On a Parabolic–Elliptic system with chemotaxis and logistic type growth

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Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$ (2016, background): “and χ(u,v) as above and let f(u,v) = au − buκ with suitable κ < 2 (for instance, κ = 3/2) and g(u,v) = u− 1 |Ω| R Ω u(x)dx, then ﬁnite-time blow-up is possible. The reader is referred to [2], [7], [14], [40], [43], [45], [46], [47], [49], [50], [52], and references therein for other studies of (1”
Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion (2019, methodology): “Based on the commonly used choices for f and g in the literature [5, 9, 21, 24, 25], throughout this paper, we assume that f is smooth in [0,∞) satisfying f(0) ≥ 0 and there are a ≥ 0, b > 0 and θ > 1 such that f(u) ≤ a− bu for all u ≥ 0 (1.”