A Note on Quasilinear Parabolic Equations on Manifolds

TL;DR

This paper establishes fundamental existence, uniqueness, and stability results for smooth solutions to high-order quasilinear parabolic equations on closed manifolds, advancing the mathematical understanding of such equations.

Contribution

It provides the first comprehensive proof of short time existence and uniqueness for arbitrary even order quasilinear locally parabolic equations on closed manifolds.

Findings

Proved short time existence of solutions

Established uniqueness of solutions

Demonstrated continuous dependence on initial data

Abstract

We prove short time existence, uniqueness and continuous dependence on the initial data of smooth solutions of quasilinear locally parabolic equations of arbitrary even order on closed manifolds.