  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.
