Boundedness in a fully parabolic chemotaxis system with singular sensitivity

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A critical blow-up exponent in a chemotaxis system with nonlinear signal production (2018, background): “1) upon replacing the first equation therein with ut = ∆u − χ∇ · ( u v ∇v), the size of the parameter χ is known to play a crucial role with regard to the occurrence of blow-up: On the one hand, exclusively global smooth solutions exist if n ≥ 2 and χ < χ⋆(n) with some χ⋆(n) ∈ (0,∞] depending on whe”
Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion (2016, background): “) The proof of boundedness of solutions for χ < √ 2 N in [6] even relies on the second equation actually ensuring a positive pointwise lower bound for v.”
A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial data (2017, background): “015 and χ0(n) := √ 2 n for n ≥ 3, global bounded classical solutions exist for all reasonably regular positive initial data ([11], [2], [4], [24], [13], [21]); in the corresponding parabolic-elliptic analogue, the same conclusion holds with χ0(2) = ∞ ([5]) and with χ0(n) := 2 n−2 when n ≥ 3 and the ”
A new approach toward boundedness in a two‐dimensional parabolic chemotaxis system with singular sensitivity (2015, unknown): “Let us mention some more results concerning (1): It has been shown in [11] that the classical solutions for < p 2=n are global-intime bounded.”