On backward uniqueness for parabolic equations when Osgood continuity of the coefficients fails at one point

TL;DR

This paper establishes the uniqueness of solutions for backward parabolic equations with coefficients that are Osgood continuous for positive times but lack this continuity at the initial time, addressing a specific regularity failure.

Contribution

It demonstrates backward uniqueness for parabolic equations with coefficients that are discontinuous at the initial time, extending previous results to cases with Osgood continuity failure.

Findings

Proves uniqueness under Osgood continuity in positive time

Addresses the case where coefficients are not continuous at t=0

Extends the theory of backward parabolic equations

Abstract

We prove uniqueness for backward parabolic equations whose coefficients are Osgood continuous in time for $t>0$ but not at $t=0$.