May  2022, 42(5): 2499-2523. doi: 10.3934/dcds.2021200

Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China

* Corresponding author

Received  January 2021 Revised  November 2021 Published  May 2022 Early access  January 2022

Fund Project: QW acknowledges the support from Sichuan Science and Technology Program (2020YJ0060). Both authors thank the referees for their careful reading of the manuscript, the nice comments about its content, and their suggestions which greatly improve the presentation

One of the most impressive findings in chemotaxis is the aggregation that randomly distributed bacteria, when starved, release a diffusive chemical to attract and group with others to form one or several stable aggregates in a long time. This paper considers pattern formation within the minimal Keller–Segel chemotaxis model with a focus on the stability and dynamics of its multi-spike steady states. We first show that any steady-state must be a periodic replication of the spatially monotone one and they present multi-spikes when the chemotaxis rate is large; moreover, we prove that all the multi-spikes are unstable through their refined asymptotic profiles, and then find a fully-fledged hierarchy of free entropy energy of these aggregates. Our results also complement the literature by finding that when the chemotaxis is strong, the single boundary spike has the least energy hence is the most stable, the steady-state with more spikes has larger free energy, while the constant has the largest free energy and is always unstable. These results provide new insights into the model's intricate global dynamics, and they are illustrated and complemented by numerical studies which also demonstrate the metastability and phase transition behavior in chemotactic movement.

Citation: Fanze Kong, Qi Wang. Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2499-2523. doi: 10.3934/dcds.2021200
References:
[1]

A. Buttenschön and T. Hillen, Non-local cell adhesion models: Steady states and bifurcations, preprint, https://arXiv.org/abs/2001.00286.

[2]

J. A. CarrilloX. ChenQ. WangZ. Wang and L. Zhang, Phase transitions and bump solutions of the Keller–Segel model with volume exclusion, SIAM J. Appl. Math., 80 (2020), 232-261.  doi: 10.1137/19M125827X.

[3]

J. A. CarrilloJ. Li and Z. Wang, Boundary spike-layer solutions of the singular Keller–Segel system: Existence and stability, Proc. London Math. Soc., 122 (2021), 42-68.  doi: 10.1112/plms.12319.

[4]

X. ChenJ. HaoX. WangY. Wu and Y. Zhang, Stability of spiky solution of Keller–Segel's minimal chemotaxis model, J. Differential Equations, 257 (2014), 3102-3134.  doi: 10.1016/j.jde.2014.06.008.

[5]

L. ChenF. Kong and Q. Wang, Stationary ring and concentric-ring solutions of the Keller–Segel model with quadratic diffusion, SIAM J. Math. Anal., 52 (2020), 4565-4615.  doi: 10.1137/19M1298998.

[6]

L. ChenF. Kong and Q. Wang, Global and exponential attractor of the repulsive Keller–Segel model with logarithmic sensitivity, European J. Appl. Math., 32 (2021), 599-617.  doi: 10.1017/S0956792520000194.

[7]

A. ChertockA. KurganovX. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.  doi: 10.3934/krm.2012.5.51.

[8]

S. Childress and J. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.

[9]

T. CieślakP. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117.  doi: 10.4064/bc81-0-7.

[10]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[11]

M. del PinoF. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents, J. Eur. Math. Soc., 16 (2014), 1687-1748.  doi: 10.4171/JEMS/473.

[12]

M. del Pino and J. Wei, Collapsing steady states of the Keller–Segel system, Nonlinearity, 19 (2006), 661-684.  doi: 10.1088/0951-7715/19/3/007.

[13]

E. FeireislP. Laurençot and H. Petzeltová, On convergence to equilibria for the Keller–Segel chemotaxis model, J. Differential Equations, 236 (2007), 551-569.  doi: 10.1016/j.jde.2007.02.002.

[14]

Y. GuQ. Wang and G. Yi, Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect, European J. Appl. Math., 28 (2017), 141-178.  doi: 10.1017/S0956792516000206.

[15]

C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.  doi: 10.1215/S0012-7094-96-08423-9.

[16]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3.

[17]

M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller–Segel model, J. Math. Biol., 35 (1996), 177-194.  doi: 10.1007/s002850050049.

[18]

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.

[19]

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.

[20]

J. Jiang, On a repulsion Keller–Segel system with a logarithmic sensitivity, European J. Appl. Math., (2021), 1-29.  doi: 10.1017/S0956792520000443.

[21]

K. KangT. Kolokolnikov and M. Ward, The stability and dynamics of a spike in 1D Keller–Segel model, IMA J. Appl. Math., 72 (2007), 140-162.  doi: 10.1093/imamat/hxl028.

