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

الأصل · EN

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 less uniform convexity assumption than that imposed by Alt and Luckhaus in their pioneering work luckhaus:quasilinear. In fact, their assumption may fail in our equation. This class of problems includes the Fokker-Planck equation, the porous-medium equation, the fast diffusion equation, and the parabolic p-Laplacian equation.