Maximal regularity of the heat evolution equation on spatial local spaces and application to a singular limit problem of the Keller–Segel system
We consider the singular limit problem for the Cauchy problem of the (Patlak–) Keller–Segel system of parabolic-parabolic type. The problem is considered in the uniformly local Lebesgue spaces and the singular limit problem as the relaxation parameter τ goes to infinity, the solution to the Keller–S...
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Veröffentlicht in: | Mathematische annalen 2023-10, Vol.387 (1-2), p.389-431 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | We consider the singular limit problem for the Cauchy problem of the (Patlak–) Keller–Segel system of parabolic-parabolic type. The problem is considered in the uniformly local Lebesgue spaces and the singular limit problem as the relaxation parameter
τ
goes to infinity, the solution to the Keller–Segel equation converges to a solution to the drift-diffusion system in the strong uniformly local topology. For the proof, we follow the former result due to Kurokiba–Ogawa [
20
–
22
] and we establish maximal regularity for the heat equation over the uniformly local Lebesgue and Morrey spaces which are non-UMD Banach spaces and apply it for the strong convergence of the singular limit problem in the scaling critical local spaces. |
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ISSN: | 0025-5831 1432-1807 |
DOI: | 10.1007/s00208-022-02469-7 |