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2005 Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory
Martial Agueh
Adv. Differential Equations 10(3): 309-360 (2005).

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.

Citation

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Martial Agueh. "Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory." Adv. Differential Equations 10 (3) 309 - 360, 2005.

Information

Published: 2005
First available in Project Euclid: 18 December 2012

zbMATH: 1103.35051
MathSciNet: MR2123134

Subjects:
Primary: 35K55
Secondary: 35K60, 35K65

Rights: Copyright © 2005 Khayyam Publishing, Inc.

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Vol.10 • No. 3 • 2005
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