NONLINEAR DIFFUSION EQUATIONS WITH VARIABLE COEFFICIENTS AS GRADIENT FLOWS IN WASSERSTEIN SPACES

We study existence and approximation of non-negative solutions of partial differential equations of the type partial derivative(t)u - div(A(del(f(u)) + u del V)) = 0 in (0, +infinity) x R-n, (0.1) where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, f : [0, +infinity) -> [0, +infinity) is a suitable non decreasing function, V : R-n -> R is a convex function. Introducing the energy functional phi(u) = integral(Rn) F(u(x))dx + integral(Rn) V (x)u(x)dx, where F is a convex function linked to f by f(u) = uF'(u) - F(u), we show that u is the "gradient flow" of phi with respect to the 2-Wasserstein distance between probability measures on the space R-n, endowed with the Riemannian distance induced by A(-1). 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.