Conditional stability for backward parabolic equations with $\rm{Log}\rm{Lip}_t \times \rm{Lip}_x$-coefficients

TL;DR

This paper improves stability results for backward parabolic equations with coefficients that are Log-Lipschitz in time and Lipschitz in space, removing the previous need for higher regularity in space.

Contribution

It extends previous stability results by replacing the $C^2$ regularity in space with Lipschitz continuity, using advanced harmonic analysis tools.

Findings

Removed the $C^2$ regularity assumption in space.

Established conditional stability under weaker regularity conditions.

Applied Littlewood-Paley theory and Bony's paraproduct techniques.

Abstract

In this paper we present an improvement of [Math. Ann. 345 (2009), 213--243], where the authors proved a result concerning continuous dependence for backward parabolic operators whose coefficients are Log-Lipschitz in $t$ and $C^2$ in $x$. The $C^2$ regularity with respect to $x$ had to be assumed for technical reasons. Here we remove this assumption, replacing it with Lipschitz-continuity. The main tools in the proof are Littlewood-Paley theory and Bony's paraproduct as well as a result of Coifman and Meyer [Ast\'erisque 57, 1978, Th. 35].