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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 |
$\left\{ \begin{array}{l}{u_t} = \nabla (D(u)\nabla u) - \nabla (\chi {u^q}\nabla v) + \mu u(1 - {u^\alpha }),\;\;\;\;\;\;\;\;& x \in \Omega ,{\mkern 1mu} {\mkern 1mu} t > 0,\\0 = \Delta v - v + {u^\gamma },\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&x \in \Omega ,{\mkern 1mu} {\mkern 1mu} t > 0\end{array} \right.$ |
$q≥ 1$ |
$α>0$ |
$γ≥ 1$ |
$D(u)≥ c_D u^{m-1}$ |
$(m≥ 1)$ |
$u>0$ |
$D(u)>0$ |
$u≥ 0$ |
$Ω\subset\mathbb{R}^N$ |
$(N≥ 1)$ |
$ m>q+γ-\frac{2}{N}, \, \, \mathbf{or}$ |
$ α>q+γ-1, \, \, \mathbf{or}$ |
$α = q+γ-1\, \, {\rm{and}}\, \, μ>μ^*$ |
$ {\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.$ |
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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