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 . Moreover, we give sufficient conditions for the existence of a non-trivial travelling solution on , which we call a bimodal travelling wave solution.

近年来，在实验的基础上引入了自走材料集体运动的数学模型，该模型的数值分析在对集体运动机理的理论认识上取得了进展。另一方面，对数学模型的数学严格研究很少。在这项研究中，为了为数学模型及其数值分析提供数学证明，我们证明了自走运动数学模型的唯一解的全局存在性。此外，我们给出了在 上存在非平凡行波解的充分条件，我们称之为双峰行波解。