Boundedness in a Keller-Segel system with external signal production

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:446

Article

Black, Tobias¹ 

[1] Univ Paderborn, Inst Math, Warburger Str 100, D-33098 Paderborn, Germany

关键词: Chemotaxis;    Boundedness;    External signal production;    Critical mass;   

DOI  :  10.1016/j.jmaa.2016.08.049

来源: Elsevier

【 摘 要 】

We study the Neumann initial-bounchry problem for the chemotaxis system {u(t) = Delta u - del.(u del v), x is an element of Omega, t > 0, v(t) = Delta v - v + u+ f(x,t), x is an element of Omega, t > 0, partial derivative u/partial derivative v = partial derivative v/partial derivative v = 0, x is an element of partial derivative Omega, t > 0, u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), x is an element of Omega in a smooth, bounded domain Omega subset of R-n with n >= 2 and f is an element of L-infinity([0, not similar or equal to); Ln/2+delta 0 (Omega)) boolean AND C-alpha (Omega x (0, infinity)) with some alpha > 0 and delta(0) is an element of (0,1). First we prove local existence of classical solutions for reasonably regular initial values. Afterwards we show that in the case of n = 2 and f being constant in time, requiring the nonnegative initial data u(0) to fulfill the property integral(Omega) u(0) dx < 4 pi 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 equivalent to 0 in the system above) to the case f not equivalent to 0. Under certain smallness conditions imposed on the initial data and f we furthermore show that for more general space dimension n >= 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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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