Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions
Authors:
Guangyi Hong and Zhi-an Wang
Journal:
Quart. Appl. Math. 79 (2021), 717-743
MSC (2020):
Primary 35K51, 35B40, 35Q92, 92C17
DOI:
https://doi.org/10.1090/qam/1599
Published electronically:
June 29, 2021
MathSciNet review:
4328145
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Additional Information
Abstract: In this paper, we consider the exogenous chemotaxis system with physical mixed zero-flux and Dirichlet boundary conditions in one dimension. Since the Dirichlet boundary condition can not contribute necessary estimates for the cross-diffusion structure in the system, the global-in-time existence and asymptotic behavior of solutions remain open up to date. In this paper, we overcome this difficulty by employing the technique of taking anti-derivative so that the Dirichlet boundary condition can be fully used, and show that the system admits global strong solutions which exponentially stabilize to the unique stationary solution as time tends to infinity against some suitable small perturbations. To the best of our knowledge, this is the first result obtained on the global well-posedness and asymptotic behavior of solutions to the exogenous chemotaxis system with physical boundary conditions.
References
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- L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 733–737. MR 208360
- Takaaki Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications Mathématiques d’Orsay, No. 78-02, Département de Mathématique, Université de Paris-Sud, Orsay, 1978. MR 0481578
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- Idan Tuval, Luis Cisneros, Christopher Dombrowski, Charles W Wolgemuth, John O Kessler, and Raymond E Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA 102 (2005), no. 7, 2277–2282.
- G.H. Wadhams and J.P. Armitage, Making sense of it all: bacterial chemotaxis, Nat Rev Mol Cell Biol 5 (2004), no. 12, 1024–1037.
- Y. Wang, C.-L. Chen, and M. Iijima, Signaling mechanisms for chemotaxis, Develop Growth Differ. 53 (2011), no. 4, 495–502.
- Zhi-An Wang, Mathematics of traveling waves in chemotaxis—review paper, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), no. 3, 601–641. MR 3007746, DOI 10.3934/dcdsb.2013.18.601
- Michael Winkler, Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, J. Math. Pures Appl. (9) 112 (2018), 118–169 (English, with English and French summaries). MR 3774876, DOI 10.1016/j.matpur.2017.11.002
References
- J. Adler, Chemotaxis in bacteria, Science 153 (1966), 708–716.
- N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015), no. 9, 1663–1763. MR 3351175, DOI 10.1142/S021820251550044X
- J.T. Bonner, The cellular slime molds, 2nd ed, Princeton University Press, 1967.
- Marcel Braukhoff and Johannes Lankeit, Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions for the oxygen, Math. Models Methods Appl. Sci. 29 (2019), no. 11, 2033–2062. MR 4022367, DOI 10.1142/S0218202519500398
- Jose A. Carrillo, Jingyu Li, and Zhi-An Wang, Boundary spike-layer solutions of the singular Keller-Segel system: existence and stability, Proc. Lond. Math. Soc. (3) 122 (2021), no. 1, 42–68. MR 4210256, DOI 10.1112/plms.12319
- Myeongju Chae and Kyudong Choi, Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion, J. Differential Equations 268 (2020), no. 7, 3449–3496. MR 4053596, DOI 10.1016/j.jde.2019.09.061
- A. Chertock, K. Fellner, A. Kurganov, A. Lorz, and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, J. Fluid Mech. 694 (2012), 155–190. MR 2897678, DOI 10.1017/jfm.2011.534
- Kyudong Choi, Moon-Jin Kang, and Alexis F. Vasseur, Global well-posedness of large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model, J. Math. Pures Appl. (9) 142 (2020), 266–297 (English, with English and French summaries). MR 4149692, DOI 10.1016/j.matpur.2020.03.002
- P. N. Davis, P. van Heijster, and R. Marangell, Absolute instabilities of travelling wave solutions in a Keller-Segel model, Nonlinearity 30 (2017), no. 11, 4029–4061. MR 3718730, DOI 10.1088/1361-6544/aa842f
- S. Dehaene, The neural basis of the Weber–Fechner law: a logarithmic mental number line, Trends Cogn. Sci. 7 (2003), no. 4, 145–147.
- T. Hillen and K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), no. 1-2, 183–217. MR 2448428, DOI 10.1007/s00285-008-0201-3
- Dirk Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), no. 3, 103–165. MR 2013508
- Qianqian Hou, Cheng-Jie Liu, Ya-Guang Wang, and Zhian Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one-dimensional case, SIAM J. Math. Anal. 50 (2018), no. 3, 3058–3091. MR 3814021, DOI 10.1137/17M112748X
- Qianqian Hou and Zhian Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl. (9) 130 (2019), 251–287 (English, with English and French summaries). MR 4001634, DOI 10.1016/j.matpur.2019.01.008
- Y.V. Kalinin, L. Jiang, Y. Tu, and M. Wu, Logarithmic sensing in escherichia coli bacterial chemotaxis, Biophys. J. 96 (2009), no. 6, 2439–2448.
