American Institute of Mathematical Sciences

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September  2003, 9(5): 1081-1104. doi: 10.3934/dcds.2003.9.1081

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, Norway
2. 

Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

Received  June 2002 Revised  February 2003 Published  June 2003

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.
Keywords: rough coefficient, entropy solution, uniqueness, stability., Degenerate parabolic equation.
Mathematics Subject Classification: 35K65, 35L6.
Citation: Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081
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