On the Existence of Self-Similar Solutions in the Thermostatted Kinetic Theory with Unbounded Activity Domain

Abstract

This paper is devoted to the mathematical analysis of a spatially homogeneous thermostatted kinetic theory framework with an unbounded activity domain. The framework consists of a partial integro-differential equation with quadratic nonlinearity where the domain of the activity variable is the whole real line. Specifically the mathematical analysis refers firstly to the existence and uniqueness of the solution for the related initial boundary value problem; Secondly the investigations are addressed to the existence of a class of self-similar solutions by employing the Fourier transform method. In particular the main result is obtained for a nonconstant interaction rate and a nonconstant force field. Conclusions and perspectives are discussed in the last section of the paper.