# American Institute of Mathematical Sciences

May  2017, 16(3): 1037-1058. doi: 10.3934/cpaa.2017050

## Global existence of solutions to an attraction-repulsion chemotaxis model with growth

 1 Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China 2 Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

Received  June 2016 Revised  November 2016 Published  February 2017

Fund Project: S. Wu is supported in part by a grant from China Scholarship Council; J. Shi is partially supported by US-NSF grant DMS-1313243 and B. Wu is partially supported by National Natural Science Foundation of China grant No. 11271100.

An attraction-repulsion chemotaxis model with nonlinear chemotactic sensitivity functions and growth source is considered. The global-in-time existence and boundedness of solutions are proved under some conditions on the nonlinear sensitivity functions and growth source function. Our results improve the earlier ones for the linear sensitivity functions.

Citation: Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050
##### References:

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##### References:
Regions in (p, q) plane where the global existence and boundedness of solutions to (1.9) are proved. The regions labelled by (), (), () and (iv) correspond to the ones defined in Theorems 1.1 and 1.2, and for the region labelled with?, the result is not known. Left: n ≤ 2; Right: n > 2

1. $n = 1, 2$, and $0\le p, q\le 2/n$, or $2/n\le \max\{p, q\}\le r-1$ and $b$ large; or
2. $n\ge 3$, and $0\le p, q\le 2/n$, or $1\le \max\{p, q\}\le r-1$ and $b$ large.

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