Equivalent histories, states and a variational problem for a rigid linear conductor with memory

The model of a rigid linear heat conductor which exibits a constitutive equation with memory for the heat flux can be characterized by processes and states. The equivalence between histories is introduced in order to consider minimal states. The inversion of the constitutive equation, in which the heat flux appears as a linear functional of the history of the temperature gradient, allows to consider states expressed in terms of the heat flux vector instead of the temperature gradient. A variational formulation of an evolution problem with mixed initial boundary conditions is also given.