On Carleman-type Formulas for Solutions to the Heat Equation

Abstract

We apply the method of integral representations to study the ill-posed Cauchy problem for the heat equa- tion. More precisely we recover a function, satisfying the heat equation in a cylindrical domain, via its values and the values of its normal derivative on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural (anisotropic) spaces (Sobolev and H¨older spaces, etc). Finally, we obtain a uniqueness theorem for the problem and a criterion of its solvability and a Carleman-type formula for its solution.