Materials with memory: free energies \& solution exponential decay
The temperature evolution within a rigid heat conductor with memory is here considered. Precisely speaking an evolution problem which describes the time evolution of the temperature distribution within a rigid heat conductor with memory is studied. He adopted constitutive equations depending on the history of the material, more precisely speaking the dependence of the heat flux on the history of the temperature gradient is presented by an integral term. Such approach is based on the model proposed by \textit{M. E. Gurtin} and \textit{A. C. Pipkin} [Arch. Ration. Mech. Anal. 31, 113--126 (1968; Zbl 0164.12901)]. So, he considered the evolution problem which is an integro-differential one with assigned initial and boundary condition. The aim of this paper is to prove the exponential decay of solutions to the formulated evolution problem. In the proof the author used the semigroup theory. To show the exponential decay of the solution the author used the free energy functional (so called Graffi-Volterra free energy).
scientific article; zbMATH DE number 5821311