Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source

Chunhua Jin

doi:10.3934/dcds.2018150

Article Contents

2018, Volume 38, Issue 7: 3547-3566. Doi: 10.3934/dcds.2018150

This issue Previous Article Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes Next Article On critical Choquard equation with potential well

Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source

Chunhua Jin^,

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author: jinchhua@126.com
* Corresponding author: jinchhua@126.com

Received: July 31, 2017

Revised: December 31, 2017

Published: June 2018

This work is supported by NSFC(11471127, 11571380), Guangdong Natural Science Funds for Distinguished Young Scholar (2015A030306029).

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we deal with a coupled chemotaxis-fluid model with logistic source $γ n-μ n^2$ . We prove the existence of global classical solution for the chemotaxis-Stokes system in a bounded domain $Ω\subset \mathbb R^3$ for any large initial data. On the basis of this, we further prove that if $γ>0$ , the zero solution is not stable; if $γ = 0$ , the zero solution is globally asymptotically stable; and if $ 0 < γ < 16μ^2$ , the nontrivial steady state $\left(\fracγμ, \fracγμ, 0\right)$ is globally asymptotically stable.

Keywords:

Mathematics Subject Classification: Primary: 92C17, 35M10; Secondary: 35K10, 76D07.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	N. Bellomo, A. Bellouquid and N. Chouhad, From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid, Math. Models Methods Appl. Sci., 26 (2016), 2041-2069. doi: 10.1142/S0218202516400078.
[2]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.
[3]	J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in ${\mathbb R}^2$, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616. doi: 10.1016/j.crma.2004.08.011.
[4]	E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Analysis: Real World Applications, 21 (2015), 110-126. doi: 10.1016/j.nonrwa.2014.07.001.
[5]	G. P. Galdi, An introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.
[6]	H. Hajajej, X. Yu and Z. Zhai, Fractional Gagliardo-Nirenberg and Hardy inequalities under lorentz norms, J. Math. Anal. Appl., 396 (2012), 569-577. doi: 10.1016/j.jmaa.2012.06.054.
[7]	M. A. Herrero, E. Medina and J. J. L. Velazquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754. doi: 10.1088/0951-7715/10/6/016.
[8]	M. Hieber and J. Pruss, Heat kernels and maximal $L^p-L^q$ estimates for parabolic evolution equations, Comm. Partial Diff. Eqs., 22 (1997), 1647-1669. doi: 10.1080/03605309708821314.
[9]	S. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.
[10]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[11]	H. Kozono and T. Yanagisawa, Leray's problem on the stationary Navier-Stokes equations with inhomogeneous boundary data, Math. Z., 262 (2009), 27-39. doi: 10.1007/s00209-008-0361-2.
[12]	J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109. doi: 10.1142/S021820251640008X.
[13]	J. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. I. H. Poincaré -AN, 28 (2011), 643-652. doi: 10.1016/j.anihpc.2011.04.005.
[14]	J. Liu and Y. Wang, Boundedness and decay property in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations, 261 (2016), 967-999. doi: 10.1016/j.jde.2016.03.030.
[15]	A. Lorz, Coupled chemotaxis fluid model, Math. Mod. Meth. Appl. Sci., 20 (2010), 987-1004. doi: 10.1142/S0218202510004507.
[16]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
[17]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[18]	H. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.
[19]	T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367. doi: 10.4310/MAA.2001.v8.n2.a9.
[20]	C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X.
[21]	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.
[22]	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), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.
[23]	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.
[24]	Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis-Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254. doi: 10.3934/dcdsb.2015.20.3235.
[25]	Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving atensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609. doi: 10.1016/j.jde.2015.08.027.
[26]	M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865.
[27]	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.
[28]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[29]	J. Zheng, Boundedness in a three-dimensional chemotaxis-fluid system involving tensorvalued sensitivity with saturation, J. Math. Anal. Appl., 442 (2016), 353-375. doi: 10.1016/j.jmaa.2016.04.047.