Boundedness in a Keller-Segel system with external signal production

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• 作者：Tobias Black ^{tblack@math.upb.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Chemotaxis ; Boundedness ; External signal production ; Critical mass
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：436-455
• 全文大小：501 K

文摘

We study the Neumann initial-boundary problem for the chemotaxis system in a smooth, bounded domain Ω⊂Rⁿmi mathvariant="normal">Ωmi>⊂<mi mathvariant="double-struck">Rmi><mi>nmi> with n≥2mi>nmi>≥2 and mi>fmi>∈<mi mathvariant="normal">Lmi>∞([0,∞);<mi mathvariant="normal">Lmi><mi>nmi>2+<mi>δmi>0(<mi mathvariant="normal">Ωmi>))∩<mi>Cmi><mi>αmi>(<mi mathvariant="normal">Ωmi>×(0,∞)) with some 241337eb338ce91c58cb1c670c" title="Click to view the MathML source">α>0mi>αmi>>0 and δ₀∈(0,1)mi>δmi>0∈(0,1). First we prove local existence of classical solutions for reasonably regular initial values. Afterwards we show that in the case of n=2mi>nmi>=2 and f   being constant in time, requiring the nonnegative initial data u₀mi>umi>0 to fulfill the property mi mathvariant="normal">Ωmi><mi>umi>0<mi mathvariant="normal">dmi><mi>xmi><4<mi>πmi> ensures that the solution is global and remains bounded uniformly in time. Thereby we extend the well known critical mass result by Nagai, Senba and Yoshida for the classical Keller–Segel model (coinciding with f≡0mi>fmi>≡0 in the system above) to the case f≢0mi>fmi>≢0. Under certain smallness conditions imposed on the initial data and f   we furthermore show that for more general space dimension n≥2mi>nmi>≥2 and f not necessarily constant in time, the solutions are also global and remain bounded uniformly in time. Accordingly we extend a known result given by Winkler for the classical Keller–Segel system to the present situation.