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

Chen, Gui-Qiang; Perthame, Benoît

doi:10.1090/S0002-9939-09-09898-0

Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations
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by Gui-Qiang Chen and Benoît Perthame PDF

Proc. Amer. Math. Soc. 137 (2009), 3003-3011 Request permission

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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