Asymptotically Autonomous Robustness of Random Attractors for 3D BBM Equations Driven by Nonlinear Colored Noise

.The aim of this paper is to establish the asymptotically autonomous robustness of pullback random attractors of nonautonomous Benjamin–Bona–Mahony equations driven by nonlinear colored noise defined on 3D unbounded channels. We first prove the existence, uniqueness, and backward compactness of a special kind of pullback random attractor by the methods of spectral decomposition inside bounded domains as well as the uniform tail-estimates of solutions outside bounded domains over the infinite time interval in order to surmount the difficulties caused by lack of compact Sobolev embedding on unbounded domains and weak dissipative structure of the equation. The measurability of such an attractor is proven by showing that the defined two kinds of attractors with respect to two different universes are equal. Finally, the asymptotically autonomous upper semicontinuity of the attractors is investigated by assuming that the time-dependent external forcing term converges to the time-independent external force as the time-parameter tends to negative infinity. This work is a continuation of our previous work [Chen et al., Math. Ann., 386 (2023), pp. 343–373], which considered the existence and uniqueness of usual tempered pullback random attractors.Keywordsasymptotically autonomous robustnessBenjamin–Bona–Mahony equationpullback random attractorsbackward compactnessnonlinear colored noise3D unbounded channelMSC codes35B4037B5537L5560H15