Stability results for solutions of a linear parabolic noncharacteristic Cauchy problem

A noncharacteristic Cauchy problem for a one-dimensional linear parabolic equation with coefficients depending on the time is studied. In general, the solutions of the problem do not depend continuously on the Cauchy data. However, it is proved in the paper that solutions that are bounded in the supremum norm depend continuously on the Cauchy data. Furthermore, several stability estimates characterizing this continuity are established. Similar results are also obtained for regular level lines of such solutions.