Asymptotic behavior for doubly degenerate parabolic equations

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

(1)$\text{?ρ}\text{?t}\text{=}\text{div}\text{ρ?c}{}^1\text{?}\text{F′(ρ)+V}\text{in}\text{(0,∞)×Ω,}\text{and}\text{ρ(t=0)=ρ}{}_0\text{in}\text{{0}×Ω,}$

where $\text{Ω}$ is $\text{R}{}^{\mathrm{n}}$, or a bounded domain of $\text{R}{}^{\mathrm{n}}$ in which case $\text{ρ?c}{}^1\text{[?(F′(ρ)+V)]·ν=0}$ on $\text{(0,∞)×?Ω}$. 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. To cite this article: M. Agueh, C. R. Acad. Sci. Paris, Ser. I 337 (2003).