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

Dipartimento di Scienze e Tecnologie Avanzate, Università degli Studi del Piemonte Orientale, Italy. stefano.lisini@unipv.it

Received: 15 May 2007
Revised: 4 February 2008

Abstract

We study existence and approximation of non-negative solutions of partial differential equations of the type 
 where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, is a suitable non decreasing function, is a convex function. Introducing the energy functional , where F is a convex function linked to f by , we show that u is the “gradient flow” of ϕ with respect to the 2-Wasserstein distance between probability measures on the space , endowed with the Riemannian distance induced by In the case of uniform convexity of V, long time asymptotic behaviour and decay rate to the stationary state for solutions of equation (0.1) are studied. A contraction property in Wasserstein distance for solutions of equation (0.1) is also studied in a particular case.