Global existence and boundedness of solutions to achemotaxis system with singular sensitivity and logistic-type source

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Zhao, Xiangdong¹  Zheng, Sining² 

[1] Liaoning Normal Univ, Sch Math Sci, Dalian 116029, Peoples R China

[2] Dalian Univ Technol, Sch Math Sci, Dalian 116024, Peoples R China

关键词: Keller-Segel system;    Singular sensitivity;    Logistic source;    Boundedness;   

DOI  :  10.1016/j.jde.2019.01.026

来源: Elsevier

【 摘 要 】

We consider the fully parabolic Keller Segel system with singular sensitivity and logistic-type source: u(t) = Delta u - chi del(u/v del v) + ru - mu u(k), v(t) = Delta v - v + u under the non-flux boundary conditions in a smooth bounded convex domain Omega subset of R-n, x, r, mu > 0, k > 1. A global very weak so 2, lution for the system with n >= 2 is obtained under one of the following conditions: (i) r > chi(2)/4 for 0 < chi <= 2, or r > max{chi(2)/4 (1 - p(0)(2)), x - 1} for chi > 2 with p(0) = 4(k-1)/4+(2k-k)k chi(2) if k epsilon (2 - 1/n, 2]; (ii) chi(2) < min {2(r+r(2)/k), 4/k(k-1)(k-2) if k > 2. Furthermore, this global very weak solution should be globally bounded in fact provided r/mu and the initial data vertical bar vertical bar u(0)vertical bar vertical bar(L2(Omega)),vertical bar vertical bar del v vertical bar vertical bar(L4(Omega)) suitably small for n = 2, 3. In addition, if k > 3 (n+2)/n+4 replaces k > 2 in the condition (ii), the system admits globally bounded classical solutions. All these describe the influence of the exponent k > 1 in the logistic-type source ru - mu u(k) to the behavior of solutions for the considered fully parabolic Keller-Segel system with singular sensitivity. (C) 2019 Elsevier Inc. All rights reserved.

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