Asymptotic behavior for doubly degenerate parabolic equations

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We use mass transportation inequalities to study the asymptotic behavior for a class of doubly degenerate parabolic equations of the form (1) ∂ρ ∂t = div ρ∇c ∗ ∇ F′(ρ)+V in (0,∞)×Ω, andρ(t=0)=ρ 0 in {0}×Ω, where Ω is R n , or a bounded domain of R n in which case ρ∇c ∗[∇(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. To cite this article: M. Agueh, C. R. Acad. Sci. Paris, Ser. I 337 (2003).