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 ap partial derivativerho/partial derivativet = div{rhodelc*[del(F'(rho)+V)]} in(0, infinity) x Omega, and rho(t = 0) = rho(0) in{0} x Omega, where Q is R-n, or a bounded domain of R-n in which case rhodelc*[del(F'(rho) + V)].nu = 0 on (0, infinity) x partial derivativeOmega. 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. (C) 2003 Academie des sciences. Published by Editions scientitiques et medicales Elsevier SAS. All rights reserved.