Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source

Lankeit, Johannes

doi:10.1016/j.jde.2014.10.016

Cited by 222 publications

(127 citation statements)

References 24 publications

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“…219,119 A different aspect in the dynamics of chemotaxis-growth models has recently been addressed for parabolic-elliptic variants of (3.43)-(3.44) involving small cell diffusion. To be more precise, let us consider:…”

Section: Theorem 38 (Lankeit) Let N ≥ 3 and ω ⊂ R N Be A Bounded Domentioning

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.

902

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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Section: Theorem 38 (Lankeit) Let N ≥ 3 and ω ⊂ R N Be A Bounded Domentioning

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.

902

818

View full text Add to dashboard Cite

show abstract

“…To the best of our knowledge, it is yet unclear whether for n ≥ 3 and small values of μ > 0 certain initial data may enforce finite-time blow-up of solutions (cf. [22] for a high-dimensional blow-up result in a simplified chemotaxis system with superlinear logistic-type degradation), although at least certain global weak solutions are known to exist for arbitrary μ > 0 [9]. The steady-state example (u, v) ≡ ( r μ , r μ ), however, trivially shows that some global classical solutions exist regardless of the size of μ > 0.…”

Section: Introductionmentioning

confidence: 99%

Persistence of mass in a chemotaxis system with logistic source

Tao

Winkler

2015

Journal of Differential Equations

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This paper studies the dynamical properties of the chemotaxis systemx ∈ , t > 0, ( ) under homogeneous Neumann boundary conditions in bounded convex domains ⊂ R n , n ≥ 1, with positive constants χ , r and μ. Numerical simulations but also some rigorous evidence have shown that depending on the relative size of r, μ and | |, in comparison to the well-understood case when χ = 0, this problem may exhibit quite a complex solution behavior, including unexpected effects such as asymptotic decay of the quantity u within large subdomains of .The present work indicates that any such extinction phenomenon, if occurring at all, necessarily must be of spatially local nature, whereas the population as a whole always persists. More precisely, it is shown that for any nonnegative global classical solution (u, v) of ( ) with u ≡ 0 one can find m > 0 such thatThe proof is based on an, in this context, apparently novel analysis of the functional ln u, deriving a lower bound for this quantity along a suitable sequence of times by appropriately exploiting a differential inequality for a suitable linear combination of ln u, u and v 2 .

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“…The same result is true for τ > 0 and γ = 2 provided that n ≤ 2 or n ≥ 3 and b > b 0 with b 0 sufficiently large [12,20]. Also, in this case, for n ≥ 3, it is proved that there exists at least one global weak solution for arbitrary b > 0 [8]. Moreover, in this case when the ratio b χ is sufficiently large, it is shown that for any choice of suitably regular nonnegative initial data, there exists a unique global classical solution (u, v) such that u(., [23].…”

Section: Introductionmentioning

confidence: 56%

Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source

Khelghati

Baghaei

2015

Comptes Rendus. Mathématique

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Global existence and boundedness of classical solutions in a quasilinear parabolic-elliptic chemotaxis system with logistic source Existence globale et bornes pour les solutions classiques d'un système quasi linéaire, parabolique-elliptique, de chimiotaxie avec source logistique Presented by the Editorial BoardWe consider the quasilinear parabolic-elliptic chemotaxis systemunder homogeneous Neumann boundary conditions in a smooth bounded domain ⊂ R n , n ≥ 1. We assume that the functions D and g are smooth and satisfyWe prove that the classical solutions to the above system are uniformly in-time-bounded without any restrictions on m and b. This result extends one of the recent results by Wang et al. (2014) [16], which assert the boundedness of solutions for γ > 2 under the condition b > b * with b * = 0 for m ≥ 2 − 2 n and b * = (2−m)n−2 (2−m)n χ for m < 2 − 2 n . © 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m é Nous considérons le système quasi linéaire, parabolique-elliptique, de chimiotaxieavec des conditions au bord homogènes de Neumann, dans un domaine lisse, borné ⊂ R n , n ≥ 1. Nous supposons que les fonctions D et D(s) > 0 pour s ≥ 0, D(s) ≥ C D s m−1 pour s > 0, g(0) ≥ 0, g(s) ≤ a − bs γ , s > 0 JID:CRASS1 AID:5570 /FLA Doctopic: Partial differential equations [m3G; v1.160; Prn:1/10/2015; 8:00] P.2 (1-5) 2 A. Khelghati, K. Baghaei / C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••-••• pour certaines constantes C D > 0, m ≥ 1, a ≥ 0, b > 0 et γ > 2. Nous démontrons que les solutions classiques du système ci-dessus sont uniformément bornées en temps, sans restriction sur m et b. Ceci étend un résultat récent de Wang et al. (2014) [16], qui borne les solutions pour γ > 2 sous la condition b > b * , où b * = 0 si m ≥ 2 − 2 n et b * = (2−m)n−2 (2−m)n χ si m < 2 − 2 n .

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Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source

Cited by 222 publications

References 24 publications

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

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

Persistence of mass in a chemotaxis system with logistic source

Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source

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