[22]

G. Karch and K. Suzuki, Spikes and diffusion waves in a one-dimensional model of chemotaxis, Nonlinearity, 23 (2010), 3119-3137.  doi: 10.1088/0951-7715/23/12/007.

[23]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation view as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[24]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[25]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[26]

T. KolokolnikovJ. Wei and A. Alcolado, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, SIAM J. Appl. Math., 74 (2014), 1375-1396.  doi: 10.1137/130914851.

[27]

K. Kurata and K. Morimoto, Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation, Discrete Contin. Dyn. Syst., 31 (2011), 139-164.  doi: 10.3934/dcds.2011.31.139.

[28]

X. LaiX. ChenC. Qin and Y. Zhang, Existence, uniqueness and stability of steady state solution of chemotaxis model, Discrete Contin. Dyn. Syst., 36 (2016), 805-832.  doi: 10.3934/dcds.2016.36.805.

[29]

H. Li, Spiky steady states of a chemotaxis system with singular sensitivity, J. Dyn. Differential Equations, 30 (2018), 1775-1795.  doi: 10.1007/s10884-017-9621-3.

[30]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitute stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[31]

F.-H. LinW.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.

[32]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42 (1973), 63-105.  doi: 10.1016/0022-5193(73)90149-5.

[33]

W.-M. Ni and I. Takagi, On the shape of least enery solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.

[34]

W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.

[35]

J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$ Fredholm mappings of index 0, J. Anal. Math., 76 (1998), 289-319.  doi: 10.1007/BF02786939.

[36]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[37]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.

[38]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.

[39]

B. SleemanM. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.  doi: 10.1137/S0036139902415117.

[40]

Q. Wang, Boundary spikes of a Keller–Segel chemotaxis system with saturated logarithmic sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1231-1250.  doi: 10.3934/dcdsb.2015.20.1231.

[41]

Q. WangC. Gai and J. Yan, Qualitative analysis of a Lotka–Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.  doi: 10.3934/dcds.2015.35.1239.

[42]

Q. WangY. Song and L. Shao, Nonconstant positive steady states and pattern formation of 1D prey-taxis systems, J. Nonlinear Sci., 27 (2017), 71-97.  doi: 10.1007/s00332-016-9326-5.

[43]

Q. Wang, J. Yan and C. Gai, Qualitative analysis of stationary Keller–Segel chemotaxis models with logistic growth, Z. Angew. Math. Phys., 67 (2016), Art. 51, 25 pp. doi: 10.1007/s00033-016-0648-9.

[44]

Q. WangJ. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller–Segel chemotaxis model: Effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3547-3574.  doi: 10.3934/dcdsb.2017179.

[45]

Q. WangL. ZhangJ. Yang and J. Hu, Global existence and steady states of a two competing species Keller–Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807.  doi: 10.3934/krm.2015.8.777.

[46]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560.  doi: 10.1137/S0036141098339897.

[47]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.  doi: 10.1007/s00285-012-0533-x.

[48]

J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133.  doi: 10.1006/jdeq.1996.3218.

[49]

Y. ZhangX. ChenJ. HaoX. Lai and C. Qin, An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model, J. Math. Anal. Appl., 420 (2014), 684-704.  doi: 10.1016/j.jmaa.2014.06.005.

[50]

Y. ZhangX. ChenJ. HaoX. Lai and C. Qin, Dynamics of spike in a Keller–Segel's minimal chemotaxis model, Discrete Contin. Dyn. Syst., 37 (2017), 1109-1127.  doi: 10.3934/dcds.2017046.

[51]

Y. ZhangX. ChenJ. HaoX. Lai and C. Qin, Spectral analysis for stability of bubble steady states of a Keller–Segel's minimal chemotaxis model, J. Math. Anal. Appl., 446 (2017), 1105-1132.  doi: 10.1016/j.jmaa.2016.09.034.

show all references

References:
[1]

A. Buttenschön and T. Hillen, Non-local cell adhesion models: Steady states and bifurcations, preprint, https://arXiv.org/abs/2001.00286.

[2]

J. A. CarrilloX. ChenQ. WangZ. Wang and L. Zhang, Phase transitions and bump solutions of the Keller–Segel model with volume exclusion, SIAM J. Appl. Math., 80 (2020), 232-261.  doi: 10.1137/19M125827X.

[3]

J. A. CarrilloJ. Li and Z. Wang, Boundary spike-layer solutions of the singular Keller–Segel system: Existence and stability, Proc. London Math. Soc., 122 (2021), 42-68.  doi: 10.1112/plms.12319.

