非线性高阶发展方程整体解的渐近性态

The project will investigate the long time behavior of solutions for the fourth-order beam equation and IMBq(Improved Modified Boussinesq) type equation which describes, respectively, the dynamic deformation of elastic beam and the propagation of small amplitude long waves on water. By comprehensive use of the theories from functional analysis, Sobolev space and modern partial differential equations, together with the semigroup methods and the techniques of energy estimates, we will study in this projection the existence of solutions, the existence, regularity, dimensional estimates for the global attractor and the exponential attractors. . For the first equation, we will focus on the dynamical behavior of solutions to the singularly perturbed damped beam equation with supercritical nonlinearities, and the main difficulty is to establish the existence of global solutions and some compactness of the semigroup. For the second equation, we will study the large time behavior of solutions for the IMBq type equation which contains the origin model, the difficulty lies in the treatment of the dispersive term and complicated nonlinear term. These problems not only have concrete background in application, but also play an important role in the theory of nonlinear partial differential equations.

本项目主要研究描述弹性梁的动力学变形的四阶梁方程，水面上小振幅长波传播的IMBq(Improved Modified Boussinesq)型方程解的长时间性态。综合运用泛函分析理论，索伯列夫空间理论，现代偏微分方程理论，半群方法和能量估计技巧，研究这两类方程解的存在性，整体吸引子以及指数吸引子的存在性、正则性、维数估计等问题。. 对于前一类方程，重点研究带有弱阻尼，超临界指数非线性项的奇异扰动梁方程解的动力学行为，主要困难在于如何得到整体解的存在性和半群的某种紧性。对于后一类方程，重点研究包含原始模型的IMBq型方程解的大时间行为，主要困难在于色散项和复杂非线性项的处理。这两类方程不仅有具体的应用背景，而且作为对非线性偏微分方程的研究，在理论上也有重要意义。

本项目研究非线性高阶发展方程解的长时间性态。综合运用泛函分析理论，无穷维动力系统理论，Sobolev空间理论，研究了在科学技术中提出的IMBq型方程和Kirchhoff型方程的整体适定性及解的长时间动力学行为。取得的主要结果如下：(1)关于IMBq型方程，我们主要讨论了非自治情形，当外力项满足一类非平移紧的条件时，我们得到了一致吸引子的存在性。(2)我们证明了一个推广的Gronwall型不等式，这将有助于得到无穷维动力系统的耗散性，有助于证明带有超临界Sobolev指数项的非线性发展方程的整体吸引子的存在性。(3)作为新的Gronwall型不等式的应用，我们研究了一类具强阻尼的Kirchhoff型方程解的渐近行为，得到了整体吸引子的存在性。