Non-linear heat-transport equation beyond the Fourier law: The influence of non-linear effects in steady states

In the framework of Extended Irreversible Thermodynamics and in agreement with the second law of thermodynamics, in this paper we derive a heat-transport equation which generalizes the classical Fourier law and contains non-local and non-linear contributions related to the heat flux. In order to delve mainly into the possible influence of the non-linear effects on the heat transfer at nanoscale, only meant as the direct consequence of the non-linear terms appearing in the heat-transport equation, we focus on the longitudinal propagation of heat flux in a thin layer in steady states. The analysis is performed by also employing enhanced boundary conditions which allow to account for the phonon-boundary scattering. A comparison with the results arising from another non-local and non-linear proposal of heat-transport equation is performed as well.