This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space $\mathbb {R}^n$. The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ is caused by the lack of compact Sobolev embeddings on $\mathbb {R}^n$, as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46–57, 2018).

中文翻译：

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无界域上一类弱耗散随机波动方程的随机吸引子的渐近自主鲁棒性

本文关注一类非自治随机非线性波动方程的渐近行为，该方程具有由定义在整个空间上的算子类型噪声驱动的色散和粘性耗散项$\mathbb {R}^n$. 存在性、唯一性、时间半均匀紧致性和渐近自治鲁棒性拉回随机吸引子的证明在$H^1(\mathbb {R}^n)\次 H^1(\mathbb {R}^n)$当非线性的增长率在亚临界范围内时，噪声的密度是适当可控的，时间相关的力在某种意义上收敛为与时间无关的函数。建立的主要困难时间半均匀解的回拉渐近紧致性$H^1(\mathbb {R}^n)\次 H^1(\mathbb {R}^n)$是由于缺乏紧凑的 Sobolev 嵌入$\mathbb {R}^n$，以及方程的弱耗散性在均匀尾估计和谱分解方法的思想下被克服。随机吸引子的可测量性通过使用考虑由 Wang 和 Li 开发的两个吸引宇宙的论证来证明（物理 D382: 46–57, 2018)。