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A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations. (English) Zbl 1072.35545

Summary: Following the lead of J. Carrillo [Arch. Ration. Mech. Anal. 147, 269–361 (1999; Zbl 0935.35056)], recently several authors have used Kruzhkov’s device of “doubling the variables” to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order differential operator is not allowed to depend explicitly on the spatial variable, which certainly restricts the range of applications of entropy solution theory. The purpose of this paper is to extend a version of Carrillo’s uniqueness result to a class of degenerate parabolic equations with spatially dependent second order differential operator. The class is large enough to encompass several interesting nonlinear partial differential equations coming from the theory of porous media flow and the phenomenological theory of sedimentation-consolidation processes.

MSC:

35K65 Degenerate parabolic equations
35B25 Singular perturbations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K15 Initial value problems for second-order parabolic equations
35M10 PDEs of mixed type

Citations:

Zbl 0935.35056
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References:

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