几类非线性色散波方程解的性质研究

本项目主要研究来源于流体力学、晶格动力学、弹性力学等实际领域中出现的几类分别具有高阶导数、高次非线性行和多个分支的色散波方程。首先探讨这几类方程在不同函数空间(特别是临界Besov)中的Cauchy问题、初边值问题以及周期问题解的局部适定性；其次研究方程的解在适当假设条件下的全局存在性、爆破现象以及在加权Sobolev空间中的持续性；再次考虑色散波方程的弱解和Peakon解；最后通过数值方法来探究方程解的长时间动力学行为。希望本课题的研究能丰富和发展非线性色散波方程理论，并对相关科学技术领域的发展起促进作用。

In project, we mainly study several nonlinear dispersive wave equations with higher-order derivative, higher-order nonlinearities and multi-component, respectively, which come from fluid mechanics, lattice dynamics, elastic mechanics and other science. The local well-posedness of solutions for the Cauchy problem, initial boundary problem and periodic problem in different spaces(especially in critical Besov space) are studied. Under some assuptions, the global solutions, blow-up scenario and the persistence properties in weighted spaces for the solutions are given. Then we consider the weak solutions and peakon solutions. Finally, we show the long time dynamics behavior of solutions by numerical method. We hope that this project can enrich and develop the theorem of nonlinear dispersive wave equations, and promote the development of the related applied sciences.