Conditional stability up to the final time for backward-parabolic equations with Log-Lipschitz coefficients

TL;DR

This paper establishes logarithmic conditional stability up to the final time for backward-parabolic equations with coefficients that are Log-Lipschitz in time and Lipschitz in space, extending previous results.

Contribution

It proves the first stability result up to the final time for such equations with Log-Lipschitz coefficients, filling a gap in the existing literature.

Findings

Proves logarithmic conditional stability up to the final time.

Extends previous stability results to Log-Lipschitz coefficients.

Complements earlier work by Del Santo and Prizzi.

Abstract

We prove logarithmic conditional stability up to the final time for backward-parabolic operators whose coefficients are Log-Lipschitz continuous in $t$ and Lipschitz continuous in $x$. The result complements previous achievements of Del Santo and Prizzi (2009) and Del Santo, Jaeh and Prizzi (2015), concerning conditional stability (of a type intermediate between Hoelder and logarithmic), arbitrarily closed, but not up to the final time.