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July  2020, 19(7): 3843-3883. doi: 10.3934/cpaa.2020170

Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

2. 

Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

3. 

Institute of Artificial Intelligence, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author

Received  January 2020 Revised  February 2020 Published  April 2020

Fund Project: The authors were supported by NSFC grant 11971185

In this paper, we investigate the chemotaxis-fluid system with singular sensitivity and logistic source in bounded convex domain with smooth boundary. We present the global existence of very weak solutions under appropriate regularity assumptions on the initial data. Then, we show that system possesses a global bounded classical solution. Finally, we present a unique globally bounded classical solution for a fluid-free system. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature, and partially results are new.

Citation: Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170
References:
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T. Black, Global solvability of chemotaxis-fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions, Nonlinear Anal., 180 (2019), 129-153.  doi: 10.1016/j.na.2018.10.003.  Google Scholar

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E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 27 (1971), 235-248.   Google Scholar

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S. Kurima and M. Mizukami, Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier-Stokes system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 98-115.  doi: 10.1016/j.nonrwa.2018.09.011.  Google Scholar

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G. Ren and B. Liu, Boundedness of solutions for a quasilinear chemotaxis-haptotaxis model, Hakkaido Math. J., (2019), Accepted. doi: 10.1002/mma.4126.  Google Scholar

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G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.  Google Scholar

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show all references

References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[2]

T. Black, Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D, J. Differ. Equ., 265 (2018), 2296-2339.  doi: 10.1016/j.jde.2018.04.035.  Google Scholar

[3]

T. Black, Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion, SIAM J. Math. Anal., 50 (2018), 4087-4116.  doi: 10.1137/17M1159488.  Google Scholar

[4]

T. Black, Global solvability of chemotaxis-fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions, Nonlinear Anal., 180 (2019), 129-153.  doi: 10.1016/j.na.2018.10.003.  Google Scholar

[5]

T. BlackJ. Lankeit and M. Mizukami, Singular sensitivity in a Keller-Segel-fluid system, J. Evol. Equ., 18 (2018), 561-581.  doi: 10.1007/s00028-017-0411-5.  Google Scholar

[6]

M. DingW. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009.  Google Scholar

[7]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[8]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[9]

C. Jin, Global solvability and boundedness to a coupled chemotaxis-fluid model with arbitrary porous medium diffusion, J. Differ. Equ., 265 (2018), 332-353.  doi: 10.1016/j.jde.2018.02.031.  Google Scholar

[10]

C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 137. doi: 10.1007/s00033-017-0882-9.  Google Scholar

[11]

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

[12]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 27 (1971), 235-248.   Google Scholar

[13]

S. Kurima and M. Mizukami, Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier-Stokes system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 98-115.  doi: 10.1016/j.nonrwa.2018.09.011.  Google Scholar

[14]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equation of Parabolic Type, Amer. Math. Soc. Transl., vol. 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[15]

E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0.  Google Scholar

[16]

B. Liu and G. Ren, Global existence and asymptotic behavior in a three-dimensional two-species chemotaxis-Stokes system with tensor-valued sensitivity, J. Korean Math. Soc., 57 (2020), 215-247.  doi: 10.4134/JKMS.j190028.  Google Scholar

[17]

J. Liu and Y. Wang, Boundedness and decay property in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differ. Equ., 261 (2016), 967-999.  doi: 10.1016/j.jde.2016.03.030.  Google Scholar

[18]

J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differ. Equ., 262 (2017), 5271-5305.  doi: 10.1016/j.jde.2017.01.024.  Google Scholar

[19]

H. Matthias and P. Jan, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Commun. Partial Differ. Equ., 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.  Google Scholar

[20]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.   Google Scholar

[21]

G. Ren and B. Liu, Boundedness of solutions for a quasilinear chemotaxis-haptotaxis model, Hakkaido Math. J., (2019), Accepted. doi: 10.1002/mma.4126.  Google Scholar

[22]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.  Google Scholar

[23]

G. Ren and B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Differ. Equ., 268 (2020), 4320-4373.  doi: 10.1016/j.jde.2019.10.027.  Google Scholar

[24]

G. Ren and B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differ. Equ., 269 (2020), 1484-1520. doi: 10.1016/j.jde.2020.01.008.  Google Scholar

[25]

G. Ren and B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 545-582.  doi: 10.1016/j.nonrwa.2018.09.020.  Google Scholar

[26]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.  Google Scholar

[27]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. Henri Poincare Anal. Non Lineaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[28]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[29]

Y. Tao and M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.  Google Scholar

[30]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), 1-23.  doi: 10.1007/s00033-016-0732-1.  Google Scholar

[31]

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

[32]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. 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

[33]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[34]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis-system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.  Google Scholar

[35]

Y. Wang, Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivity, Bound. Value Probl., 2016 (2016), 177. doi: 10.1186/s13661-016-0687-3.  Google Scholar

[36]

Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Meth. Appl. Sci., 27 (2017), 2745-2780.  doi: 10.1142/S0218202517500579.  Google Scholar

[37]

Y. WangM. Winkler and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 18 (2018), 421-466.  doi: 10.1109/tps.2017.2783887.  Google Scholar

[38]

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