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Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes
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 |
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.
References:
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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. |
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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