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

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July 2018, 38(7): 3547-3566. doi: 10.3934/dcds.2018150

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

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

* Corresponding author: jinchhua@126.com

Received August 2017 Revised January 2018 Published April 2018

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

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: Chemotaxis-Stokes system, global classical solution, stability.

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

Citation: Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150

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.
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[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.

show all references

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.

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