Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

We study existence and approximation of non-negative solutions of a class of nonlinear diffusion equations with variable coefficients. The results are obtained interpreting this kind of equations as "gradient flow" of a suitable energy functional with respect to a suitable Wasserstein distance. More precisely the Wasserstein distance between probability measures on the euclidean space endowed with the Riemannian distance induced by the inverse matrix of the coefficients of the equation. Long time asymptotic behavior and rate decay to stationary state for solutions of the equation are studied. A contraction property in Wasserstein distance for solutions of the equation is studied in a particular case.