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

 $\frac{\partial \rho }{\partial t}=\mathrm{div}\left\{\rho \nabla {c}^{*}\left[\nabla \left(F\text{'}\left(\rho \right)+V\right)\right]\right\}\phantom{\rule{4pt}{0ex}}\mathrm{in}\phantom{\rule{3.30002pt}{0ex}}\left(0,\infty \right)×\Omega ,\mathrm{and}\rho \left(t=0\right)={\rho }_{0}\mathrm{in}\phantom{\rule{3.30002pt}{0ex}}\left\{0\right\}×\Omega ,$ (1)
where $\Omega$ is ${ℝ}^{n}$, or a bounded domain of ${ℝ}^{n}$ in which case $\rho \nabla {c}^{*}\left[\nabla \left(F\text{'}\left(\rho \right)+V\right)\right]\phantom{\rule{0.277778em}{0ex}}·\phantom{\rule{0.277778em}{0ex}}\nu =0$ on $\left(0,\infty \right)×\partial \Omega$. 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 $\Omega$ est soit ${ℝ}^{n}$, ou un domaine borné de ${ℝ}^{n}$ auquel cas $\rho \nabla {c}^{*}\left[\nabla \left(F\text{'}\left(\rho \right)+V\right)\right]\phantom{\rule{0.277778em}{0ex}}·\phantom{\rule{0.277778em}{0ex}}\nu =0$ sur $\left(0,\infty \right)×\partial \Omega$. 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.

Accepted:
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
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00352-2/

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