The parabolic-parabolic Keller-Segel model in R2

Cálvez, Vincent; Corrias, Lucilla

doi:10.4310/cms.2008.v6.n2.a8

Cited by 167 publications

(186 citation statements)

References 26 publications

(47 reference statements)

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“…For τ > 0, according to [7], solutions globally exist for any M < 8 π. However, it has not yet been proved that explosion occurs in finite time as soon as M > 8 π, for instance under some additional assumptions like a smallness condition on R R 2 |x| 2 n 0 (x) dx.…”

Section: Introductionmentioning

confidence: 99%

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

2010

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In two space dimensions, the parabolic-parabolic Keller-Segel system shares many properties with the parabolic-elliptic Keller-Segel system. In particular, solutions globally exist in both cases as long as their mass is less than 8 π. However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above 8 π always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller-Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above 8 π, which is forbidden in the parabolic-elliptic case.

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Section: Introductionmentioning

confidence: 99%

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

2010

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“…This macroscopic model has been investigated by many authors (e.g. see [2,6,11,18]). If one looks at the kinetic level, a good model is the so-called Othmar-Dunbar-Alt system [20] which is based on velocity-jump processes and leads to the transport equation (1.2) for v ∈ V ⊂ R d where f (t, x, v) denotes the cellular density.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…There are many choices for the evolution of the chemoattractant, one of them is to use a parabolic diffusion equation, (e.g. see [6,8,10,11,17,21])…”

Section: T [S](t X V → V) = χ[V · ∇S(t X)] + mentioning

confidence: 99%

One-dimensional chemotaxis kinetic model

tabar

2010

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper, we study a variation of the equations of a chemotaxis kinetic model and investigate it in one dimension. In fact, we use fractional diffusion for the chemoattractant in the Othmar-Dunbar-Alt system (Othmer in J Math Biol 26(3): 1988). This version was exhibited in Calvez in Amer Math Soc, pp [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] 2007 for the macroscopic well-known Keller-Segel model in all space dimensions. These two macroscopic and kinetic models are related as mentioned in Bournaveas, Ann Inst H Poincaré

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“…For mass greater than 8π , solutions blowup (under an additional assumption on the second moment [10 …”

Section: Resultsmentioning

confidence: 99%

“…Since much attention has been paid to the blowup problems for this system, recall that 8π is the critical value of initial mass ( u 0 (x) dx) for the existence problem. Namely, below this value, one can construct global solutions to the problem while for mass above 8π (with the second momentum |x| 2 u 0 (x)dx small enough, i.e., smaller than the value of g(M)-a monotone increasing function of mass M), the blowup occurs (see [10,Theorem 1.2…”

Section: )mentioning

confidence: 99%