How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:459

Article

Xiang, Tian¹ 

[1] Renmin Univ China, Inst Math Sci, Beijing 100872, Peoples R China

关键词: Chemotaxis systems;    Logistic damping;    Boundedness;    Global existence;    Blow-up;    Long time dynamics;   

DOI  :  10.1016/j.jmaa.2017.11.022

来源: Elsevier

【 摘 要 】

We study nonnegative solutions of parabolic parabolic Keller-Segel minimal-chemotaxis-growth systems with prototype by {u(t) = del .(d(1) del u- chi u del v) + kappa u - mu u(2), x is an element of Omega, t > 0, v(t) = d(2)Delta v + beta v + alpha u, x is an element of Omega, t > 0 in a smooth bounded smooth but not necessarily convex domain Omega subset of R-n (n >= 3) with nonnegative initial data u(0), v(0) and homogeneous Neumann boundary data, where d(1), d(2), alpha,beta, mu > 0, chi, kappa is an element of R. We provide quantitative and qualitative descriptions of the competition between logistic damping and other ingredients, especially, chemotactic aggregation to guarantee boundedness and convergence. Specifically, we first obtain an explicit formula mu(0) = mu(0)(n, d(1), d(2), alpha, chi) for the logistic damping rate tz such that the system has no blow-ups whenever mu > mu(0). In particular, for Omega subset of R-3, we get a clean formula for mu(0): mu(0) = mu(0)(3, d(1), d(2), alpha, chi) = { 3/ 4d(1) alpha chi, if d(1) = d(2), chi > 0 and Omega is convex, 3/root 10-2 (1/d(1) + 2/d(2))alpha vertical bar chi vertical bar, otherwise. This offers a quantized effect of the logistic source on the prevention of blow-ups. Our result extends the fundamental boundedness principle by Winkler [42] with d(1) = 1, d(2) = alpha = beta := 1/tau, Omega being convex and sufficiently large values of mu beyond a certain number not explicitly known (except the simple case tau = 1 and chi > 0) and quantizes the qualitative result of Yang et al. [52]. Besides, in non-convex domains, since mu(0) (3, 1, 1, 1, chi) = (7.743416 ...)chi, the recent boundedness result, mu > 20 chi, of Mu and Lin [25] is greatly improved. Then we derive another explicit formula: mu(1) = mu(1)(d(1), d(2), alpha, beta, kappa, chi) = alpha vertical bar chi vertical bar/4 root kappa+/d(1)d(2)beta for the logistic damping rate so that convergence of bounded solutions is ensured and the respective convergence rates are explicitly calculated out whenever mu > mu(1) Recent convergence results of He and Zheng [9] are therefore complemented and refined. (C) 2017 Elsevier Inc. All rights reserved.

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