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Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

## 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, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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