[4]

X. ChenJ. HaoX. WangY. Wu and Y. Zhang, Stability of spiky solution of Keller–Segel's minimal chemotaxis model, J. Differential Equations, 257 (2014), 3102-3134.  doi: 10.1016/j.jde.2014.06.008.

[5]

L. ChenF. Kong and Q. Wang, Stationary ring and concentric-ring solutions of the Keller–Segel model with quadratic diffusion, SIAM J. Math. Anal., 52 (2020), 4565-4615.  doi: 10.1137/19M1298998.

[6]

L. ChenF. Kong and Q. Wang, Global and exponential attractor of the repulsive Keller–Segel model with logarithmic sensitivity, European J. Appl. Math., 32 (2021), 599-617.  doi: 10.1017/S0956792520000194.

[7]

A. ChertockA. KurganovX. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.  doi: 10.3934/krm.2012.5.51.

[8]

S. Childress and J. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.

[9]

T. CieślakP. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117.  doi: 10.4064/bc81-0-7.

[10]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[11]

M. del PinoF. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents, J. Eur. Math. Soc., 16 (2014), 1687-1748.  doi: 10.4171/JEMS/473.

[12]

M. del Pino and J. Wei, Collapsing steady states of the Keller–Segel system, Nonlinearity, 19 (2006), 661-684.  doi: 10.1088/0951-7715/19/3/007.

[13]

E. FeireislP. Laurençot and H. Petzeltová, On convergence to equilibria for the Keller–Segel chemotaxis model, J. Differential Equations, 236 (2007), 551-569.  doi: 10.1016/j.jde.2007.02.002.

[14]

Y. GuQ. Wang and G. Yi, Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect, European J. Appl. Math., 28 (2017), 141-178.  doi: 10.1017/S0956792516000206.

[15]

C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.  doi: 10.1215/S0012-7094-96-08423-9.

[16]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3.

[17]

M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller–Segel model, J. Math. Biol., 35 (1996), 177-194.  doi: 10.1007/s002850050049.

[18]

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.

[19]

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.

[20]

J. Jiang, On a repulsion Keller–Segel system with a logarithmic sensitivity, European J. Appl. Math., (2021), 1-29.  doi: 10.1017/S0956792520000443.

[21]

K. KangT. Kolokolnikov and M. Ward, The stability and dynamics of a spike in 1D Keller–Segel model, IMA J. Appl. Math., 72 (2007), 140-162.  doi: 10.1093/imamat/hxl028.

[22]

G. Karch and K. Suzuki, Spikes and diffusion waves in a one-dimensional model of chemotaxis, Nonlinearity, 23 (2010), 3119-3137.  doi: 10.1088/0951-7715/23/12/007.

[23]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation view as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[24]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[25]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[26]

T. KolokolnikovJ. Wei and A. Alcolado, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, SIAM J. Appl. Math., 74 (2014), 1375-1396.  doi: 10.1137/130914851.

[27]

K. Kurata and K. Morimoto, Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation, Discrete Contin. Dyn. Syst., 31 (2011), 139-164.  doi: 10.3934/dcds.2011.31.139.

[28]

X. LaiX. ChenC. Qin and Y. Zhang, Existence, uniqueness and stability of steady state solution of chemotaxis model, Discrete Contin. Dyn. Syst., 36 (2016), 805-832.  doi: 10.3934/dcds.2016.36.805.

[29]

H. Li, Spiky steady states of a chemotaxis system with singular sensitivity, J. Dyn. Differential Equations, 30 (2018), 1775-1795.  doi: 10.1007/s10884-017-9621-3.

[30]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitute stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[31]

F.-H. LinW.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.

[32]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42 (1973), 63-105.  doi: 10.1016/0022-5193(73)90149-5.

[33]

W.-M. Ni and I. Takagi, On the shape of least enery solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.

[34]

W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.

[35]

J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$ Fredholm mappings of index 0, J. Anal. Math., 76 (1998), 289-319.  doi: 10.1007/BF02786939.

[36]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[37]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.

[38]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.

[39]

B. SleemanM. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.  doi: 10.1137/S0036139902415117.

[40]

Q. Wang, Boundary spikes of a Keller–Segel chemotaxis system with saturated logarithmic sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1231-1250.  doi: 10.3934/dcdsb.2015.20.1231.

[41]

Q. WangC. Gai and J. Yan, Qualitative analysis of a Lotka–Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.  doi: 10.3934/dcds.2015.35.1239.

