Blow-up, Concentration Phenomenon and Global Existence for the Keller–Segel Model in High Dimension

Cálvez, Vincent; Corrias, Lucilla; Ebde, Mohammed Abderrahman

doi:10.1080/03605302.2012.655824

Cited by 64 publications

(38 citation statements)

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“…For p > d 2 local well-posedness in L p and regularity for positive times are well-established in the community (see e.g. [BHN94] for results on bounded domains and [CCE12] for results on the whole space assuming sufficient decay at infinity), and at any (positive) level of mass (= L 1 -norm for non-negative solutions) there exist smooth solutions which blow up in finite time [BHN94,BK10,CCE12]. Moreover, for global regularity it suffices to globally control the L p -norm of the solution, and statements analogous to those established in Section 3 hold true whenever p > d 2 .…”

Section: Remarksmentioning

confidence: 99%

Aggregation equations with fractional diffusion: Preventing concentration by mixing

Hopf

Rodrigo

2018

Communications in Mathematical Sciences

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We investigate a class of aggregation-diffusion equations with strongly singular kernels and weak (fractional) dissipation in the presence of an incompressible flow. Without the flow the equations are supercritical in the sense that the tendency to concentrate dominates the strength of diffusion and solutions emanating from sufficiently localised initial data may explode in finite time. The main purpose of this paper is to show that under suitable spectral conditions on the flow, which guarantee good mixing properties, for any regular initial datum the solution to the corresponding advectionaggregation-diffusion equation is global if the prescribed flow is sufficiently fast. This paper can be seen as a partial extension of Kiselev and Xu (Arch. Rat. Mech. Anal. 222(2), 2016), and our arguments show in particular that the suppression mechanism for the classical 2D parabolic-elliptic Keller-Segel model devised by Kiselev and Xu also applies to the fractional Keller-Segel model (where △ is replaced by −Λ γ ) requiring only that γ > 1. In addition, we remove the restriction to dimension d < 4.Conservation of mean. First note that formally for any solution to equation (1.1) the mean value is conserved in time:In fact, all evolution equations which we shall consider here enjoy this property, and in this context we will abbreviateρ = T d ρ 0 . In applications ρ usually describes a density, and for the sake of exposition, we will henceforth assume ρ 0 ≥ 0, a property, which by the maximum principle (see e.g. [LR10] for a proof in a related setting) is preserved in time for any sufficiently regular solution to (1.1). It will, however, be obvious that (apart from the blowup proof in Appendix A) our results remain valid without the assumption of positivity.Scaling. Let us for the moment replace T d by R d and consider the scaling properties of the equation obtained by substituting in (1.1) the kernel ∇K for its homogeneous approximation near the origin, i.e. x |x| 2+a . This equation is invariant under the scaling

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Section: Remarksmentioning

confidence: 99%

Aggregation equations with fractional diffusion: Preventing concentration by mixing

Hopf

Rodrigo

2018

Communications in Mathematical Sciences

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“…One of the most considered models is the Patlak-Keller-Segel system [17,23], where the evolution of the density of cells is described by a parabolic equation of driftdiffusion type, and the concentration of a chemoattractant is generally given by a parabolic or elliptic equation, depending on the different regimes to be described. The behavior of this system is quite well known now, at least for linear diffusions: in the one-dimensional case, the solution is always global in time [19], while in two and more dimensions the solutions exist globally in time or blow up according to the size of the initial data, see [4,5]. However, a drawback of this model is that the diffusion leads alternatively to a fast dissipation or an explosive behavior, while in general we are interested in the creation of patterns, like in the vasculogenesis process.…”

Section: Introductionmentioning

confidence: 99%

Stationary solutions with vacuum for a one-dimensional chemotaxis model with nonlinear pressure

Berthelin

Chiron

Ribot

2016

Communications in Mathematical Sciences

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In this article, we study a one-dimensional hyperbolic quasilinear model of chemotaxis with a non-linear pressure and we consider its stationary solutions, in particular with vacuum regions. We study both cases of the system set on the whole line R and on a bounded interval with no-flux boundary conditions. In the case of the whole line R, we find only one stationary solution, up to a translation, formed by a positive density region (called bump) surrounded by two regions of vacuum. However, in the case of a bounded interval, an infinite of stationary solutions exists, where the number of bumps is limited by the length of the interval. We are able to compare the value of an energy of the system for these stationary solutions. Finally, we study the stability of these stationary solutions through numerical simulations.

