AI 中文总结
本文针对三维有界域上具有能量依赖阻尼和次五次源项的波动方程，结合Galerkin逼近与Strichartz估计证明全局适定性，并建立全局吸引子及其有限维性、更高正则性和广义指数吸引子的存在性。
AI 中文摘要
本文研究了一个有界三维区域中的波动方程，该方程具有依赖于系统能量的退化非局部阻尼机制以及次五次增长的源项。这类模型受到光学物理应用的启发。我们通过结合Galerkin逼近和有界区域上的时空Strichartz估计，建立了Shatah--Struwe意义上的全局存在性。我们的主要贡献涉及长时间动力学，这对于具有缺乏局部Lipschitz正则性的强迫项的三维波动而言尤其具有挑战性。具体而言，我们建立了紧致全局吸引子的存在性，并推导了其Kolmogorov ε-熵的上界。此外，当阻尼系数非退化时，我们采用了一种新的可容许Strichartz对的组合，以适应强迫项的正则性。这种方法使我们能够建立拟稳定性，从而得到全局吸引子的有限维性和更高正则性，以及广义指数吸引子的存在性。
英文摘要
In this paper, we consider a wave equation in a bounded three-dimensional domain with a degenerate nonlocal damping mechanism depending on the system's energy and a source term with subquintic growth. This kind of model is motivated by applications to optical physics. We establish global existence in the Shatah--Struwe sense by combining Galerkin approximations with space-time Strichartz estimates on bounded domains. Our main contributions concern long-time dynamics, which are particularly challenging for three-dimensional waves with forcing terms without local Lipschitz regularity. Specifically, we establish the existence of a compact global attractor and derive an upper bound for its Kolmogorov $\varepsilon$-entropy. Furthermore, when the damping coefficient is non-degenerate, we employ a novel combination of admissible Strichartz pairs, adapted to the regularity of the forcing term. This approach allows us to establish quasi-stability and, consequently, the finite dimensionality and higher regularity of the global attractor, as well as the existence of a generalized exponential attractor.
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