Anomalous Heat Conduction

Since more than two centuries the phenomenon of heat transport in matter is a classical research topic in Physics and applied Mathematics. Such a long time lapse testifies to the difficulties encountered in developing a convincing and general theory. Already at the end of the XVIII century the design of the first steam engines attracted the interest of the scientists on such a phenomenon. At the same time, Thermodynamics was going to gain a prominent position in Science, by occupying a seat of equal importance as Mechanics. Besides the attention induced by technological applications that led to the industrial revolution, the problem of heat transport played a central role in the scientific debate concerning the geological discoveries, which were unveiling the secrets about the formation of the solar system. In fact, the observation that a thermal gradient is present inside the Earth led many scientists to conjecture that our planet was originally a fire ball, that had progressively cooled down, while keeping in its interior track of its original nature. The Moon was also assumed having undergone the same fate, although its reduced size with respect to the Earth had fastened its cooling. Great efforts and many debates were devoted to this challenging problem, which concerned aspects inherent the very creation of the Universe. The first basic contributions to the problem were provided by the French scientist J.B.J. Fourier already in 1808. His scientific masterpiece entitled “Theorie Analytique de la Chaleur” was finally published in 1822. It contains the first mathematical theory of heat transport in matter. In particular, Fourier provided a convincing explanation for the existence of a thermal gradient inside the Earth. The basic ingredient of Fourier approach was that the phenomenon of heat flow through matter can be described by a diffusion equation, which contains a first order derivative in time and a