A class of chemotaxis systems with growth source and nonlinear secretion

Wang, Zhi‐An; Xiang, Tian

doi:10.48550/arxiv.1510.07204

Cited by 6 publications

(5 citation statements)

References 45 publications

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“…For other aspects, we mention, in 3-D bounded, smooth and convex domains, that logistic damping guarantees the existence of global weak solutions to (1.3) [18], and that sufficiently strong logistic damping can enhance 'expected' results such as global existence and stabilization toward constant equilibria, as well as more 'unexpected' behavior witnessing a certain strength of chemotactic destabilization for (1.3), cf. [2,9,6,19,29,31,36,37,38,41].…”

Section: Introductionmentioning

confidence: 99%

Sub-logistic source can prevent blow-up in the 2D minimal Keller-Segel chemotaxis system

Xiang

2018

Journal of Mathematical Physics

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It is well-known that the Neumann initial-boundary value problem for the minimal-chemotaxis-logistic systemin a bounded smooth domain Ω ⊂ R 2 doesn't have any blow-ups for any a ∈ R, τ ≥ 0, χ > 0 and b > 0. Here, we obtain the same conclusion by replacing the logistic source au−bu 2 with a kinetic term fη>0 sup{f (s) + ηs : s > 0} η . In this setup, it is shown that this problem doesn't have any blow-up by ensuring all solutions are global-in-time and uniformly bounded. Clearly, f covers super-, logistic, sub-logistic sources like f (s) = as − bs θ with b > 0 and θ ≥ 2, f (s) = as− bs 2 ln γ (s+1) with b > 0 and γ ∈ (0, 1), and f (s) = as− bs 2 ln(ln(s+e)) with b > 0 etc. This indicates that logistic damping is not the weakest damping to guarantee boundedness for the 2D Keller-Segel minimal chemotaxis model. 2000 Mathematics Subject Classification. Primary: 35K59, 35K51, 35K57, 92C17; Secondary: 35B44, 35A01.

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Section: Introductionmentioning

confidence: 99%

Sub-logistic source can prevent blow-up in the 2D minimal Keller-Segel chemotaxis system

Xiang

2018

Journal of Mathematical Physics

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“…Whereas, in the case n ≥ 3, the global existence and boundedness were first obtained for a parabolic-elliptic simplification of (1.1) under µ > (n−2) n χ [30]. Nowadays, this result has been improved to the borderline case µ ≥ n−2 n χ [9,13,29]. Moreover, with a very slow self-diffusion of cells, the u component can exceed the carrying capacity r µ to an arbitrary extent at some intermediate time [17,35].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

Jin

Xiang

2018

Comptes Rendus. Mathématique

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“…Lately, we turn our eyes into the chemotaxis system with nonlinear productions. Wang-Xiang [9] proved for the chemotaxis system,…”

Section: Introductionmentioning

confidence: 99%

Global boundedness in an attraction-repulsion chemotaxis system with nonlinear productions and logistic source

Wang,

Yan

2023

Preprint

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This paper deals with the quasilinear(\(\tau =0\)) and fully parabolic(\(\tau =1\)) attraction-repulsion chemotaxis system with nonlinear productions and logistic source, \(u_t = \newnabla{D(u)}{u} - \newnabla{\Phi (u)}{v} + \newnabla{\Psi (u)}{w} + f(u), v_t = \Delta v+\alpha {{u}^{k}}-\beta v, \tau w_t = \Delta w+\gamma {{u}^{l}}-\delta w, \tau \in \{0,1\},\) in bounded domain \(\Omega \subset {{\mathbb{R}}^{n}} \text{ } \newbrac{n \ge 1},\) subject to the homogeneous Neumann boundary conditions and initial conditions, \(D,\Phi ,\Psi \in {{C}^{2}}[0,\infty )\) nonnegative with \(D(s)\ge {{(s+1)}^{p}}\text{ for }s\ge 0,\) \(\Phi (s)\le \chi {{s}^{q}},\) \(\xi {{s}^{g}}\le \Psi (s),\text{ }s\ge {{s}_{0}}\) for \({{s}_{0}}>1.\) And the logistic source satisfying\(f(s)\le s(a-b{{s}^{d}}), \text{ } s>0, \text{ } f(0)\ge 0,\) and the nonlinear productions for the attraction and repulsion chemicals are described via \(\alpha {{u}^{k}} \text{ and } \gamma {{u}^{l}}\) respectively. When \(k=l=1\) , it is known that above system possesses a globally bounded solution in some cases. However, there has been no work in the case that \(k,l>0\). This paper develops global boundedness of the solution to the above system in some cases. And extends the global boundedness criteria established by Tian-He-Zheng(2016) for the quasilinear attraction-repulsion chemotaxis system.

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A class of chemotaxis systems with growth source and nonlinear secretion

Cited by 6 publications

References 45 publications

Sub-logistic source can prevent blow-up in the 2D minimal Keller-Segel chemotaxis system

Sub-logistic source can prevent blow-up in the 2D minimal Keller-Segel chemotaxis system

Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

Global boundedness in an attraction-repulsion chemotaxis system with nonlinear productions and logistic source

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