Expanding solutions of quasilinear parabolic equations

TL;DR

This paper develops a Taylor-like expansion for solutions of quasilinear parabolic equations on closed manifolds, linking the local geometric properties to the solution's behavior via advanced regularity and singular analysis techniques.

Contribution

It introduces a novel expansion method that explicitly incorporates local geometric data for quasilinear parabolic equations on manifolds.

Findings

Explicit powers of the expansion depend on local geometry.

The expansion describes solutions' local behavior with high precision.

Method bridges geometric analysis and PDE regularity theory.

Abstract

By using the theory of maximal $L^{q}$-regularity and methods of singular analysis, we show a Taylor's type expansion--with respect to the geodesic distance around an arbitrary point--for solutions of quasilinear parabolic equations on closed manifolds. The powers of the expansion are determined explicitly by the local geometry, whose reflection to the solutions is established through the local space asymptotics.