  We initiate the study of non-Archimedean reaction-ultradiffusion equations
and their connections with models of complex hierarchic systems. From a
mathematical perspective, the equations studied here are the p-adic counterpart
of the integro-differential models for phase separation introduced by Bates and
Chmaj. Our equations are also generalizations of the ultradiffusion equations
on trees studied in the 80's by Ogielski, Stein, Bachas, Huberman, among
others, and also generalizations of the master equations of the Avetisov et al.
models, which describe certain complex hierarchic systems. From a physical
perspective, our equations are gradient flows of non-Archimedean free energy
functionals and their solutions describe the macroscopic density profile of a
bistable material whose space of states has an ultrametric structure. Some of
our results are p-adic analogs of some well-known results in the Archimedean
settting, however, the mechanism of diffusion is completely different due to
the fact that it occurs in an ultrametric space.
