Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model

By Michael Winkler. 2010

1,401 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/ad1d02d0e78d09a32badac5c660bf7572f6cdf2e

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Citations in the corpus

Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source (2010, neutral): “To this end, according to well-known estimates for the Neumann heat semigroup ( e t Δ ) t ≥0 (see [ 22 , Lemma 1.”
Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement (2017, positive): “13) Furthermore, let us fix C2 > 0 and C3 > 0 such that in accordance with a well-known regularization feature of the Neumann heat semigroup ([42]) and the Gagliardo-Nirenberg inequality we have ‖∇eφ‖L2q(Ω) ≤ C2‖φ‖W 1,2q(Ω) for all φ ∈W 1,2q(Ω) and any t > 0 (6.”
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening (2014, neutral): “According to known smoothing estimates for the Neumann heat semigroup in Ω ([25], [31]), we can choose c1(p) > 0 such that ‖∇eφ‖Lp(Ω) ≤ c1(p)τ 1 2 ‖φ‖L∞(Ω) for all τ > 0 and any φ ∈ L∞(Ω).”
A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up (2016, background): “ u(·, 0) is small, or if n = 1, then for various types of initial-boundary value problems, global bounded solutions are known to exist ([13, 14, 27, 28, 32]).”
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source (2014, neutral): “3 of [25] such that ∥∇eτ∆w ∥∥ L(Ω) ≤ C1 ‖∇w‖L∞(Ω) (4) for all w ∈ W (Ω) and ∥∇eτ∆w ∥∥ L(Ω) ≤ C2(1 + τ 1 2− n 2q ) ‖w‖Lq(Ω) (5) for all w ∈ L(Ω) as well as ∥eτ∆∇ · w ∥∥ L(Ω) ≤ C3(1 + τ 1 2− n 2q ) ‖w‖Lq(Ω) (6) for w ∈ L(Ω,R).”