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Communications in Mathematical Physics

, Volume 367, Issue 2, pp 665–681 | Cite as

Blow-up Profiles for the Parabolic–Elliptic Keller–Segel System in Dimensions \({n\geq 3}\)

  • Philippe Souplet
  • Michael WinklerEmail author
Article

Abstract

We study the blow-up asymptotics of radially decreasing solutions of the parabolic–elliptic Keller–Segel–Patlak system in space dimensions \( {n \geq 3}\). In view of the biological background of this system and of its mass conservation property, blowup is usually interpreted as a phenomenon of concentration or aggregation of the bacterial population. Understanding the asymptotic behavior of solutions at the blowup time is thus meaningful for the interpretation of the model. Under mild assumptions on the initial data, for \({n \geq 3}\), we show that the final profile satisfies \({{C}_{1}|{x}|^{-2}\leq u(x, T ) \leq {C}_{2}|x|^{-2}}\), with convergence in L1 as \({t\rightarrow T}\).This is in sharp contrast with the two-dimensional case, where solutions are known to concentrate to a Dirac mass at the origin (plus an integrable part). We also obtain refined space–time estimates of the form u(x, t) ≤  C(Tt + |x|2)−1 for type I blowup solutions. Previous work had shown that radial, self-similar blowup solutions (which satisfy the above estimates) exist in dimensions \( {n \geq 3}\) and do not exist in dimension 2. Our results thus reveal that the final profile displayed by these special solutions actually corresponds to a much more general phenomenon.

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Notes

Acknowledgments

The first author is partially supported by the Labex inflamex (ANR project 10-LABX-0017), and by the Labex MME-DII (ANR project 11-LBX-0023-01). The second author acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Analysis of chemotactic cross-diffusion in complex frameworks.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire Analyse, Géométrie et ApplicationsCNRS UMR 7539, Université Paris 13, Sorbonne Paris CitéVilletaneuseFrance
  2. 2.Institut für MathematikUniversität PaderbornPaderbornGermany

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