标题： 时间二阶非保守动力学的吸引子与非线性阻尼 显示英文标题

标题： Attractors for Second Order in Time Non-Conservative Dynamics with Nonlinear Damping

摘要： 研究一种受非保守和非耗散效应以及非线性阻尼影响的非线性板模型解的长期行为。 所研究的模型是气体不稳定流动作用下悬索桥的原型。 为了抑制不希望的振动，应用了非线性性质的阻尼机制。 从非线性偏微分方程的角度来看，我们处理的是一个非耗散且非线性的时间二阶双曲型动力系统，该系统受到非线性阻尼的影响。 第一个目标是建立所有解的终极耗散性，这将意味着弱吸引子的存在。 非耗散强迫与非线性阻尼的综合效应导致过阻尼，从而在证明吸收集的存在性方面带来了重大挑战。 基于能量均分的方法不足以解决这个问题。 将开发一种基于“屏障”方法的通用新方法来解决这一问题及相关问题。 最终将证明弱吸引子会变为强吸引子，并且非线性偏微分方程系统具有协调的有限维渐近行为。 显示英文摘要

摘要： A long-time behavior of solutions to a nonlinear plate model subject to non-conservative and non-dissipative effects and nonlinear damping is considered. The model under study is a prototype for a suspension bridge under the effects of unstable flow of gas. To counteract the unwanted oscillations a damping mechanism of a nonlinear nature is applied. From the point of view of nonlinear PDEs, we are dealing with a non-dissipative and nonlinear second order in time dynamical system of hyperbolic nature subjected to nonlinear damping. One of the first goals is to establish ultimate dissipativity of all solutions, which will imply an existence of a weak attractor. The combined effects of non-dissipative forcing with nonlinear damping-leading to an overdamping-give rise to major challenges in proving an existence of an absorbing set. Known methods based on equipartition of the energy do not suffice. A rather general novel methodology based on ``barrier's'' method will be developed to address this and related problems. Ultimately, it will be shown that a weak attractor becomes strong, and the nonlinear PDE system has a coherent finite-dimensional asymptotic behavior.