Advances in Differential Equations

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

Martial Agueh

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.


Export citation