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Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source
February  2020, 13(2): 321-328. doi: 10.3934/dcdss.2020018

## On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion

 1 School of Sciences, Southwest Petroleum University, Chengdu 610500, China 2 College of Electrical & Information Engineering, Shaanxi University of Science & Technology, Xian 710021, China

* Corresponding author: Yilong Wang

Received  April 2017 Revised  November 2017 Published  January 2019

Fund Project: The first author is supported by Young scholars development fund of SWPU grant 200631010065, Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications grant 18TD0013, Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems grant 2017CXTD02 and the NNSF of China grant 11701461. The second author is supported by 2016 Google Nurturing Project for Young Researchers in West China

This paper considers the following parabolic-elliptic chemotaxis-growth system with nonlinear diffusion
 $\left\{ \begin{array}{l}{u_t} = \nabla (D(u)\nabla u) - \nabla (\chi {u^q}\nabla v) + \mu u(1 - {u^\alpha }),\;\;\;\;\;\;\;\;& x \in \Omega ,{\mkern 1mu} {\mkern 1mu} t > 0,\\0 = \Delta v - v + {u^\gamma },\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&x \in \Omega ,{\mkern 1mu} {\mkern 1mu} t > 0\end{array} \right.$
under homogeneous Neumann boundary conditions for some constants
 $q≥ 1$
,
 $α>0$
and
 $γ≥ 1$
, where
 $D(u)≥ c_D u^{m-1}$
 $(m≥ 1)$
for all
 $u>0$
and
 $D(u)>0$
for all
 $u≥ 0$
, and
 $Ω\subset\mathbb{R}^N$
 $(N≥ 1)$
is a bounded domain with smooth boundary. It is shown that when
 $m>q+γ-\frac{2}{N}, \, \, \mathbf{or}$
 $α>q+γ-1, \, \, \mathbf{or}$
 $α = q+γ-1\, \, {\rm{and}}\, \, μ>μ^*$
, where
 ${\mu ^*} = \left\{ \begin{array}{l}\begin{array}{*{20}{l}}{\frac{{(\alpha + 1 - m)N - 2}}{{(\alpha + 1 - m)N + 2(\alpha - \gamma )}}\chi ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}~~{\mkern 1mu} {\mkern 1mu} m \le q + \gamma - \frac{2}{N},}\end{array}\\0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}~~{\mkern 1mu} {\mkern 1mu} m > q + \gamma - \frac{2}{N},\end{array} \right.$
then the above system possesses a global bounded classical solution for any sufficiently smooth initial data. The results improve the results by Wang et al. (J. Differential Equations 256 (2014)) and generalize the results of Zheng (J. Differential Equations 259 (2015)) and Galakhov et al. (J. Differential Equations 261 (2016)).
Citation: Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 321-328. doi: 10.3934/dcdss.2020018
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