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Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel system

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Abstract

The fully parabolic Keller-Segel system is considered in n-dimensional balls with n ≥ 2. Pointwise time-independent estimates are derived for arbitrary radially symmetric solutions. These are firstly used to assert that any radial classical solution which blows up in finite time possesses a uniquely determined blow-up profile which satisfies an associated pointwise upper inequality. Secondly, in conjunction with additional regularity features implied by a very weak but temporally and spatially global quasi-entropy property, these estimates are seen to ensure global extensibility of any such solution within a suitable framework of renormalized solutions.

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Acknowledgement

The author is very grateful to the anonymous reviewer for numerous helpful remarks. The author moreover acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Analysis of chemotactic cross-diffusion in complex frameworks.

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  1. Institut für Mathematik, Universität Paderborn, 33098, Paderborn, Germany

    Michael Winkler

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  1. Michael Winkler
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Correspondence to Michael Winkler.

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Winkler, M. Blow-up profiles and life beyond blow-up in the fully parabolic Keller-Segel system. JAMA 141, 585–624 (2020). https://doi.org/10.1007/s11854-020-0109-4

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  • DOI: https://doi.org/10.1007/s11854-020-0109-4

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