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

Author: 

Chen, G
Perthame, B

Publication Date: 

1 September 2009

Journal: 

Proceedings of the American Mathematical Society

Last Updated: 

2019-06-08T03:53:16.627+01:00

Issue: 

9

Volume: 

137

DOI: 

10.1090/S0002-9939-09-09898-0

page: 

3003-3011

abstract: 

We are interested in the large-time behavior of periodic entropy solutions in L∞ 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. © 2009 American Mathematical Society.

Symplectic id: 

203544

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article