The Keller-Segel system with logistic growth and signal-dependent motility

Hai-Yang Jin; Zhi-An Wang

doi:10.3934/dcdsb.2020218

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June 2021, 26(6): 3023-3041. doi: 10.3934/dcdsb.2020218

The Keller-Segel system with logistic growth and signal-dependent motility

Hai-Yang Jin ^1, and Zhi-An Wang ^2,,

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

^*Corresponding author: Zhi-An Wang

Received February 2020 Revised May 2020 Published June 2021 Early access July 2020

Fund Project: The research of H.Y. Jin was supported by the NSF of China (No. 11871226), Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515010140), Guangzhou Science and Technology Program No.202002030363 and the Fundamental Research Funds for the Central Universities. The research of Z.A. Wang was supported by the Hong Kong RGC GRF grant 15303019 (Project ID P0030816)

The paper is concerned with the following chemotaxis system with nonlinear motility functions

$\begin{equation}\label{0-1}\begin{cases}u_t = \nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, \\ 0 = \Delta v+ u-v, & x\in \Omega, ~~t>0, \\u(x, 0) = u_0(x), & x\in \Omega, \end{cases}~~~~(\ast)\end{equation}$

subject to homogeneous Neumann boundary conditions in a bounded domain

$ \Omega\subset \mathbb{R}^2 $

with smooth boundary, where the motility functions

$ \gamma(v) $

and

$ \chi(v) $

satisfy the following conditions

●

$ (\gamma, \chi)\in [C^2[0, \infty)]^2 $

with

$ \gamma(v)>0 $

and

$ \frac{|\chi(v)|^2}{\gamma(v)} $

is bounded for all

$ v\geq 0 $

By employing the method of energy estimates, we establish the existence of globally bounded solutions of ($\ast$) with

$ \mu>0 $

for any

$ u_0 \in W^{1, \infty}(\Omega) $

with

$ u_0 \geq (\not\equiv) 0 $

. Then based on a Lyapunov function, we show that all solutions

$ (u, v) $

of ($\ast$) will exponentially converge to the unique constant steady state

$ (1, 1) $

provided

$ \mu>\frac{K_0}{16} $

with

$ K_0 = \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)} $

Keywords: Chemotaxis, density-dependent motility, global boundedness, exponential decay.

Mathematics Subject Classification: Primary: 35A01, 35B40, 35K57, 35Q92; Secondary: 92C17.

Citation: Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934/dcdsb.2020218

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	C. Liu, Sequtential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241. doi: 10.1126/science.1209042. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	C. Liu, Sequtential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238-241. doi: 10.1126/science.1209042. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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The Keller-Segel system with logistic growth and signal-dependent motility

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