# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021162
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## Time periodic solution to a two-species chemotaxis-Stokes system with $p$-Laplacian diffusion

 Department of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author

Received  April 2021 Revised  August 2021 Early access September 2021

Fund Project: This work is supported by the Jilin Scientific and Technological Development Program (no. 20210101466JC)

In this paper, we consider a two-species chemotaxis-Stokes system with $p$-Laplacian diffusion in two-dimensional smooth bounded domains. It is proved that the existence of time periodic solution for any $\frac{15}{7}\leq p<3$ and any large periodic source $g_1(x,t)$ and $g_2(x,t)$.

Citation: Chengxin Du, Changchun Liu. Time periodic solution to a two-species chemotaxis-Stokes system with $p$-Laplacian diffusion. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021162
##### References:
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##### References:
 [1] X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154.  doi: 10.1002/mma.4807.  Google Scholar [2] J. Han and C. Liu, Global existence for a two-species chemotaxis-Navier-Stokes system with p-Laplacian, Electron. Res. Arch., http://dx.doi.org/10.3934/era.2021050. doi: 10.3934/era.2021050.  Google Scholar [3] J. Huang and C. Jin, Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion, Discrete Contin. Dyn. Syst., 40 (2020), 5415-5439.  doi: 10.3934/dcds.2020233.  Google Scholar [4] C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 24pp. doi: 10.1007/s00033-017-0882-9.  Google Scholar [5] C. Jin, Large time periodic solution to the coupled chemotaxis-Stokes model, Math. Nachr., 290 (2017), 1701-1715.  doi: 10.1002/mana.201600180.  Google Scholar [6] C. Jin, Periodic pattern formation in the coupled chemotaxis-(Navier-)Stokes system with mixed nonhomogeneous boundary conditions, Proc. Roy. Soc. Edinb. Sect. A, 150 (2020), 3121-3152.  doi: 10.1017/prm.2019.62.  Google Scholar [7] C. Liu and P. Li, Global existence for a chemotaxis-haptotaxis model with p-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 1399-1419.  doi: 10.3934/cpaa.2020070.  Google Scholar [8] C. Liu and P. Li, Time periodic solutions for a two-species chemotaxis-Navier-Stokes system, Discret. Contin. Dynam. Syst. Series B., 26 (2021), 4567-4585.  doi: 10.3934/dcdsb.2020303.  Google Scholar [9] J. Liu, Boundedness in a Chemotaxis-Navier-Stokes System modeling coral fertilization with slow p-Laplacian diffusion, J. Math. Fluid Mech., 22 (2020), 31 pp. doi: 10.1007/s00021-019-0469-7.  Google Scholar [10] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar [11] W. Tao and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow p-Laplacian diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 26-52.  doi: 10.1016/j.nonrwa.2018.06.005.  Google Scholar [12] W. Tao and Y. Li, Boundedness of weak solutions of a chemotaxis-Stokes system with slow p-Laplacian diffusion, J. Differ. Equ., 268 (2020), 6872-6919.  doi: 10.1016/j.jde.2019.11.078.  Google Scholar [13] I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar [14] M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.  Google Scholar [15] M. Winkler, Global large-data solutions in a chemotaxis-Navier-Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.  Google Scholar [16] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.  doi: 10.1090/tran/6733.  Google Scholar [17] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar [18] J. Yin and C. Jin, Periodic solutions of the evolutionary p-Laplacian with nonlinear sources, J. Math. Anal. Appl., 368 (2010), 604-622.  doi: 10.1016/j.jmaa.2010.03.006.  Google Scholar
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