Evolution Problems in Materials with Memory: Exponential Decay of Solutions and Free Energies

The model of a rigid linear heat conductor with memory conductor is considered and some new results are given. 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 model proposed by Fabrizio, Gentili and Reynolds [2] and the constitutive equations therein, depends on the history of the material. When initial and boundary conditions are assigned, the evolution problem turns out to be an integro-differential one whose kernel depends on time through the present time as well as the past history of the material. Hence, the choice of suitable expressions of the minimum free energy and of the thermal work [1] is crucial in studying evolution problems in materials with memory. Indeed, the functional spaces where solutions are looked for, are obtained by the requirement to be meaningful both under the physical as well as the analytic viewpoint. Then, on application of existence and uniqueness results recently proved [1] [3]