## Advances in Differential Equations

- Adv. Differential Equations
- Volume 10, Number 3 (2005), 309-360.

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

#### Abstract

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

#### Article information

**Source**

Adv. Differential Equations Volume 10, Number 3 (2005), 309-360.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867881

**Mathematical Reviews number (MathSciNet)**

MR2123134

**Zentralblatt MATH identifier**

1103.35051

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35K60: Nonlinear initial value problems for linear parabolic equations 35K65: Degenerate parabolic equations

#### Citation

Agueh, Martial. Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Adv. Differential Equations 10 (2005), no. 3, 309--360. https://projecteuclid.org/euclid.ade/1355867881