ν=0 sur (0,)×Ω. Nous examinons le cas où le potentiel V est uniformément c-convexe, et le cas dégénéré où V=0. Dans ces deux cas, nous montrons une décroissance exponentielle de la différence d'entropies et de la distance de Wasserstein – suivant le coût c – des solutions de l'équation et de sa solution stationnaire, et nous précisons les taux de convergence. En particulier, nous généralisons à tous les p>1 les inégalités HWI obtenues dans Otto et Villani (J. Funct. Anal. 173 (2) (2000) 361–400) lorsque p=2. Cette classe d'équations contient les équations de Fokker–Planck, des milieux poreux et du p-Laplacien.

Published online:
DOI: 10.1016/S1631-073X(03)00352-2

Martial Agueh 1

1 Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada
     author = {Martial Agueh},
     title = {Asymptotic behavior for doubly degenerate parabolic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {331--336},
     publisher = {Elsevier},
     volume = {337},
     number = {5},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00352-2},
     language = {en},
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JO  - Comptes Rendus. Mathématique
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PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00352-2
LA  - en
ID  - CRMATH_2003__337_5_331_0
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%A Martial Agueh
%T Asymptotic behavior for doubly degenerate parabolic equations
%J Comptes Rendus. Mathématique
%D 2003
%P 331-336
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Martial Agueh. Asymptotic behavior for doubly degenerate parabolic equations. Comptes Rendus. Mathématique, Volume 337 (2003) no. 5, pp. 331-336. doi : 10.1016/S1631-073X(03)00352-2.

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