Chapter 3 -- Uniqueness and Stability in the Cauchy Problem

Abstract

In this chapter we formulate and in many cases prove results on the uniqueness and stability of solutions of the Cauchy problem for general partial differential equations. One of the basic tools is Carleman-type estimates. In Section 3.1 we describe the results for a simplest problem of this kind (the backward parabolic equation), where a choice of the weight function in Carleman estimates is obvious, and the method is equivalent to that of the logarithmic convexity. In Section 3.2 we formulate general conditional Carleman estimates and their simplifications for second-order equations, and we apply the results to the general Cauchy problem and give numerous counterexamples showing that the assumptions of positive results are quite sharp.