Exponential attractor for a chemotaxis-growth system of equations

Osaki, Koichi; Tsujikawa, Tohru; Yagi, Akira; Mimura, Mamoru

doi:10.1016/s0362-546x(01)00815-x

Cited by 379 publications

(258 citation statements)

References 18 publications

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“…Here the kinetic term can control the local population growth due to migration. Our model (M8) falls into the class of models studied by Osaki et al [74]. They demonstrate that certain generalisations of (M8) have global existing solutions and an exponential attractor (a compact attracting set of finite fractal dimension, that attracts bounded sets exponentially).…”

Section: Lemma 5 Solutions To Model (M3b) Exist Globally In Timementioning

confidence: 70%

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A user’s guide to PDE models for chemotaxis

2008

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Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225-234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display "auto-aggregation", has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

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Section: Lemma 5 Solutions To Model (M3b) Exist Globally In Timementioning

confidence: 70%

“…2. We will discuss the theoretical results of Osaki and Yagi [74], Wrzosek [110], and Wang and Hillen [105] in detail in the section on global existence.…”

Section: (M8) the Cell Kinetics Modelmentioning

confidence: 99%

A user’s guide to PDE models for chemotaxis

2008

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“…On the contrary, if the growth term includes quadratic degradation (e.g. f ðuÞ ¼ u À mu 2 ) and the secretion gðuÞ is linear, then blow-up does not occur and global existence of solutions is assured even if n b 2 and w and ku 0 k L 1 are large: this fact has been shown for n ¼ 2 by one of the authors et al [10] and for n b 1 with convex W and large m by Winkler [18].…”

Section: Introductionmentioning

confidence: 90%

“…In this section we shall list some well-known results in the theories of function spaces and linear operators [10,13,15,20].…”

Section: Preliminariesmentioning

confidence: 99%

Lp-Estimates of Solutions to n-Dimensional Parabolic-Parabolic System for Chemotaxis with Subquadratic Degradation

Nakaguchi

Osaki

2016

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“…Growth terms may prevent blow up in chemotaxis systems, as shown by the numerous examples existing in the literature. For instance, in Osaki, Tsujikawa, Yagi and Mimura [8], the logistic growth in a two dimensional parabolic-parabolic chemotaxis system drives the solution to an exponential attractor in a suitable space. In Winkler [9], growth terms/ e W|Q C°°( R) satisfying/(S) < a -jis 2 for fi > /x 0 (a) shows global existence of solutions with no dimensional restrictions, i.e.…”

Section: Introductionmentioning

confidence: 99%

On a parabolic–elliptic chemotactic system with non-constant chemotactic sensitivity

Negreanu

Tello

2013

Nonlinear Analysis: Theory, Methods & Applications

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We study a parabolic-elliptic chemotactic system describing the evolution of a population's density "u" and a chemoattractant's concentration "v". The system considers a nonconstant chemotactic sensitivity given by "/(JV -u)", for JV > 0, and a source term of logistic type "Xu(\ -u)". The existence of global bounded classical solutions is proved for any / > 0, JV > 0 and X > 0. By using a comparison argument we analyze the stability of the constant steady state u = \,v = 1, for a range of parameters.-For JV > 1 and NX > 2/, any positive and bounded solution converges to the steady state. -For JV < 1 the steady state is locally asymptotically stable and for /JV < X, the steady state is globally asymptotically stable.

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Exponential attractor for a chemotaxis-growth system of equations

Cited by 379 publications

References 18 publications

A user’s guide to PDE models for chemotaxis

A user’s guide to PDE models for chemotaxis

<i>L</i><sub><i>p</i></sub>-Estimates of Solutions to <i>n</i>-Dimensional Parabolic-Parabolic System for Chemotaxis with Subquadratic Degradation

On a parabolic–elliptic chemotactic system with non-constant chemotactic sensitivity

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