Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory

Abstract

We obtain solutions of the nonlinear degenerate parabolic equation ∂ρ/∂t = div {ρ∇c*[∇(F'(ρ) + V)]} as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires fewer uniform convexity assumptions than those imposed by Alt and Luckhaus in their pioneering work [4]. In fact, their assumptions may fail in our equation. This class of equations includes the Fokker-Planck equation, the porous-medium equation, the fast diffusion equation and the parabolic p-Laplacian equation.