Kirchhoff型波动方程的吸引子问题的研究

The Kirchhoff type wave equation originates from Kirchhoff's study of elastic string vibration, and is an important quasi-linear wave equation. Recently, mathematical researchers have made some breakthroughs on this model. However, since the strong solution u(x,t) can not be more regular than the initial space in the supercritical case, and the degeneration of the principal part causes the difficulties with compactness verification, there are few results in the above-mentioned problems. In the degenerate case, the conclusion on the attractors for Kirchhoff wave models has never been seen. In addition, the study on the infinite dimension phenomenon of the attractors of the autonomous system on the bounded region has just started in recent years. Many issues still remain untouched..The project focuses on the problem of attractors for Kirchhoff type wave models on bounded domains. It mainly includes the existence of the attractors, the estimation on the dimension of the attractors for strong solutions in the case of supercritical source, the existence of the global attractors when the principal part degenerates, and the influence of the degeneration on the fractal dimension and the Hausdorff dimension. Firstly, we are going to combine the existing dimension estimation methods to find the method that guarantees the finite-dimensionality of the attractors with lower regularity, and to characterize the infinite-dimensionality in the degenerate case respectively. On this basis, we will study the relationship between the degree of degeneration of the principal part of Kirchhoff equation and the dimension of the attractor. Finally, we will apply the Strichartz type estimates on the research of the Kirchhoff equation in the supercritical case. The research of this project will further reveal the properties of the Kirchhoff equation, and the method established above will be an important progress in the theory of the dimension of the attractor.

Kirchhoff型波方程起源于Kirchhoff对弹性弦振动的研究，是一类重要的拟线性波动方程。近年来，关于该方程的研究取得了一些有意义的成果。然而超临界情形在强解范数下关于函数u正则性的缺失，和主部退化给紧性验证带来的困难，导致这两种情形下的成果相对较少，特别是主部退化时，常规空间中吸引子相关结论从来未见。另一方面，对有界区域上自治系统吸引子维数无限的研究也刚起步。.本项目以有界区域上Kirchhoff波方程为对象，主要研究该方程当非线性项超临界增长时在强解范数下全局吸引子的存在性和维数估计，以及主部退化时全局吸引子的存在性及其退化程度对分形维数、Hausdorff维数的影响。我们拟将现有的维数估计方法相结合，首先在理论层面，分别给出需要更低正则性的刻画维数有限的方法，及在退化情形刻画分形维数为无限的方法。在此基础上，研究Kirchhoff方程主部的退化程度和吸引子维数之间的关系。我们也将引入Strichartz估计方法研究Kirchhoff方程的超临界情形。这些研究将进一步揭示该方程的性质，这些方法的建立是对吸引子维数理论的重要推进。

本项目主要研究了有界区域上Kirchhoff型波方程的适定性和全局吸引子、拉回吸引子问题。我们首先采用一个特殊的能量估计方法，结合非紧性测度和Z2指标，证明了退化Kirchhoff方程全局吸引子的存在性，和方程具有对称性时全局吸引子分形维数无限。这一结果填补了退化Kirchhoff方程在吸引子理论方面的空白，也揭示了主部退化对吸引子维数的本质影响。在此基础上，我们对相应的非自治问题的拉回吸引子进行了研究。为了分离对指数吸引速率和维数的研究，我们提出了一个全新的构造一致指数吸引集的方法，该方法可适用于吸引子维数无穷维时对其指数吸引性质的研究。. 对非退化Kirchhoff方程，我们研究了当非线性项超临界增长时解在强解范数下的适定性和吸引子的存在性，并对二维情形下非线性项指数增长时问题的适定性和全局吸引子的存在性进行了研究。最后，我们将建立在解算子光滑性基础上的体积压缩的方法和建立在拟稳定估计上的维数估计方法相结合，提出了一个更具一般性的维数估计方法，并给出了相应的验证指数吸引子的存在性的方法。这些研究将进一步揭示Kirchhoff方程的性质，上述方法的建立对吸引子及其维数理论的研究具有推进作用。