Quantitative uniqueness of solutions to parabolic equations

TL;DR

This paper studies how solutions to parabolic equations uniquely vanish on compact manifolds, providing bounds on their vanishing order based on potential and coefficient norms, with new Carleman estimates and inequalities.

Contribution

It introduces new quantitative Carleman estimates and characterizes the vanishing order of solutions in terms of potential and coefficient norms on manifolds.

Findings

Established bounds on the vanishing order of solutions.

Derived new Carleman estimates for parabolic equations.

Linked vanishing rate to norms of potential and coefficient functions.

Abstract

We investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth manifolds. Quantitative uniqueness is a quantitative form of strong unique continuation property. We characterize quantitative uniqueness by the rate of vanishing. We can obtain the vanishing order of solutions by $C^{1, 1}$ norm of the potential functions, as well as the $L^\infty$ norm of the coefficient functions. Some quantitative Carleman estimates and three cylinder inequalities are established.