  We introduce a functional framework which is specially suited to formulate
several classes of anisotropic evolution equations of tempered diffusion type.
Under an amenable set of hypothesis involving a very natural potential
function, these models can be shown to belong to the entropy solution framework
devised by 4, 5, therefore ensuring well-posedness. We connect the properties
of this potential with those of the associated cost function, thus providing a
link with optimal transport theory and a supply of new examples of relativistic
cost functions. Moreover, we characterize the anisotropic spreading properties
of these models and we determine the Rankine-Hugoniot conditions that rule the
temporal evolution of jump hypersurfaces under the given anisotropic flows.
