On equivalence of entropy and viscosity solutions to degenerate parabolic equations and applications

TL;DR

This paper proves the equivalence of entropy and viscosity solutions for certain degenerate parabolic equations and applies this to analyze their long-term behavior in periodic settings.

Contribution

It establishes the equivalence of entropy and viscosity solutions for anisotropic degenerate equations, enabling new insights into their large-time behavior.

Findings

Proved equivalence of entropy and viscosity solutions.

Analyzed large-time behavior of solutions.

Applied results to periodic degenerate equations.

Abstract

Here, we consider anisotropic degenerate parabolic-hyperbolic equations and degenerate quasilinear Hamilton-Jacobi equations. We prove the equivalence of two notions of entropy and viscosity solutions of two equations, and apply it to obtain a large-time behavior of viscosity solutions to quasilinear Hamilton-Jacobi equations, and entropy solutions to degenerate parabolic-hyperbolic equations in a periodic setting.