Close-to-Fourier heat conduction equation of solids of constant mass density

Heat conduction close-to-Fourier means that we look for a minimal extension of heat conduction theory using the usual Fourier expression of the heat flux density and modifying that of the internal energy as minimally as possible by choosing the minimal state space. Applying Liu's procedure results in the class of materials and a dierential equation both belonging to the close-to-Fourier case of heat conduction.