doi: 10.3934/dcdsb.2021011

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

* Corresponding author: yrchen@szu.edu.cn

Received  August 2020 Revised  November 2020 Published  December 2020

Fund Project: Z. Liu was partially supported by the National Natural Science Foundation of China (No. 11971176 and No. 12026608). Y.Chen was partially supported by the National Natural Science Foundation of China (No. 12001377 and No. 11971176) and the Outstanding Innovative Young Talents of Guangdong Province, China (No. 2019KQNCX122)

In this paper, we shall study the initial-boundary value problem of a chemotaxis model with signal-dependent diffusion and sensitivity as follows
$ \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} \;\;(*)$
in a bounded domain
$ \Omega\subset \mathbb{R}^2 $
with smooth boundary, where
$ \alpha,\beta, D $
are positive constants,
$ \theta\in \mathbb{R} $
and
$ \nu $
denotes the outward normal vector of
$ \partial \Omega $
. The functions
$ \chi(v),\gamma(v) $
and
$ F(v) $
satisfy
$ (\gamma(v),\chi(v))\in [C^2[0,\infty)]^2 $
with
$ \gamma(v)>0,\gamma'(v)<0 $
and
$ \frac{|\chi(v)|+|\gamma'(v)|}{\gamma(v)} $
is bounded;
$ 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). $
We first prove that the existence of globally bounded solution of system (*) based on the method of weighted energy estimates. Moreover, by constructing Lyapunov functional, we show that the solution
$ (u,v,w) $
will converge to
$ (0,0,w_*) $
in
$ L^\infty $
with some
$ w_*\geq0 $
as time tends to infinity in the case of
$ \theta\leq 0 $
, while if
$ \theta>0 $
, the solution
$ (u,v,w) $
will asymptotically converge to
$ (\frac{\theta}{\beta},\frac{\theta}{\beta},0) $
in
$ L^\infty $
-norm provided
$ D>\max\limits_{0\leq v\leq \infty}\frac{\theta|\chi(v)|^2}{16\beta^2\gamma(v)} $
.
Citation: Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021011
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.  Google Scholar

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N. BellomoA. BellouquidY. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[10]

T HillenK. 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.  Google Scholar

[11]

H. Y. JinY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Y. JinS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. KutoK. OsakiT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

K. OsakiT. TsujikawaA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. Smith-RobergeD. 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.  Google Scholar

[26]

C. StinnerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

S. WangJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

S. WuJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

H. Amann, Dynamic theory of quasilinear parabolic equations, Ⅱ. Reaction-diffusion systems, Differ. Integral Equ., 3 (1990), 13-75.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

N. BellomoA. BellouquidY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T HillenK. 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.  Google Scholar

[11]

H. Y. JinY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Y. JinS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. KutoK. OsakiT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

K. OsakiT. TsujikawaA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. Smith-RobergeD. 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.  Google Scholar

[26]

C. StinnerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

S. WangJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

S. WuJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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