Inverse Problems for the Heat Equation with Memory

Abstract

We study inverse boundary problems for a one dimensional linear integro-differential equation of the Gurtin--Pipkin type with the Dirichlet-to-Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator, we give the explicit formula for the solution of the problem with the observation on the semiaxis $t>0.$ For the observation on finite time interval, we prove the uniqueness result, which is similar to the local Borg--Marchenko theorem for the Schr\"odinger equation.