On long time behavior of periodic entropy solutions of a degenerate non-linear parabolic equation

TL;DR

This paper proves that space-periodic entropy solutions of a one-dimensional degenerate parabolic equation asymptotically converge to a traveling wave, revealing linear flux and constant diffusion in the limit.

Contribution

It establishes the long-time behavior and asymptotic convergence of solutions, providing new insights into the structure of the limit profile.

Findings

Solutions converge to a traveling wave asymptotically

Flux function becomes linear with the wave speed

Diffusion function remains constant in the limit

Abstract

We prove the asymptotic convergence of a space-periodic entropy solution of a one-dimensional degenerate parabolic equation to a traveling wave. It is also shown that on a segment containing the essential range of the limit profile the flux function is linear (with the slope equaled to the speed of the traveling wave) and the diffusion function is constant.