A new result on backward uniqueness for parabolic operators

TL;DR

This paper improves a backward uniqueness result for parabolic operators by replacing the regularity condition on coefficients from ${ m C}^2$ to Lipschitz continuity in the spatial variable, using Bony's paramultiplication.

Contribution

It demonstrates that backward uniqueness holds under weaker regularity assumptions on coefficients, specifically replacing ${ m C}^2$ with Lipschitz regularity in $x$, using advanced microlocal analysis techniques.

Findings

Backward uniqueness is valid with Lipschitz regularity in $x$

Bony's paramultiplication effectively improves previous results

The regularity condition on coefficients is relaxed from ${ m C}^2$ to Lipschitz

Abstract

Using Bony's paramultiplication we improve a result obtained in in a previous paper for operators having coefficients non-Lipschitz-continuous with respect to $t$ but ${\mathcal C}^2$ with respect to $x$, showing that the same result is valid when ${\mathcal C}^2$ regularity is replaced by Lipschitz regularity in $x$.