Materials with memory: Free energies & solution exponential decay

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.