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

We obtain solutions of the nonlinear degenerate parabolic equation $$\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}=div\{\rho \mathrm{\nabla }{c}^{\star }[\mathrm{\nabla }(F^{\mathrm{\prime }}(\rho )+V)]\}$$ 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 \cite{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.