Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect

By Michael Winkler, Kianhwa C. Djie. 2010

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Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction (2011, background): “Ω |x|u(x, t)dx to an ODI analysis ([15], [1]), or the circumstance that (radial) solutions can be transformed into solutions of a single scalar parabolic equation allowing for a comparison principle ([12], [3], [4]).”
Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source (2015, background): “Namely, if q − p < 2 n , then all solutions are global and uniformly bounded [24, 18]; however, if q− p > 2 n , unbounded solutions do exist [22], even finite-time blow-up may occur under some additional conditions n ≥ 3 and q ≥ 1 [4, 24].”
Dissipative Gradient Nonlinearities Prevent δ$\delta$ ‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models (2024, unknown): “…effects (i.e., with a general flux G m 1 + H m 2 and h k ≡ 0 ) and linear production (i.e., P 1 ( v ) = 0 ), among other findings established in [55] we indicate that for m 1 ≤ 1 , m 2 > 0 , and m 2 > m 1 + 2 n − 1 , scenarios with unbounded solutions at certain finite times T max can be…”
Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source (2015, background): “Thus, the result (R6) is not surprising; however, there are volume-filling models that admit chemotactic collapses [51,52].”