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“…This technique was first used by Nagai [22], then by many authors in various contexts (see [2,3,12,13,11] for instance). Other strategies have been used to prove the existence of blowing-up solutions (either constructive by Herrero and Velázquez [17] or undirect [19]), however up to date this trick is the only way to provide explicit criterion and appears to be quite robust to variations around Keller-Segel [4,8].…”

Section: The Critical Mass Phenomenonmentioning

confidence: 98%

A one-dimensional Keller–Segel equation with a drift issued from the boundary

Cálvez

Meunier²

2010

Comptes Rendus. Mathématique

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We investigate in this Note the dynamics of a one-dimensional Keller-Segel type model on the half-line. On the contrary to the classical configuration, the chemical production term is located on the boundary. We prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass, they blow-up in finite time above the critical mass, and they converge to some equilibrium at the critical mass. Entropy techniques are presented which aim at providing quantitative convergence results for the subcritical case. This Note is completed with a brief introduction to a more realistic model (still one-dimensional).r é s u m é Nous étudions dans cette Note la dynamique d'un modèle unidimensionnel de type KellerSegel posé sur une demi-droite. Dans le cas présent, la production du signal chimique est localisée sur le bord, au lieu d'être répartie à l'intérieur du domaine comme dans le cas classique. On démontre, sous des hypothèses convenables, la dichotomie suivante qui rappelle le système de Keller-Segel en dimension deux d'espace. Les solutions sont globales si la masse est sous-critique, elles explosent en temps fini si la masse dépasse la masse critique. Enfin, les solutions convergent vers un état d'équilibre lorsque la masse est égale à la valeur critique. Des méthodes d'entropie sont développées, dans le but d'obtenir des résultats de convergence quantitatifs. Cette Note est enrichie d'une brève introduction à un modèle plus réaliste (à nouveau unidimensionnel).Dans cette Note nous allons étudier le comportement mathématique en dimension un de l'équation aux dérivées partielles suivante : ∂ t n(t, x) = ∂ xx n(t, x) + n(t, 0)∂ x n(t, x), t > 0, x ∈ (0, +∞), avec la condition initiale : n(t = 0, x) = n 0 (x) 0. Nous imposons au bord une condition de flux nul : ∂ x n(t, 0) + n(t, 0) 2 = 0, de sorte que la masse est conservée au cours du temps (au moins formellement) :(2) Ce modèle a été proposé dans [15] pour décrire synthétiquement la polarisation des cellules de levure. Une caractéris-tique intéressante de (1) réside dans le fait que la solution peut devenir non bornée en temps fini. Dans cette Note nous allons montrer l'alternative suivante : Théorème 1 (Existence globale vs. explosion). Supposons que n 0 est continue sur [0, +∞) et que n 0 ∈ L 1 + ((1 + x 2 ) dx). Si M 1 alors il existe une solution faible de (1) qui est globale en temps. Au contraire si M > 1, en supposant en outre que n 0 est décroissante, alors toute solution forte explose en temps fini. Nous annonçons également les résultats suivants concernant le comportement asymptotique de la solution lorsque M 1 : Théorème 2 (Comportement asymptotique). Dans le cas critique M = 1, il existe une famille d'états stationnaires pour (1) paramétrée par α > 0. La solution converge (au sens de l'entropie relative (14)) vers l'équilibre tel que α −1 = x>0 xn 0 (x) dx.Dans le cas sous-critique M < 1, la solution décroît vers zéro, et converge (au sens de l'e...

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Blow-up, Concentration Phenomenon and Global Existence for the Keller–Segel Model in High Dimension

Cited by 64 publications

References 50 publications

Aggregation equations with fractional diffusion: Preventing concentration by mixing

Aggregation equations with fractional diffusion: Preventing concentration by mixing

Stationary solutions with vacuum for a one-dimensional chemotaxis model with nonlinear pressure

A one-dimensional Keller–Segel equation with a drift issued from the boundary

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