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Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations


Authors: Gui-Qiang Chen and Benoît Perthame
Journal: Proc. Amer. Math. Soc. 137 (2009), 3003-3011
MSC (2000): Primary 35K65, 35K15, 35B10, 35B40, 35D99; Secondary 35K10, 35B30, 35B41, 35M10, 35L65
DOI: https://doi.org/10.1090/S0002-9939-09-09898-0
Published electronically: April 10, 2009
MathSciNet review: 2506459
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Abstract: We are interested in the large-time behavior of periodic entropy solutions in $ L^\infty$ to anisotropic degenerate parabolic-hyperbolic equations of second order. Unlike the pure hyperbolic case, the nonlinear equation is no longer self-similar invariant, and the diffusion term in the equation significantly affects the large-time behavior of solutions; thus the approach developed earlier, based on the self-similar scaling, does not directly apply. In this paper, we develop another approach for establishing the decay of periodic solutions for anisotropic degenerate parabolic-hyperbolic equations. The proof is based on the kinetic formulation of entropy solutions. It involves time translations and a monotonicity-in-time property of entropy solutions and employs the advantages of the precise kinetic equation for the solutions in order to recognize the role of nonlinearity-diffusivity of the equation.


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Additional Information

Gui-Qiang Chen
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China – and – Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730
Email: gqchen@math.northwestern.edu

Benoît Perthame
Affiliation: Université Pierre et Marie Curie, Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005, Paris, France – and – Institut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, France
Email: benoit.perthame@upmc.fr

DOI: https://doi.org/10.1090/S0002-9939-09-09898-0
Keywords: Periodic solutions, entropy solutions, decay, large-time behavior, kinetic formulation, degenerate parabolic equations, anisotropic diffusion, nonlinearity-diffusivity
Received by editor(s): October 16, 2008
Published electronically: April 10, 2009
Additional Notes: The first author’s research was supported in part by the National Science Foundation under Grants DMS-0244473, DMS-0807551, DMS-0720925, and DMS-0505473; an Alexander von Humboldt Foundation Fellowship; and the Natural Science Foundation of China under Grant NSFC-10728101. This paper was written as part of the International Research Program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–09.
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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