The one‐dimensional chemotaxis model: global existence and asymptotic profile

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The fractional Keller–Segel model (2006, methodology): “This system is known to have finite time blowing up solutions for large enough initial conditions in dimensions d ≥ 2, but all the solutions are regular for d = 1; there is a large literature on the analysis of this problem for the Keller-Segel model as well as for some simplifications [3, 4, 5, 6, ”
Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source (2015, background): “The key phenomena are: no blow-up in 1-D [38,19,56], except in some extreme nonlinear diffusion models [13,10], critical mass blow-up in 2-D [21,17], and generic blow-up in ≥3-D [54].”
How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? (2016, background): “A finite/infinite time blow-up never occurs in 1-dimension [15, 39, 60] (except in some extreme nonlinear denerate diffusion model [8]), a critical mass blow-up occurs in 2-dimension: when the initial mass lies below the threshold solutions exist globally, while above the threshold solutions blow up”