具阻尼Kirchhoff型波动方程的长时间动力学行为

发布时间：2018-05-27 00:34

本文选题：Kirchhoff型方程 + 初值问题　；参考：《郑州大学》2017年博士论文

【摘要】：本文主要研究以下三种类型的具有阻尼项的自治Kirchhoff型波动方程的整体适定性和长时间动力学行为:1.无界域上具有强阻尼的Kirchhoff型波动方程:2.无界域上具有变系数和强阻尼的Kirchhoff型波动方程:3.有界域上具Dirichlet边界条件和结构阻尼的Kirchhoff型波动方程:对方程(0.0.1),综合利用能量方法和单调性方法得到了解在自然能量空间H=H~1(R~N)×L~2(R~N)的整体适定性;利用新的截断函数得到了解的一致尾部估计,并利用近年发展起来的补偿紧致方法,克服了无界域上和临界非线性Sobolev空间嵌入失去紧性的本质困难,在相空间H中建立了有限维的整体吸引子和指数吸引子的存在性.见第三章.对无界域上具变系数和强阻尼的Kirchhoff型波动方程(0.0.2),在非线性项f(u)的增长指数p达到超临界范围的情况下,即,,在加权相空间证明了解的整体适定性,并且利用弱拟稳定估计和补偿紧致方法建立了部分强拓扑意义下的有限维整体吸引子和指数吸引子的存在性.特别地,在非超临界情况下,即,,在相空间存在强拓扑意义下的有限维整体吸引子和指数吸引子,并且强拓扑意义下的整体吸引子与部分强拓扑意义下的整体吸引子保持一致.见第四章.对于有界域上具Dirichlet边界条件和结构阻尼的方程(0.0.3),发现了新的临界指标,即,唯一性指标.证明当时解在相空间的整体适定性,并且当时解具有抛物方程的性质;利用弱拓扑空间的稳定性估计和紧性恢复的方法,建立了H中的有限维整体吸引子和指数吸引子的存在性;在超临界情况,即,,证明了上述问题极限解的存在性,构造了极限解类G(弱解的子类)并利用构造性方法证明了极限解类G存在弱整体吸引子.见第五章.
[Abstract]:In this paper, the following three types of autonomous Kirchhoff wave equations with damping terms are studied: the global fitness and the long-time dynamical behavior: 1. Kirchhoff type wave equation with strong damping on unbounded domain: 2. Kirchhoff type wave equation with variable coefficient and strong damping on unbounded domain: 3. The Kirchhoff wave equation with Dirichlet boundary condition and structural damping on the bounded domain: for the equation 0.0.1, the global fitness of the solution in natural energy space is obtained by using energy method and monotonicity method. By using the new truncation function, the uniform tail estimation of the solution is obtained, and the compensatory compactness method developed in recent years is used to overcome the essential difficulty of embedding loss of compactness in unbounded domain and in critical nonlinear Sobolev space. The existence of finite dimensional global attractors and exponential attractors in phase space H is established. See chap. III. For the Kirchhoff type wave equation with variable coefficient and strong damping in unbounded domain, if the growth exponent p of the nonlinear term fu) reaches the supercritical range, the global suitability of the solution is proved in the weighted phase space. The existence of finite-dimensional global attractor and exponential attractor in the sense of partial strong topology is established by using weak quasi-stability estimation and compensatory compactness method. In particular, in the case of non-supercritical condition, that is, the finite-dimensional global attractor and exponential attractor exist in the phase space in the sense of strong topology, and the global attractor in the sense of strong topology is consistent with the global attractor in the sense of partial strong topology. See chap. IV. For the equation with Dirichlet boundary condition and structural damping on bounded domain, a new critical index, that is, uniqueness index, is found. It is proved that the present solution has the property of parabolic equation in the phase space, and the method of stability estimation and compactness recovery of weak topological space is used. The existence of finite dimensional global attractor and exponential attractor in H is established. In this paper, a class of limit solutions G (a subclass of weak solutions) is constructed and the existence of weak global attractors for class G of limit solutions is proved by using the constructive method. See chapter V.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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