- Evelyn F. Keller and Garrett M. Odell, Necessary and sufficient conditions for chemotactic bands, Math. Biosci. 27 (1975), no. 3-4, 309–317. MR 411681, DOI 10.1016/0025-5564(75)90109-1
- Evelyn F. Keller and Lee A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), no. 3, 399–415. MR 3925816, DOI 10.1016/0022-5193(70)90092-5
- E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol. 30 (1971), no. 2, 235–248.
- Chiun-Chang Lee, Zhi-An Wang, and Wen Yang, Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis, Nonlinearity 33 (2020), no. 10, 5111–5141. MR 4143968, DOI 10.1088/1361-6544/ab8f7c
- Hyun Geun Lee and Junseok Kim, Numerical investigation of falling bacterial plumes caused by bioconvection in a three-dimensional chamber, Eur. J. Mech. B Fluids 52 (2015), 120–130. MR 3342441, DOI 10.1016/j.euromechflu.2015.03.002
- Howard A. Levine, Brian D. Sleeman, and Marit Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. The role of protease inhibitors in preventing angiogenesis, Math. Biosci. 168 (2000), no. 1, 77–115. MR 1788960, DOI 10.1016/S0025-5564(00)00034-1
- Huicong Li and Kun Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations 258 (2015), no. 2, 302–338. MR 3274760, DOI 10.1016/j.jde.2014.09.014
- Jingyu Li, Tong Li, and Zhi-An Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci. 24 (2014), no. 14, 2819–2849. MR 3269780, DOI 10.1142/S0218202514500389
- Tong Li and Zhi-An Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math. 70 (2009/10), no. 5, 1522–1541. MR 2578681, DOI 10.1137/09075161X
- Tong Li and Zhi-An Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations 250 (2011), no. 3, 1310–1333. MR 2737206, DOI 10.1016/j.jde.2010.09.020
- Alexander Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci. 20 (2010), no. 6, 987–1004. MR 2659745, DOI 10.1142/S0218202510004507
- Toshitaka Nagai and Tsutomu Ikeda, Traveling waves in a chemotactic model, J. Math. Biol. 30 (1991), no. 2, 169–184. MR 1138847, DOI 10.1007/BF00160334
- L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 733–737. MR 208360
- Takaaki Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Département de Mathématique, Université de Paris-Sud, Orsay, 1978. Publications Mathématiques d’Orsay, No. 78-02. MR 0481578
- Hongyun Peng and Zhi-An Wang, Nonlinear stability of strong traveling waves for the singular Keller-Segel system with large perturbations, J. Differential Equations 265 (2018), no. 6, 2577–2613. MR 3804725, DOI 10.1016/j.jde.2018.04.041
- Hartmut R. Schwetlick, Travelling fronts for multidimensional nonlinear transport equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 4, 523–550 (English, with English and French summaries). MR 1782743, DOI 10.1016/S0294-1449(00)00127-X
- Idan Tuval, Luis Cisneros, Christopher Dombrowski, Charles W Wolgemuth, John O Kessler, and Raymond E Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA 102 (2005), no. 7, 2277–2282.
- G.H. Wadhams and J.P. Armitage, Making sense of it all: bacterial chemotaxis, Nat Rev Mol Cell Biol 5 (2004), no. 12, 1024–1037.
- Y. Wang, C.-L. Chen, and M. Iijima, Signaling mechanisms for chemotaxis, Develop Growth Differ. 53 (2011), no. 4, 495–502.
- Zhi-An Wang, Mathematics of traveling waves in chemotaxis—review paper, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), no. 3, 601–641. MR 3007746, DOI 10.3934/dcdsb.2013.18.601
- Michael Winkler, Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, J. Math. Pures Appl. (9) 112 (2018), 118–169 (English, with English and French summaries). MR 3774876, DOI 10.1016/j.matpur.2017.11.002
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Additional Information
Guangyi Hong
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of China—and—School of Mathematics, South China University of Technology, Guangzhou, 510641, China
MR Author ID:
1084645
Email:
gyhmath05@outlook.com
Zhi-an Wang
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People’s Republic of China
MR Author ID:
686941
Email:
mawza@polyu.edu.hk
Keywords:
Exogenous chemotaxis,
steady state,
asymptotic behavior,
anti-derivative,
energy method
Received by editor(s):
February 1, 2021
Received by editor(s) in revised form:
May 5, 2021
Published electronically:
June 29, 2021
Additional Notes:
The first author was partially supported by the CAS AMSS-POLYU Joint Laboratory of Applied Mathematics postdoctoral fellowship scheme. The second author was supported in part by the Hong Kong Research Grant Council General Research Fund No. PolyU 153031/17P (Q62H) and internal grant No. ZZHY from HKPU. The second author is the corresponding author.
Article copyright:
© Copyright 2021
Brown University
