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Strong solutions to the Keller‐Segel system with the weak L n /2 initial data and its application to the blow‐up rate
Author(s) -
Kozono Hideo,
Sugiyama Yoshie
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200610835
Subject(s) - mathematics , interval (graph theory) , combinatorics , space (punctuation) , weak solution , characterization (materials science) , mathematical analysis , physics , philosophy , linguistics , optics
We shall show an exact time interval for the existence of local strong solutions to the Keller‐Segel system with the initial data u 0 in L n /2 w (ℝ n ), the weak L n /2 ‐space on ℝ n . If ‖ u 0 ‖ L n /2 w(ℝ n )is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in L n /2 w (ℝ n ) stems from obtaining a self‐similar solution which does not belong to any usual L p (ℝ n ). Furthermore, the characterization of local existence of solutions gives us an explicit blow‐up rate of ‖ u ( t )‖ L p (ℝ n )for n /2 < p < ∞ as t → T max , where T max denotes the maximal existence time (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)