Anisotropic tempered diffusion equations

Abstract

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 [F. Andreu, V. Caselles, J. M. Mazo ́n, Nonlinear Anal. 61 (2005), J. Eur. Math. Soc. 7 (2005)], 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.