Time periodic solution to a two-species chemotaxis-Stokes system with porous medium diffusion

Hui Tang; Chengxin Du; Hui Tang; Chengxin Du

doi:10.3934/era.2025224

Electronic Research Archive

2025, Volume 33, Issue 8: 4984-5021. doi: 10.3934/era.2025224

Previous Article Next Article

Research article

Time periodic solution to a two-species chemotaxis-Stokes system with porous medium diffusion

Hui Tang ^{1
,
,},
Chengxin Du ^{2
,
,}

1.
College of Mathematics, Changchun Normal University, Changchun 130032, China
2.
Information Engineering University, Zhengzhou 450001, China

Received: 26 May 2025 Revised: 09 August 2025 Accepted: 13 August 2025 Published: 25 August 2025

In this paper, we considered a two-species chemotaxis-Stokes system with porous medium diffusion $ \Delta n_1^m $ in two-dimensional smooth bounded domains. We proved the existence of a time periodic solution for any $ m\geq\frac{6}{5} $ and any large periodic sources $ g_1(x, t) $ and $ g_2(x, t) $.
- two-species chemotaxis-Stokes system,
- time periodic solution,
- double-level approximation
Citation: Hui Tang, Chengxin Du. Time periodic solution to a two-species chemotaxis-Stokes system with porous medium diffusion[J]. Electronic Research Archive, 2025, 33(8): 4984-5021. doi: 10.3934/era.2025224

Related Papers:

Abstract

In this paper, we considered a two-species chemotaxis-Stokes system with porous medium diffusion $ \Delta n_1^m $ in two-dimensional smooth bounded domains. We proved the existence of a time periodic solution for any $ m\geq\frac{6}{5} $ and any large periodic sources $ g_1(x, t) $ and $ g_2(x, t) $.

References

[1]	I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler, R. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277–2282. https://doi.org/10.1073/pnas.0406724102 doi: 10.1073/pnas.0406724102
[2]	C. Huang, X. Ding, Q. Wang, Positive periodic stability to a neutral Nicholson's Blowflies model, Qual. Theory Dyn. Syst., 24 (2025), 1–22. https://doi.org/10.1007/s12346-025-01295-x doi: 10.1007/s12346-025-01295-x
[3]	M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329–1352. https://doi.org/10.1016/j.anihpc.2015.05.002 doi: 10.1016/j.anihpc.2015.05.002
[4]	M. Winkler, Global large-data solutions in a chemotaxis-Navier-Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equations, 37 (2012) 319–351. https://doi.org/10.1080/03605302.2011.591865
[5]	M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Am. Math. Soc., 369 (2017), 3067–3125. https://doi.org/10.1090/tran/6733 doi: 10.1090/tran/6733
[6]	H. Xu, L. Zhang, C. Jin, Global solvability andlarge time behavior to a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinear Anal. Real World Appl., 46 (2019), 238–256. https://doi.org/10.1016/j.nonrwa.2018.09.019 doi: 10.1016/j.nonrwa.2018.09.019
[7]	J. Liu, J. Zheng, Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), 1–33. https://doi.org/10.1007/s00033-016-0620-8 doi: 10.1007/s00033-016-0620-8
[8]	X. Cao, S. Kurima, M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Methods Appl. Sci., 41 (2018), 3138–3154. https://doi.org/10.1002/mma.4807 doi: 10.1002/mma.4807
[9]	J. Han, C. Liu, Global existence for a two-species chemotaxis-Navier-Stokes system with p-Laplacian, Electron. Res. Arch., 29 (2021), 3509–3533. https://doi.org/10.3934/era.2021050 doi: 10.3934/era.2021050
[10]	C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 1–24. https://doi.org/10.1007/s00033-017-0882-9 doi: 10.1007/s00033-017-0882-9
[11]	C. Jin, Large time periodic solution to the coupled chemotaxis-Stokes model, Math. Nachr., 290 (2017), 1701–1715. https://doi.org/10.1002/mana.201600180 doi: 10.1002/mana.201600180
[12]	J. Huang, C. Jin, Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion, Discrete Contin. Dyn. Syst., 40 (2020), 5415–5439. https://doi.org/10.3934/dcds.2020233 doi: 10.3934/dcds.2020233
[13]	C. Liu, P. Li, Time periodic solutions for a two-species chemotaxis-Navier-Stokes system, Discrete Contin. Dyn.-B, 26 (2021), 4567–4585. https://doi.org/10.3934/dcdsb.2020303 doi: 10.3934/dcdsb.2020303
[14]	C. Du, C. Liu, Time periodic solution to a two-species chemotaxis-Stokes system with p-Laplacian diffusion, Commun. Pure Appl. Anal., 20 (2021), 4321–4345. https://doi.org/10.3934/cpaa.2021162 doi: 10.3934/cpaa.2021162
[15]	C. Du, C. Liu, Time periodic solution to a mechanochemical model in biological patterns, Evol. Equations Control Theory, 12 (2023), 502–524. https://doi.org/10.3934/eect.2022039 doi: 10.3934/eect.2022039
[16]	C. Du, C. Liu, M. Mei, Time-periodic solution to a three-phase model of viscoelastic fluid flow, Discrete Cont. Dyn., 43 (2023), 276–308. https://doi.org/10.3934/dcds.2022149 doi: 10.3934/dcds.2022149
[17]	T. Eiter, On the spatially asymptotic structure of time-periodic solutions to the Navier-Stokes equations, Proc. Am. Math. Soc., 149 (2021), 3439–3451. https://doi.org/10.1090/proc/15482 doi: 10.1090/proc/15482
[18]	T. Eiter, M. Kyed, Y. Shibata, On periodic solutions for one-phase and two-phase problems of the Navier-Stokes equations, J. Evol. Equations, 21 (2021), 2955–3014. https://doi.org/10.1007/s00028-020-00619-5 doi: 10.1007/s00028-020-00619-5
[19]	Y. Yang, Y. Zhou, Time-periodic solution to a two-phase model with magnetic field in a periodic domain, J. Math. Anal. Appl., 489 (2020), 1–22. https://doi.org/10.1016/j.jmaa.2020.124146 doi: 10.1016/j.jmaa.2020.124146
[20]	C. Jin, Periodic pattern formation in the coupled chemotaxis-(Navier-)Stokes system with mixed nonhomogeneous boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 3121–3152. https://doi.org/10.1017/prm.2019.62 doi: 10.1017/prm.2019.62
[21]	N. Mizoguchi, P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851–875. https://doi.org/10.1016/j.anihpc.2013.07.007 doi: 10.1016/j.anihpc.2013.07.007
[22]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equations, 248 (2010), 2889–2905. https://doi.org/10.1016/j.jde.2010.02.008 doi: 10.1016/j.jde.2010.02.008
[23]	Z. Wu, J.Yin, C. Wang, Elliptic and Parabolic Equations, World Scientific Publishing, 2006. https://doi.org/10.1142/6238
[24]	Z. Wu, J. Zhao, J. Yin, H. Li, Nonlinear Diffusion Equations, World Scientific Publishing, 2001. https://doi.org/10.1142/4782
[25]	J. Yin, C. Jin, Periodic solutions of the evolutionary p-Laplacian with nonlinear sources, J. Math. Anal. Appl., 368 (2010), 604–622. https://doi.org/10.1016/j.jmaa.2010.03.006 doi: 10.1016/j.jmaa.2010.03.006

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)