Mathematical aspects of non-Fourier heat equations

Due to technological advancement, as materials with complex structures (e.g., metamaterials and foams) appear in practice there is a need to develop advanced thermal models. These are called non-Fourier equations, and all have particular mathematical properties differing from the conventional attributes of Fourier's law. The present paper discusses the thermodynamic origin of non-Fourier equations and their consequences. The second law of thermodynamics influences the relations among the material parameters, and therefore, it restricts how the temperature-dependent properties can be included in the model. Furthermore, we present the properties of initial and boundary conditions, since these are crucial in solving any practical problems and are different from the usual interpretation used for the Fourier equation.