On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

Kenneth Hvistendahl Karlsen; Nils Henrik Risebro

doi:10.3934/dcds.2003.9.1081

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2003, Volume 9, Issue 5: 1081-1104. Doi: 10.3934/dcds.2003.9.1081

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On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

Kenneth Hvistendahl Karlsen^1, and
Nils Henrik Risebro^2,

1.
Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N–5008 Bergen
2.
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo

Received: June 2002

Revised: February 2003

Published: September 2003

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Abstract

We study nonlinear degenerate parabolic equations where the flux function $f(x,t,u)$ does not depend Lipschitz continuously on the spatial location $x$. By properly adapting the "doubling of variables" device due to Kružkov [25] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form $k(x)f(u)$, where $k(x)$ is a vector-valued function and $f(u)$ is a scalar function.

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Mathematics Subject Classification: 35K65, 35L65.

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