The evolution of the entropy production in solids due to heat transfer is usually associated with the Prigogine’s minimum entropy production principle. In this paper, we propose a critical review of the results of Prigogine and some comments on the succeeding literature. We suggest a characterization of the evolution of the entropy production of the system through the generalized Fourier modes, showing that they are the only states with a time independent entropy production. The variational approach and a Lyapunov functional of the temperature, monotonically decreasing with time, are discussed. We describe the analytic properties of the entropy production as a function of time in terms of the generalized Fourier coefficients of the system. Analytical tools are used throughout the paper and numerical examples will support the statements.