Global existence of a unique solution and a bimodal travelling wave solution for the 1D particle-reaction-diffusion system

Abstract

In recent years, a mathematical model for collective motions of a self-propelled material have been introduced on the basis of the experiments and numerical analysis for the model have made progress in theoretical understanding of the mechanism of collective motions. On the other hand, there are few mathematically rigorous studies for the mathematical model. In this study, to provide mathematical justification for the mathematical model and its numerical analysis, we show the global existence of a unique solution of the mathematical model for a self-propelled motion on R . Moreover, we give sufficient conditions for the existence of a non-trivial travelling solution on R , which we call a bimodal travelling wave solution.