Quantitative unique continuation for a parabolic equation

TL;DR

This paper establishes bounds on the order of vanishing for solutions to a class of parabolic equations, extending the understanding of quantitative unique continuation from elliptic to parabolic cases.

Contribution

It provides the first quantitative bounds on the order of vanishing for parabolic equations with bounded coefficients, matching known elliptic bounds.

Findings

Order of vanishing is bounded by a constant involving norms of v and w.

The bounds match previously known elliptic case results.

The results apply to solutions with bounded coefficients in parabolic equations.

Abstract

We address the quantitative uniqueness properties of the solutions of the parabolic equation $ \partial_t u - \Delta u = w_j (x,t) \partial_j u + v(x,t) u $ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of vanishing is bounded by $C(\Vert v\Vert_{L^\infty}^{2/3}+\Vert w\Vert_{L^\infty}^2)$ matching the upper bound previously established in the elliptic case. in the elliptic case.