In this paper, we study the dynamics of an abstract evolution equation with fading memory with a critical growing nonlinearity. By use of some new methods and asymptotic estimate techniques, we first verify the asymptotic compact of solution semigroup and then prove the existence of global attractors in weak topological space and strong topological space, while the forcing term only belongs to −1 (Ω) or 2 (Ω), respectively. The results are new and appear to be optimal.

The spatiotemporal representation of task-based brain functional networks is a key topic in functional magnetic resonance image (fMRI) research. At present, deep learning has been more powerful and flexible in brain functional network research than traditional methods. However, the dominant deep learning models failed in capturing the long-distance dependency (LDD) in task-based fMRI images (tfMRI) due to the time correlation among different task stimuli, the nature between temporal and spatial dimensions, which resulting in inaccurate brain pattern extraction. To address this issue, this paper proposes a spatiotemporal attention neural architecture search (NAS) model for task-based brain functional networks representation (SA-NAS-BFNR), where attention mechanism and gate recurrent unit (GRU) are integrated into a novel framework and GRU structure is searched by the differentiable neural architecture search. This model can not only achieve meaningful brain functional networks (BFNs) by addressing the LDD, but also simplify the existing recurrent structure models in tfMRI. Experiments show that the proposed model is capable of improving the fitting ability between time series and task stimulus sequence, and extracting the BFNs effectively as well. CCS CONCEPTS CCS CONCEPTS•Computing methodologies~Artificial intelligence~Knowledge representation and reasoning~Temporal reasoning