具有非局部耗散梁(板)方程解的适定性及长时间动力学行为的研究

The wave equation describing the nonlinear vibration of beam and plate components is a type of important nonlinear evolution equation. It is highly desirable to investigate the longtime dynamical behavior and dynamical stability of the components for its theoretical and practical significance. Because of nonlinearity of the equation and complexity of the described phenomena, the mathematical theory has been the focus of the research in nonlinear partial differential equations. Recently, there are some good researches on beam(plate) equation with nonlocal damping terms. Even so, there are still many problems to be solved. In this background , this program mainly studies the following two aspects: (1) extensiblebeam and plate equation with nonlocal damping term; (2) p-Laplacian Euler-Bernoulli beam and plate equation with (nonlocal) damping term. We mainly study the global well-posedness and longtime dynamical behavior for the above mentioned equations under certain conditions ,we also study the existence of the global attractors and the exponential attractors and the estimates on their Hausdorff dimensions for the related infinite dimensional dynamical system. The deep and meticulous study of these problems is meaningful to promote the development and application of science and technology and mathematics itself.

描述梁(板)构件非线性振动的波动方程是一类重要的非线性发展方程，研究这类梁(板)方程的长时间动力行为及动力稳定性具有重要的理论价值和实际意义。由于方程的非线性程度和描述现象的复杂性，其数学理论研究是非线性偏微分方程的一个研究焦点。近年来具有非局部耗散的梁(板)方程的研究并取得了一些好的结果，但还有许多问题亟待解决。本项目拟研究以下两个方面的问题：(1)研究几类具有非局部耗散项的可伸缩梁(板)方程；(2)研究具有(非局部)耗散项的p-Laplacian Euler-Bernoulli 梁(板)方程，在一定条件下研究这两类问题整体适定性和长时间动力学行为，研究解能量的衰减估计以及对应无穷维动力系统的整体吸引子和指数吸引子的存在性、它们的维数估计等。对这些问题深入细致的研究对科学技术以及数学自身的发展和应用都具有重要的科学意义。

描述梁(板)构件非线性振动的波动方程是一类重要的非线性发展方程，研究这类梁(板)方程的适定性及长时间动力行为具有重要的理论价值和实际意义。本项目在此相关方向展开工作，经过项目组成员的共同努力，取得了以下主要成果：(1) 具有非局部耗散项的p-Laplacian Euler-Bernoulli梁(板)方程，具有记忆项和时滞项的悬浮桥梁(板)方程，在一定条件下得到了这两类方程初边值问题整体适定性和长时间动力学行为，研究解能量的衰减估计以及对应无穷维动力系统的整体吸引子和指数吸引子的存在性、它们的维数估计等；(2)对于具有非局部耗散项和多项式源项的梁(板)方程、非局部源项及非线性对数项源项的梁(板)方程等，得到了解的适定性、能量的衰减估计和有限时刻爆破；（3) 同时我们还研究了具有非局部耗散的拟抛物方程、非局部源项的拟抛物方程解的适定性、能量的衰减估计和爆破理论。对这些问题深入细致的研究对科学技术以及数学自身的发展和应用都具有重要的科学意义。