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February  2020, 13(2): 321-328. doi: 10.3934/dcdss.2020018

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

Yilong Wang 1,, and  Xuande Zhang 2,

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)).
Keywords: Chemotaxis, global existence, boundedness, parabolic-elliptic systems.
Mathematics Subject Classification: Primary: 35K57, 35Q92, 35A01, 92C17.
Citation: Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 321-328. doi: 10.3934/dcdss.2020018
References:
[1]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

E. Galakhov, O. Salieva and J. I. Tello, On a Parabolic-Elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.

[3]

J. Gao, P. Zhu and A. Alsaedi, et al., A new switching control for finite-time synchronization of memristor-based recurrent neural networks, Neural Networks, 86 (2017), 1–9. doi: 10.1016/j.neunet.2016.10.008.

[4]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[5]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165. 

[6]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien, 106 (2004), 51-69. 

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[8]

B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.  doi: 10.1016/j.aml.2016.08.003.

[9]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.

[10]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[11]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198.  doi: 10.1093/imamat/hxv033.

[12]

X. Li and Z. Xiang, Boundedness in quasilinear Keller–Segel equations with nonlinear sensitivity and logistic source, Discrete Continuous Dynam. Systems - A, 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503.

[13]

E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic- parabolic system for chemotaxis with subquadratic degradation, Discrete Continuous Dynam. Systems - B, 18 (2013), 2627-2646.  doi: 10.3934/dcdsb.2013.18.2627.

[14]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543. 

[15]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[16]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[17]

L. C. Wang, Y. H. Li and C. L. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Continuous Dynam. Systems - A, 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[18]

L. C. Wang, C. L. Mu and P. Zheng, On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.

[19]

Y. Wang, A quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259-292.  doi: 10.1016/j.jmaa.2016.03.061.

[20]

Y. Wang, Global existence and boundedness in a quasilinear attraction–repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Probl., 2016 (2016), 1-22.  doi: 10.1186/s13661-016-0518-6.

[21]

Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attractionrepulsion chemotaxis system, Discrete Continuous Dynam. Systems - B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.

[22]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[23]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal.-Theor.Methods Appl., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.

[24]

X. Wu, X. Ding, T. Lu and J. Wang, Topological dynamics of Zadeh's extension on upper semi-continuous fuzzy sets, Int. J. Bifurcation and Chaos, 27 (2017), 1750165, 13pp. doi: 10.1142/S0218127417501656.

[25]

X. Wu, X. Ma, Z. Zhu and T. Lu, Topological ergodic shadowing and chaos on uniform spaces, Int. J. Bifurcation and Chaos, 28 (2018), 1850043, 9pp. doi: 10.1142/S0218127418500438.

[26]

C. Yang, X. Cao, Z. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic KellerSegel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093.

[27]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.

show all references

References:
[1]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

E. Galakhov, O. Salieva and J. I. Tello, On a Parabolic-Elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.

[3]

J. Gao, P. Zhu and A. Alsaedi, et al., A new switching control for finite-time synchronization of memristor-based recurrent neural networks, Neural Networks, 86 (2017), 1–9. doi: 10.1016/j.neunet.2016.10.008.

[4]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[5]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165. 

[6]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien, 106 (2004), 51-69. 

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[8]

B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.  doi: 10.1016/j.aml.2016.08.003.

[9]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.

[10]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[11]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198.  doi: 10.1093/imamat/hxv033.

[12]

X. Li and Z. Xiang, Boundedness in quasilinear Keller–Segel equations with nonlinear sensitivity and logistic source, Discrete Continuous Dynam. Systems - A, 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503.

[13]

E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic- parabolic system for chemotaxis with subquadratic degradation, Discrete Continuous Dynam. Systems - B, 18 (2013), 2627-2646.  doi: 10.3934/dcdsb.2013.18.2627.

[14]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543. 

[15]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[16]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[17]

L. C. Wang, Y. H. Li and C. L. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Continuous Dynam. Systems - A, 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[18]

L. C. Wang, C. L. Mu and P. Zheng, On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.

[19]

Y. Wang, A quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259-292.  doi: 10.1016/j.jmaa.2016.03.061.

[20]

Y. Wang, Global existence and boundedness in a quasilinear attraction–repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Probl., 2016 (2016), 1-22.  doi: 10.1186/s13661-016-0518-6.

[21]

Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attractionrepulsion chemotaxis system, Discrete Continuous Dynam. Systems - B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.

[22]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[23]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal.-Theor.Methods Appl., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.

[24]

X. Wu, X. Ding, T. Lu and J. Wang, Topological dynamics of Zadeh's extension on upper semi-continuous fuzzy sets, Int. J. Bifurcation and Chaos, 27 (2017), 1750165, 13pp. doi: 10.1142/S0218127417501656.

[25]

X. Wu, X. Ma, Z. Zhu and T. Lu, Topological ergodic shadowing and chaos on uniform spaces, Int. J. Bifurcation and Chaos, 28 (2018), 1850043, 9pp. doi: 10.1142/S0218127418500438.

[26]

C. Yang, X. Cao, Z. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic KellerSegel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093.

[27]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.

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