Tempered diffusion: A transport process with propagating fronts and inertial delay

The speed of sound is the highest admissible free velocity in a medium. This property is lost in the classical transport theory that predicts the unphysical divergence of the flux with gradients. Keeping the acoustic speed provides the means to control the growth of the flux and enables us to derive a better transport theory; flux saturates as the gradients became unbounded. Initial discontinuities persist for a finite time and diffusion fronts are convected with a finite speed. Various applications are considered (heat or mass transfer, plasma diffusion, and hydrodynamics).