A new approach toward boundedness in a two‐dimensional parabolic chemotaxis system with singular sensitivity

By Johannes Lankeit. 2015

125 citations recorded by the source corpus.

We consider the parabolic chemotaxis model ut=Δu−χ∇·uv∇vvt=Δv−v+u in a smooth, bounded, convex two‐dimensional domain and show global existence and boundedness of solutions for χ∈(0,χ0) for some χ0>1, thereby proving that the value χ = 1 is not critical in this regard.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/fd75d707495c90ed22a546903aeb606b0c4f5c66

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Citations in the corpus

Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion (2016, background): “(For this system, global solutions are known to exist if χ is sufficiently small, where the precise condition depends on the dimension as well as on whether classical ([17, 40, 4]) or weak solutions ([34, 40]) are considered and on radial symmetry of initial data ([4, 28]); but for large χ also blow”
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”
Long-term behaviour in a chemotaxis-fluid system with logistic source (2016, unknown): “ ∇· (∇logw) = ∇ · ∇w w = ∆w w − |∇w|2 w2 immediately entail ∆ √ w= 1 2 √ w∆logw+ 1 4 |∇w|2 w3 2 and thus (9). iv) Being a special case of assertions from [8], this can be found as Lemma 3.4 i) in [22]. v) This was proven as [44, Lemma 3.3]. vi) Let η >0. Part i) and Young’s inequality in combin”
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 ”
Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system (2017, result): “1) seems complex enough so as to allow for partial results identifying conditions sufficient for global boundedness up to now, with an exception formed by the particular choice χ(v) = χ0 v of a singular sensitivity consistent with the Weber-Fechner law for which besides various results on global sol”
The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties (2016, background): “2) the signal production mechanism expressed in the second equation is known to inhibit an evolution of v toward small values in such an effective manner that global existence of bounded smooth solutions can be achieved for reasonably smooth but arbitrarily large data in initial-boundary value probl”