Asymptotic Behavior for Doubly Degenerate Parabolic Equations
We use mass transportation inequalities to study the asymptotic behavior for a class of parabolic equations which contains the Heat,
Fokker-Planck, porous-medium, fast diffusion, and parabolic p-Laplacian equations, and some generalizations of these equations. We establish an exponential decay in "relative entropy" and in the p-Wasserstein distance of solutions -- or self-similar solutions -- of these equations 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) (2002) 361-400) when p=2.