Large-Time Behavior of Periodic Entropy Solutions to Anisotropic Degenerate Parabolic-Hyperbolic Equations

TL;DR

This paper investigates the long-term behavior of periodic entropy solutions to complex anisotropic degenerate parabolic-hyperbolic equations, highlighting how diffusion influences decay and developing a new approach based on kinetic formulation.

Contribution

It introduces a novel method using kinetic formulation and time translation techniques to analyze decay of solutions, overcoming limitations of previous self-similar approaches.

Findings

Established decay of periodic solutions in anisotropic degenerate equations

Demonstrated the role of nonlinearity and diffusivity in large-time behavior

Developed a new approach applicable to complex parabolic-hyperbolic equations

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.