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Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces
Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China |
$ \lambda\in\mathbb{R}, \, \mu>0 $ |
$ l>2 $ |
$\left\{ \begin{array}{l}{u_t} = \Delta (\gamma (v)u) + \lambda u - \mu {u^l},\;\;\;\;\;x \in \Omega ,{\mkern 1mu} t > 0,\;\\{v_t} = \Delta v - v + u,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \Omega ,{\mkern 1mu} t > 0,\end{array} \right.$ |
References:
[1] |
H. Amann,
Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte Math., Teubner, Stuttgart, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[2] |
X. R. Cao,
Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.
doi: 10.3934/dcdsb.2017141. |
[3] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann.
Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633–683. |
[4] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[5] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.
|
[6] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[7] |
H.-Y. Jin, Y.-J. Kim and Z.-A. Wang,
Boundedness, stabilization, and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 78 (2018), 1632-1657.
doi: 10.1137/17M1144647. |
[8] |
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. |
[9] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
[10] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[11] |
T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62. Birkhäuser Boston, Inc., Boston, MA, 2005.
doi: 10.1007/0-8176-4436-9. |
[12] |
Y. S. 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. |
[13] |
Y. S. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[14] |
Y. S. Tao and M. Winkler,
Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Methods Appl. Sci., 27 (2017), 1645-1683.
doi: 10.1142/S0218202517500282. |
[15] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[16] |
J. P. Wang and M. X. Wang, Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth, J. Math. Phys., 60 (2019), 011507, 14 pp.
doi: 10.1063/1.5061738. |
[17] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13 pp.
doi: 10.1063/1.2766864. |
[18] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[19] |
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. |
[20] |
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. |
[21] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[22] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[23] |
M. Winkler,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.
doi: 10.3934/dcdsb.2017135. |
[24] |
C. Yoon and Y.-J. Kim,
Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math., 149 (2017), 101-123.
doi: 10.1007/s10440-016-0089-7. |
show all references
References:
[1] |
H. Amann,
Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte Math., Teubner, Stuttgart, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[2] |
X. R. Cao,
Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.
doi: 10.3934/dcdsb.2017141. |
[3] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann.
Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633–683. |
[4] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[5] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.
|
[6] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[7] |
H.-Y. Jin, Y.-J. Kim and Z.-A. Wang,
Boundedness, stabilization, and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 78 (2018), 1632-1657.
doi: 10.1137/17M1144647. |
[8] |
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. |
[9] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
[10] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[11] |
T. Suzuki, Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62. Birkhäuser Boston, Inc., Boston, MA, 2005.
doi: 10.1007/0-8176-4436-9. |
[12] |
Y. S. 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. |
[13] |
Y. S. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[14] |
Y. S. Tao and M. Winkler,
Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Methods Appl. Sci., 27 (2017), 1645-1683.
doi: 10.1142/S0218202517500282. |
[15] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[16] |
J. P. Wang and M. X. Wang, Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth, J. Math. Phys., 60 (2019), 011507, 14 pp.
doi: 10.1063/1.5061738. |
[17] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13 pp.
doi: 10.1063/1.2766864. |
[18] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[19] |
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. |
[20] |
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. |
[21] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[22] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[23] |
M. Winkler,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.
doi: 10.3934/dcdsb.2017135. |
[24] |
C. Yoon and Y.-J. Kim,
Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math., 149 (2017), 101-123.
doi: 10.1007/s10440-016-0089-7. |
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