Probability Theory/Ordinary Differential Equations
Asymptotic behavior for doubly degenerate parabolic equations
[Comportement asymptotique des équations paraboliques doublement dégénérées]
Comptes Rendus. Mathématique, Tome 337 (2003) no. 5, pp. 331-336.

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.

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.

Reçu le :
Accepté le :
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DOI : 10.1016/S1631-073X(03)00352-2
Agueh, Martial 1

1 Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada
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Agueh, Martial. Asymptotic behavior for doubly degenerate parabolic equations. Comptes Rendus. Mathématique, Tome 337 (2003) no. 5, pp. 331-336. doi : 10.1016/S1631-073X(03)00352-2. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00352-2/

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