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The Keller-Segel system with logistic growth and signal-dependent motility
1. | Department of Mathematics, South China University of Technology, Guangzhou, 510640, China |
2. | Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong |
$\begin{equation}\label{0-1}\begin{cases}u_t = \nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, \\ 0 = \Delta v+ u-v, & x\in \Omega, ~~t>0, \\u(x, 0) = u_0(x), & x\in \Omega, \end{cases}~~~~(\ast)\end{equation}$ |
$ \Omega\subset \mathbb{R}^2 $ |
$ \gamma(v) $ |
$ \chi(v) $ |
$ (\gamma, \chi)\in [C^2[0, \infty)]^2 $ |
$ \gamma(v)>0 $ |
$ \frac{|\chi(v)|^2}{\gamma(v)} $ |
$ v\geq 0 $ |
$ \mu>0 $ |
$ u_0 \in W^{1, \infty}(\Omega) $ |
$ u_0 \geq (\not\equiv) 0 $ |
$ (u, v) $ |
$ (1, 1) $ |
$ \mu>\frac{K_0}{16} $ |
$ K_0 = \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)} $ |
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Commun. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[2] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Commun. Pure Appl. Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[3] |
J. Ahn and C. Yoon,
Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing, Nonlinearity, 32 (2019), 1327-1351.
doi: 10.1088/1361-6544/aaf513. |
[4] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[5] |
X. Fu, L.-H. Tang, C. Liu, J.-D. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.
doi: 10.1103/PhysRevLett.108.198102. |
[6] |
K. Fujie and J. Jiang, Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities, arXiv: 2001.01288. Google Scholar |
[7] |
K. Fujie and J. Jiang,
Global existence for a kinetic model of pattern formation with density-suppressed motilities, J. Differential Equations, 269 (2020), 5338-5378.
doi: 10.1016/j.jde.2020.04.001. |
[8] |
K. Fujie and T. Senba,
Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.
doi: 10.1088/0951-7715/29/8/2417. |
[9] |
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. |
[10] |
H.-Y. Jin and Z.-A. Wang, Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Amer. Math. Soc., DOI: https://doi.org/10.1090/proc/15124, 2020. Google Scholar |
[11] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[12] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa,
Spatial pattern formation in a chemotaxis-diffusion-growth model, Phys. D, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[13] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[14] |
K. Lin and C. Mu,
Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.
doi: 10.3934/dcds.2016018. |
[15] |
C. Liu,
Sequtential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241.
doi: 10.1126/science.1209042. |
[16] |
M. Ma, C. Ou and Z.-A. Wang,
Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math., 72 (2012), 740-766.
doi: 10.1137/110843964. |
[17] |
M. Ma, R. Peng and Z. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Phys. D, 402 (2020), 132259.
doi: 10.1016/j.physd.2019.132259. |
[18] |
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. |
[19] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[20] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. TMA, 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[21] |
K. J. Painter and T. Hillen,
Spatio-temporal chaos in a chemotaxis model, Phys. D, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[22] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[23] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 2007. |
[24] |
J. Smith-Roberge, D. Iron and T. Kolokolnikov,
Pattern formation in bacterial colonies with density-dependent diffusion, European J. Appl. Math., 30 (2019), 196-218.
doi: 10.1017/S0956792518000013. |
[25] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[26] |
Y. Tao and Z.-A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[27] |
Y. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[28] |
Y. Tao and M. Winkler,
Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.
doi: 10.1142/S0218202517500282. |
[29] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[30] |
J. Wang and M. Wang, Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth, J. Math. Phys., 60 (2019), 011507.
doi: 10.1063/1.5061738. |
[31] |
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. |
[32] |
M. Winkler,
Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.
doi: 10.1002/mma.1346. |
[33] |
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. |
[34] |
P. Xia, Y. Han, J. Tao and M. Ma, Existence and metastability of non-constant steady states in a Keller-Segel model with density-suppressed motility, Mathematics in Applied Sciences and Engineering, 1 (2020), 1-15. Google Scholar |
[35] |
T. Xiang,
Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.
doi: 10.1016/j.jde.2015.01.032. |
[36] |
C. Yoon and Y.-J. Kim,
Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.
doi: 10.1007/s10440-016-0089-7. |
show all references
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Commun. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[2] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Commun. Pure Appl. Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[3] |
J. Ahn and C. Yoon,
Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing, Nonlinearity, 32 (2019), 1327-1351.
doi: 10.1088/1361-6544/aaf513. |
[4] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[5] |
X. Fu, L.-H. Tang, C. Liu, J.-D. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.
doi: 10.1103/PhysRevLett.108.198102. |
[6] |
K. Fujie and J. Jiang, Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities, arXiv: 2001.01288. Google Scholar |
[7] |
K. Fujie and J. Jiang,
Global existence for a kinetic model of pattern formation with density-suppressed motilities, J. Differential Equations, 269 (2020), 5338-5378.
doi: 10.1016/j.jde.2020.04.001. |
[8] |
K. Fujie and T. Senba,
Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.
doi: 10.1088/0951-7715/29/8/2417. |
[9] |
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. |
[10] |
H.-Y. Jin and Z.-A. Wang, Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Amer. Math. Soc., DOI: https://doi.org/10.1090/proc/15124, 2020. Google Scholar |
[11] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[12] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa,
Spatial pattern formation in a chemotaxis-diffusion-growth model, Phys. D, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[13] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[14] |
K. Lin and C. Mu,
Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.
doi: 10.3934/dcds.2016018. |
[15] |
C. Liu,
Sequtential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241.
doi: 10.1126/science.1209042. |
[16] |
M. Ma, C. Ou and Z.-A. Wang,
Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math., 72 (2012), 740-766.
doi: 10.1137/110843964. |
[17] |
M. Ma, R. Peng and Z. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Phys. D, 402 (2020), 132259.
doi: 10.1016/j.physd.2019.132259. |
[18] |
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. |
[19] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[20] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. TMA, 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[21] |
K. J. Painter and T. Hillen,
Spatio-temporal chaos in a chemotaxis model, Phys. D, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[22] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[23] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 2007. |
[24] |
J. Smith-Roberge, D. Iron and T. Kolokolnikov,
Pattern formation in bacterial colonies with density-dependent diffusion, European J. Appl. Math., 30 (2019), 196-218.
doi: 10.1017/S0956792518000013. |
[25] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[26] |
Y. Tao and Z.-A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[27] |
Y. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[28] |
Y. Tao and M. Winkler,
Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.
doi: 10.1142/S0218202517500282. |
[29] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[30] |
J. Wang and M. Wang, Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth, J. Math. Phys., 60 (2019), 011507.
doi: 10.1063/1.5061738. |
[31] |
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. |
[32] |
M. Winkler,
Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.
doi: 10.1002/mma.1346. |
[33] |
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. |
[34] |
P. Xia, Y. Han, J. Tao and M. Ma, Existence and metastability of non-constant steady states in a Keller-Segel model with density-suppressed motility, Mathematics in Applied Sciences and Engineering, 1 (2020), 1-15. Google Scholar |
[35] |
T. Xiang,
Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.
doi: 10.1016/j.jde.2015.01.032. |
[36] |
C. Yoon and Y.-J. Kim,
Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.
doi: 10.1007/s10440-016-0089-7. |
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