Conditional stability for backward parabolic equations with Osgood coefficients

TL;DR

This paper investigates the stability of solutions to backward parabolic equations with coefficients that are Osgood continuous, extending previous results that required Log-Lipschitz continuity, and demonstrates conditions under which stability holds or fails.

Contribution

It improves existing stability results by showing stability under Osgood continuity and constructs counterexamples where Log-Lipschitz conditions fail.

Findings

Stability holds for coefficients with Osgood continuity.

Counterexamples show stability fails without Osgood or Log-Lipschitz conditions.

Provides a broader understanding of coefficient regularity needed for stability.

Abstract

The interest of the scientific community for the existence, uniqueness and stability of solutions to PDE's is testified by the numerous works available in the literature. In particular, in some recent publications on the subject an inequality guaranteeing stability is shown to hold provided that the coefficients of the principal part of the differential operator are Log-Lipschitz continuous. Herein this result is improved along two directions. First, we describe how to construct an operator, whose coefficients in the principal part are not Log-Lipschitz continuous, for which the above mentioned inequality does not hold. Second, we show that the stability of the solution is guaranteed, in a suitable functional space, if the coefficients of the principal part are Osgood continuous.