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Super fast vanishing solutions of the fast diffusion equation
Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China |
This paper is concerned with the time periodic problem to a coupled chemotaxis-fluid model with porous medium diffusion $ \Delta n^m $. The global existence of solutios for the initial and boundary value problem of this model have been studied by many authors, and in particular, the global solvability is established for $ m>\frac65 $ in dimension 3. Here, taking advantage of a double-level approximation scheme, we establish the existence of uniformly bounded time periodic solution for any $ m\ge \frac 65 $ and any large periodic source $ g(x, t) $. In particular, the energy estimates techniques we used also applicable to the proof of global existence of the initial-boundary value problem, and one can supply the existence of global solutions for $ m = \frac65 $ by this method.
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A. Blanchet, J. A. Carrillo and P. Laurençot,
Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.
doi: 10.1007/s00526-008-0200-7. |
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X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), 39pp.
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Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
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T. Cieślak and C. Stinner,
Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.
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T. Cieślak and M. Winkler,
Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal., 159 (2017), 129-144.
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E. Espejo and T. Suzuki,
Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.
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C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 24pp.
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C. Jin,
Large time periodic solution to the coupled chemotaxis-Stokes model, Math. Nachr., 290 (2017), 1701-1715.
doi: 10.1002/mana.201600180. |
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C. Jin, Periodic pattern formation in the coupled chemotaxis-(Navier-)Stokes system with mixed nonhomogeneous boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, (2019).
doi: 10.1017/prm.2019.62. |
[10] |
P. Laurençot and N. Mizoguchi,
Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 197-220.
doi: 10.1016/j.anihpc.2015.11.002. |
[11] |
Y. Li and J. Lankeit,
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.
doi: 10.1088/0951-7715/29/5/1564. |
[12] |
J.-G. Liu and A. Lorz,
A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
doi: 10.1016/j.anihpc.2011.04.005. |
[13] |
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. |
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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.
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[15] |
Y. Peng and Z. Xiang,
Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions, J. Differential Equations, 267 (2019), 1277-1321.
doi: 10.1016/j.jde.2019.02.007. |
[16] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
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[17] |
Y. Tao and M. Winkler,
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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), 23pp.
doi: 10.1007/s00033-016-0732-1. |
[19] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[20] |
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. |
[21] |
M. Winkler,
A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Funct. Anal., 276 (2019), 1339-1401.
doi: 10.1016/j.jfa.2018.12.009. |
[22] |
M. Winkler,
Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.
doi: 10.1007/s00205-013-0678-9. |
[23] |
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. |
[24] |
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. |
[25] |
J. Zheng,
Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 627-643.
doi: 10.3934/dcds.2017026. |
show all references
References:
[1] |
A. Blanchet, J. A. Carrillo and P. Laurençot,
Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.
doi: 10.1007/s00526-008-0200-7. |
[2] |
X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), 39pp.
doi: 10.1007/s00526-016-1027-2. |
[3] |
T. Cieślak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[4] |
T. Cieślak and C. Stinner,
Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.
doi: 10.1007/s10440-013-9832-5. |
[5] |
T. Cieślak and M. Winkler,
Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal., 159 (2017), 129-144.
doi: 10.1016/j.na.2016.04.013. |
[6] |
E. Espejo and T. Suzuki,
Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.
doi: 10.1016/j.nonrwa.2014.07.001. |
[7] |
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. |
[8] |
C. Jin,
Large time periodic solution to the coupled chemotaxis-Stokes model, Math. Nachr., 290 (2017), 1701-1715.
doi: 10.1002/mana.201600180. |
[9] |
C. Jin, Periodic pattern formation in the coupled chemotaxis-(Navier-)Stokes system with mixed nonhomogeneous boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, (2019).
doi: 10.1017/prm.2019.62. |
[10] |
P. Laurençot and N. Mizoguchi,
Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 197-220.
doi: 10.1016/j.anihpc.2015.11.002. |
[11] |
Y. Li and J. Lankeit,
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.
doi: 10.1088/0951-7715/29/5/1564. |
[12] |
J.-G. Liu and A. Lorz,
A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
doi: 10.1016/j.anihpc.2011.04.005. |
[13] |
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. |
[14] |
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. |
[15] |
Y. Peng and Z. Xiang,
Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions, J. Differential Equations, 267 (2019), 1277-1321.
doi: 10.1016/j.jde.2019.02.007. |
[16] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[17] |
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. |
[18] |
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), 23pp.
doi: 10.1007/s00033-016-0732-1. |
[19] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[20] |
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. |
[21] |
M. Winkler,
A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Funct. Anal., 276 (2019), 1339-1401.
doi: 10.1016/j.jfa.2018.12.009. |
[22] |
M. Winkler,
Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.
doi: 10.1007/s00205-013-0678-9. |
[23] |
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. |
[24] |
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. |
[25] |
J. Zheng,
Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 627-643.
doi: 10.3934/dcds.2017026. |
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