Preprint 2000-015

On the Uniqueness and Stability of Entropy Solutions of Nonlinear Degenerate Parabolic Equations with Rough Coefficients

Kenneth Hvistendahl Karlsen and Nils Henrik Risebro

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 Kru\v{z}kov \cite{Kruzkov} and Carrillo \cite{Carrillo}, 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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Kenneth Hvistendahl Karlsen, <>
Nils Henrik Risebro, <>
Publishing information:
Preprint updated December 8, 2000.
Submitted by:
<> April 28 2000.

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