A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity |
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Authors: | Masaaki Mizukami Tomomi Yokota |
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Affiliation: | Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601 Japan |
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Abstract: | This paper deals with the Keller–Segel system where Ω is a bounded domain in with smooth boundary , ; χ is a nonnegative function satisfying for some and . In the case that and , Fujie 2 established global existence of bounded solutions under the condition . On the other hand, when , Winkler 14 asserted global existence of bounded solutions for arbitrary . However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary . Moreover, the condition for K when cannot connect with the condition when . The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for χ and to build a mathematical bridge between the cases and . |
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Keywords: | chemotaxis sensitivity function global existence boundedness Primary: 35K51 Secondary: 35A01 35B45 92C17 |
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