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Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity
1. | School of Mathematics, South China University of Technology, Guangzhou 510640, China |
2. | College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, China |
$ \begin{cases} u_t = \nabla\cdot(\gamma(v)\nabla u-\chi(v)u\nabla v)+\alpha u F(w) +\theta u-\beta u^2, &x\in \Omega, \; \; t>0,\\ v_t = D\Delta v+u-v,& x\in \Omega, \; \; t>0,\\ w_t = \Delta w-uF(w),& x\in \Omega, \; \; t>0,\\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0,&x\in \partial\Omega, \; \; t>0,\\ u(x,0) = u_0(x), v(x,0) = v_0(x),w(x,0) = w_0(x), & x\in\Omega, \end{cases} \;\;(*)$ |
$ \Omega\subset \mathbb{R}^2 $ |
$ \alpha,\beta, D $ |
$ \theta\in \mathbb{R} $ |
$ \nu $ |
$ \partial \Omega $ |
$ \chi(v),\gamma(v) $ |
$ F(v) $ |
$ (\gamma(v),\chi(v))\in [C^2[0,\infty)]^2 $ |
$ \gamma(v)>0,\gamma'(v)<0 $ |
$ \frac{|\chi(v)|+|\gamma'(v)|}{\gamma(v)} $ |
$ F(w)\in C^1([0,\infty)), F(0) = 0,F(w)>0 \ \mathrm{in}\; (0,\infty)\; \mathrm{and}\; F'(w)>0 \ \mathrm{on}\ \ [0,\infty). $ |
$ (u,v,w) $ |
$ (0,0,w_*) $ |
$ L^\infty $ |
$ w_*\geq0 $ |
$ \theta\leq 0 $ |
$ \theta>0 $ |
$ (u,v,w) $ |
$ (\frac{\theta}{\beta},\frac{\theta}{\beta},0) $ |
$ L^\infty $ |
$ D>\max\limits_{0\leq v\leq \infty}\frac{\theta|\chi(v)|^2}{16\beta^2\gamma(v)} $ |
References:
[1] |
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. |
[2] |
N. D. Alikakos, $L^{p}$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations, 4 (1979), 827–868.
doi: 10.1080/03605307908820113. |
[3] |
H. Amann,
Dynamic theory of quasilinear parabolic equations, Ⅱ. Reaction-diffusion systems, Differ. Integral Equ., 3 (1990), 13-75.
|
[4] |
H. Amann,
Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte zur Math., Stuttgart-Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[5] |
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. |
[6] |
N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler,
Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[7] |
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. |
[8] |
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 |
[9] |
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. |
[10] |
T Hillen, K. Painter and M. Winkler,
Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Method Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[11] |
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. |
[12] |
H. Y. Jin and Z. A. Wang,
Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.
doi: 10.1016/j.jde.2016.10.010. |
[13] |
H. Y. Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., doi: 10.1017/S0956792520000248,2020.
doi: 10.1017/S0956792520000248. |
[14] |
H. Y. Jin, S. Shi and Z. A. Wang,
Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility, J. Differential Equations, 269 (2020), 6758-6793.
doi: 10.1016/j.jde.2020.05.018. |
[15] |
H. Y. Jin and Z. A. Wang,
Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Amer. Math. Soc., 148 (2020), 4855-4873.
doi: 10.1090/proc/15124. |
[16] |
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. |
[17] |
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. |
[18] |
Y. Lou and M. Winkler,
Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 1905-1941.
doi: 10.1080/03605302.2015.1052882. |
[19] |
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. |
[20] |
C. Liu, Sequtential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241. Google Scholar |
[21] |
M. Ma, R. Peng and Z. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Physica D, 402 (2020), 132259, 13pp.
doi: 10.1016/j.physd.2019.132259. |
[22] |
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. |
[23] |
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. |
[24] |
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. |
[25] |
J. Smith-Roberge, D. Iron and T. Kolokolnikov,
Pattern formation in bacterial colonies with density-dependent diffusion, Eur. J. Appl. Math., 30 (2019), 196-218.
doi: 10.1017/S0956792518000013. |
[26] |
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. |
[27] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[28] |
Y. S. 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. |
[29] |
Y. S. 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. |
[30] |
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. |
[31] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[32] |
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, 14pp.
doi: 10.1063/1.5061738. |
[33] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp.
doi: 10.1063/1.2766864. |
[34] |
S. Wang, J. Wang and J. Shi,
Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.
doi: 10.1142/S0218202518400158. |
[35] |
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. |
[36] |
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. |
[37] |
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. |
[38] |
S. Wu, J. Shi and B. Wu,
Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations, 260 (2016), 5847-5874.
doi: 10.1016/j.jde.2015.12.024. |
[39] |
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. |
[40] |
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] |
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. |
[2] |
N. D. Alikakos, $L^{p}$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations, 4 (1979), 827–868.
doi: 10.1080/03605307908820113. |
[3] |
H. Amann,
Dynamic theory of quasilinear parabolic equations, Ⅱ. Reaction-diffusion systems, Differ. Integral Equ., 3 (1990), 13-75.
|
[4] |
H. Amann,
Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte zur Math., Stuttgart-Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[5] |
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. |
[6] |
N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler,
Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[7] |
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. |
[8] |
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 |
[9] |
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. |
[10] |
T Hillen, K. Painter and M. Winkler,
Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Method Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[11] |
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. |
[12] |
H. Y. Jin and Z. A. Wang,
Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.
doi: 10.1016/j.jde.2016.10.010. |
[13] |
H. Y. Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., doi: 10.1017/S0956792520000248,2020.
doi: 10.1017/S0956792520000248. |
[14] |
H. Y. Jin, S. Shi and Z. A. Wang,
Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility, J. Differential Equations, 269 (2020), 6758-6793.
doi: 10.1016/j.jde.2020.05.018. |
[15] |
H. Y. Jin and Z. A. Wang,
Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Amer. Math. Soc., 148 (2020), 4855-4873.
doi: 10.1090/proc/15124. |
[16] |
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. |
[17] |
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. |
[18] |
Y. Lou and M. Winkler,
Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Partial Differential Equations, 40 (2015), 1905-1941.
doi: 10.1080/03605302.2015.1052882. |
[19] |
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. |
[20] |
C. Liu, Sequtential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241. Google Scholar |
[21] |
M. Ma, R. Peng and Z. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Physica D, 402 (2020), 132259, 13pp.
doi: 10.1016/j.physd.2019.132259. |
[22] |
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. |
[23] |
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. |
[24] |
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. |
[25] |
J. Smith-Roberge, D. Iron and T. Kolokolnikov,
Pattern formation in bacterial colonies with density-dependent diffusion, Eur. J. Appl. Math., 30 (2019), 196-218.
doi: 10.1017/S0956792518000013. |
[26] |
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. |
[27] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[28] |
Y. S. 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. |
[29] |
Y. S. 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. |
[30] |
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. |
[31] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[32] |
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, 14pp.
doi: 10.1063/1.5061738. |
[33] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp.
doi: 10.1063/1.2766864. |
[34] |
S. Wang, J. Wang and J. Shi,
Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.
doi: 10.1142/S0218202518400158. |
[35] |
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. |
[36] |
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. |
[37] |
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. |
[38] |
S. Wu, J. Shi and B. Wu,
Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations, 260 (2016), 5847-5874.
doi: 10.1016/j.jde.2015.12.024. |
[39] |
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. |
[40] |
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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