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 $p=2$.