The model of a rigid linear heat conductor with memory is analyzed. Specifically, an evolution problem which describes the time evolution
of the temperature distribution within a rigid heat conductor with memory
is studied. The attention is focussed on the thermodynamical state of such a
rigid heat conductor which, according to the adopted constitutive equations,
depends on the history of the material; indeed, the dependence of the heat flux
on the history of the temperature’s gradient is modeled via an integral term.
Thus, the evolution problem under investigation is an integro-differential one
with assigned initial and boundary conditions. Crucial in the present study are
suitable expressions of an appropriate free energy and thermal work, related
one to the other, which allow to construct functional spaces which are meaningful both under the physical as well as the analytic viewpoint. On the basis
of existence and uniqueness results previously obtained, exponential decay at
infinity is proved.