Behavior of radially symmetric solutions of a system related to chemotaxis

Nagai, Toshitaka; Senba, Takasi

doi:10.1016/s0362-546x(96)00256-8

Cited by 67 publications

(88 citation statements)

References 6 publications

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“…This result was extended to dimensions d ≥ 3 in the case of radial symmetric solutions by Nagai [50] who shows that blow-up may arise whatever the initial mass is, depending on the x momentum of order d of n 0 . Actually, the radial case is better understood and, in two space dimensions for large mass M (larger than the corresponding K 0 in Theorem 2.1), the type of blow-up has been specified.…”

Section: Remark 24mentioning

confidence: 84%

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Perthame

2004

Appl Math

116

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Modeling the movement of cells (bacteria, amoeba) is a long standing subject and Partial Differential equations have been used several times. The most classical and successful system was proposed by Patlak [55] and Keller & Segel [39] and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on bacteria like Escherichia Coli or amoeba like Dictyostelium Discoïdeum exhibiting pointwise concentrations.For human endothelial cells, several experiments show the formation of networks that can be interpreted as the initiation of angiogenesis. To recover such patterns a hydrodynamical model seems better adapted.The two systems can be unified by a kinetic approach that was proposed for Escherichia Coli, based on more precise experiments showing a movement by 'jump and tumble'. This nonlinear kinetic model is interesting by itself and the existence theory is not complete. It is also interesting from a scaling point of view; in a diffusion limit one recovers the Keller-Segel model and in a hydrodynamical limit one recovers the model proposed for human endothelial cells.We also mention the mathematical interest of a analyzing another degenerate parabolic system (exhibiting different properties) proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels. 1

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Section: Remark 24mentioning

confidence: 84%

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Perthame

2004

Appl Math

116

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“…In [15], the relations The following inequality is proven similarly to Lemma 13 of [9]. We have only to make use of the elliptic estimate instead of Lemma However, as is noted at the end of §3, those relations imply T m ax < +00, a contradictioin.…”

Section: Blowup Solution In the Infinite Timementioning

confidence: 90%

“…We study (1.5) on the general domain Q, with dQ, sufficiently smooth around XQ G 90, and u denotes the solution. Theorem 3 is proven by localizing the argument of [15].…”

Section: Boundary Blowup Of the Solutionmentioning

confidence: 99%

“…For a more simplified system they showed that A =||t6o|| 1 <C 1 implies T max = +oo, while ^max < H-oo can happen when A ^> 1. Later Nagai [15] proved that the conjecture holds in the affirmative for radially symmetric solutions of…”

mentioning

confidence: 97%

“…If N = 3, T max < -foo can occur regardless of A, and rather interesting features of the solution can be observed. See [15] and the references therein for those facts concerning other space dimensions.…”

mentioning

confidence: 99%

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Parabolic system of chemotaxis: Blowup in a finite and the infinite time

Senba¹,

Suzuki²

2001

Methods and Applications of Analysis

Self Cite

142

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1. Introduction. The purpose of the present paper is to study blowup mechanism for a system of parabolic equations. It arises in mathematical biology to describe the chemotactic feature of slime molds.We take the form proposed by Nanjundiah [20], simplifying the one previously given by Keller and Segel [14]. It is stated as follows, where u = u (x, t) and v = v(x, t) stand the density of slime molds and the concentration chemical substances secreted by them, respectively:

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Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ²

Blanchet

Carrillo

Masmoudi

2007

Comm Pure Appl Math

259

261

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We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R 2 . Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 π/χ. Actually, we prove that solutions blow-up as a delta Dirac at the center of mass when t → ∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t → ∞ if initially bounded.

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Behavior of radially symmetric solutions of a system related to chemotaxis

Cited by 67 publications

References 6 publications

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Parabolic system of chemotaxis: Blowup in a finite and the infinite time

Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ²

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Behavior of radially symmetric solutions of a system related to chemotaxis

Cited by 67 publications

References 6 publications

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Parabolic system of chemotaxis: Blowup in a finite and the infinite time

Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2

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Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ²