On a sideways parabolic equation

Abstract

is considered. This is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but one side is inaccessible to measurements. This problem is well known to be severely ill-posed: a small perturbation in the data, g, may cause dramatically large errors in the solution. The results available in the literature are mainly devoted to the case of constant coefficients, where one can find an explicit representation for the solution of the problem. In this paper a stability estimate of the Hölder type for the solution of this general problem is established, it is also shown how to apply the mollification method recently proposed by Dinh Nho Háo to solve the problem in a stable way.