Blow-up prevention by logistic sources in a parabolic–elliptic Keller–Segel system with singular sensitivity

Fujie, Kentarou; Winkler, Michael; Yokota, Tomomi

doi:10.1016/j.na.2014.06.017

Cited by 82 publications

(67 citation statements)

References 17 publications

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“…Remark 1 Since 3n−2 n = 2 for n = 2, we conclude by Theorem 1 with [20] that the classical solution to (1.6) for the case n = 2 must be globally bounded if k ≥ 2. In addition, Theorems 2 and 3 show that k < 2 is permitted for the global existence-boundedness of solution to (1.6).…”

Section: Introductionmentioning

confidence: 69%

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Global boundedness to a chemotaxis system with singular sensitivity and logistic source

Zhao

Zheng

2016

Z. Angew. Math. Phys.

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In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: u t = ∆u − χ∇·( u v ∇v)+ ru − µu k , 0 = ∆v − v + u under the non-flux boundary conditions in a smooth bounded convex domain Ω ⊂ R n , χ, r, µ > 0, k > 1 and n ≥ 2. It is shown that the system possesses a globally bounded classical solution if k > 3n−2 n , and r > χ 2 4 for 0 < χ ≤ 2, or r > χ − 1 for χ > 2. In addition, under the same condition for r, χ, the system admits a global generalized solution when k ∈ (2 − 1 n , 3n−2 n ], moreover this global generalized solution should be globally bounded provided r µ and the initial data u 0 suitably small.

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Section: Introductionmentioning

confidence: 69%

“…Similarity to system (1.1), the suitable smallness of chemotactic sensitive coefficient χ > 0 is necessary to establish global existence-boundedness of solutions to system (1.3). If n, k = 2, there exists a unique globally bounded classical solution [20,21], whenever r >…”

Section: Introductionmentioning

confidence: 99%

Global boundedness to a chemotaxis system with singular sensitivity and logistic source

Zhao

Zheng

2016

Z. Angew. Math. Phys.

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“…214 Somewhat going beyond this, in the case when Ω is a ball in R n , n ≥ 2, and u 0 and v 0 are radially symmetric, a more subtle analysis of solutions near the origin shows that the dampening effect on cross-diffusion, as expressed by (3.53), at least allows for the construction of certain very weak global solutions to (3.52) for arbitrary choices of χ > 0. 175 Apart from that, boundedness results have been derived for numerous related models, including parabolic-elliptic simplifications, also involving more general forms of S. 83,133,26,143,84,85 For certain parabolic-elliptic variants, even some exploding solutions could be detected. 143…”

Section: Signal-dependent Sensitivitiesmentioning

confidence: 99%

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Bellomo

Bellouquid

Tao

et al. 2015

Math. Models Methods Appl. Sci.

927

818

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This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller-Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as 1663

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“…Moreover, in virtue of additional dampening kinetic terms, Manásevich, Phan and Souplet [12] proved global existence and boundedness in a related system for all χ. As compared to the above, the parabolic-elliptic case has been studied more precisely ( [2,13,4,6,5]). Many references to earlier work on chemotaxis systems can be found in Hillen and Painter [8].…”

Section: Introductionmentioning

confidence: 99%

Boundedness in a fully parabolic chemotaxis system with singular sensitivity

Fujie

2015

Journal of Mathematical Analysis and Applications

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151

119

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Please cite this article in press as: K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, AbstractThis paper deals with a fully parabolic chemotaxis system u t = Δu−χ∇·( u v ∇v), v t = Δv−v+u with singular sensitivity χ v (χ > 0) on a bounded domain Ω ⊂ R n , n ≥ 2. The main result solves the open problem of uniform-in-time boundedness of solutions for χ < 2 n , which was conjectured by Winkler [16].

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Blow-up prevention by logistic sources in a parabolic–elliptic Keller–Segel system with singular sensitivity

Cited by 82 publications

References 17 publications

Global boundedness to a chemotaxis system with singular sensitivity and logistic source

Global boundedness to a chemotaxis system with singular sensitivity and logistic source

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Boundedness in a fully parabolic chemotaxis system with singular sensitivity

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