Conditional stability for backward parabolic equations with Osgood coefficients

in "Analysis, Probability, Applications and Computation - Proceedings of the 11th ISAAC Congress, Vaxjo (Sweden) 2017", Karl-Olof Lindahl, Torsten Lindstrom, Luigi Rodino, Joachim Toft, Patrik Wahlberg (eds.), Birkhauser (2019)) 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.