Sharp order of vanishing for parabolic equations, nodal set estimates and Landis type results

TL;DR

This paper introduces a sharp estimate for the vanishing order of solutions to parabolic equations with variable coefficients, extending nodal set estimates and Landis type results to broader contexts.

Contribution

It provides a new sharp vanishing order estimate for parabolic equations and generalizes nodal set bounds and Landis type results for solutions with real-analytic coefficients.

Findings

Sharp vanishing order estimate for parabolic solutions

Generalized nodal set estimates at specific time levels

Landis type results for global solutions

Abstract

We establish a new sharp estimate of the order of vanishing of solutions to parabolic equations with variable coefficients. For real-analytic leading coefficients, we prove a localised estimate of the nodal set, at a given time-level, that generalises the celebrated one of Donnelly and Fefferman. We also establish Landis type results for global solutions.