Stationary solutions, intermediate asymptotics and large-time behaviour of type II Streater's models

Abstract

In this paper, we study type II Streater's models. These models describe the coupled evolution of the density of a cloud of particles in an external potential and a temperature, preserving the energy, with eventually a nonlocal Poisson coupling. We introduce an entropy and consider in a bounded domain, or in an unbounded domain with a~confining external potential, the stationary solutions (with given mass and energy), for which we have existence and uniqueness results. The entropy is reinterpreted as a relative entropy which controls the convergence to the stationary solutions. We consider also the whole $\mathbb R^d$ space problems without exterior potential using time-dependent rescalings and show the existence of intermediate asymptotics in special cases.