[42]

Q. WangY. Song and L. Shao, Nonconstant positive steady states and pattern formation of 1D prey-taxis systems, J. Nonlinear Sci., 27 (2017), 71-97.  doi: 10.1007/s00332-016-9326-5.

[43]

Q. Wang, J. Yan and C. Gai, Qualitative analysis of stationary Keller–Segel chemotaxis models with logistic growth, Z. Angew. Math. Phys., 67 (2016), Art. 51, 25 pp. doi: 10.1007/s00033-016-0648-9.

[44]

Q. WangJ. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species Keller–Segel chemotaxis model: Effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3547-3574.  doi: 10.3934/dcdsb.2017179.

[45]

Q. WangL. ZhangJ. Yang and J. Hu, Global existence and steady states of a two competing species Keller–Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807.  doi: 10.3934/krm.2015.8.777.

[46]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560.  doi: 10.1137/S0036141098339897.

[47]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.  doi: 10.1007/s00285-012-0533-x.

[48]

J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133.  doi: 10.1006/jdeq.1996.3218.

[49]

Y. ZhangX. ChenJ. HaoX. Lai and C. Qin, An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model, J. Math. Anal. Appl., 420 (2014), 684-704.  doi: 10.1016/j.jmaa.2014.06.005.

[50]

Y. ZhangX. ChenJ. HaoX. Lai and C. Qin, Dynamics of spike in a Keller–Segel's minimal chemotaxis model, Discrete Contin. Dyn. Syst., 37 (2017), 1109-1127.  doi: 10.3934/dcds.2017046.

[51]

Y. ZhangX. ChenJ. HaoX. Lai and C. Qin, Spectral analysis for stability of bubble steady states of a Keller–Segel's minimal chemotaxis model, J. Math. Anal. Appl., 446 (2017), 1105-1132.  doi: 10.1016/j.jmaa.2016.09.034.

Figure 1.  Asymptotic profiles of the single interior spike (top left) and double-boundary spike (top right) with the coupled steady chemical concentrations (in the bottom) given by (1.5) and (1.6) with $ M = 2 $ and $ \chi = 20 $. We shall prove that both the single interior spike and the double-boundary spike are unstable
Figure 2.  Left: Illustration of the solution loop through a phase plane; Right: a solution with four similar profiles on the right. The maximum of $ v(x) $ are achieved at $ A $ and $ C $, and then minimum at $ B $ and $ D $. Each closed loop on the phase plane leads to a solution with a complete period, and it can be extended repeatedly. However, each loop intersects with the $ v $-axis only twice and this suggests that the solution $ v(x) $ must be similar-profiled
Figure 3.  Leading profiles of the eigenpair $ (\phi^d,\psi^d) $ of (2.2) over $ (0,\pi) $ with $ \chi = 5,10,20 $ and 100. Each function is normalized such that $ \Vert \phi^d\Vert_{L^2} = \Vert \psi^d\Vert_{L^2} = 1 $. These eigenfunctions are odd about the middle point $ x = \frac{L}{2} $ and they drive the insability of the double-boundary spike $ (u^d,v^d) $ in (1.5)
Figure 4.  Leading profiles of the eigenpair $ (\phi^i,\psi^i) $ of (2.16) over $ (0,\pi) $ with $ \chi = 5,10,20 $ and 100. Similar as in Figure 3, we normalize each eigenfunction such that $ \Vert \phi^i\Vert_{L^2} = \Vert \psi^i\Vert_{L^2} = 1 $. These eigenfunctions are odd about the middle point $ x = \frac{L}{2} $ and they drive the instability of the single interior spike $ (u^i,v^i) $ in (1.6)
Figure 5.  Staircased hierarchy of free energies of the constant solution (3.3) and the multi-spikes (3.5). Theorem 1.3 finds that when the chemotaxis rate is large, the single boundary spike is the least energy solution among all the steady states, whereas the constant solution has the largest energy; moreover, the energy increases as the number of spikes. In light of the energy dissipation (1.10), the hierarchy indicates that the single boundary spike is the most stable and the constant solution is most unstable when the chemotaxis rate is large. Accordingly, the spatial-temporal dynamics of (1.1) are dominated by the single boundary spike but for symmetric initial data. These findings will be demonstrated and verified in our coming simulations
Figure 6.  This figure demonstrates the instability of the double-boundary spike $ (u^d,v^d) $. Here we choose $ \kappa = 20 $ and the initial data $ (u_0,v_0) = (u^d,v^d) + (0.2e^{-2x^2},0) $ to be small perturbations from the double-boundary spike, with the initial cell density $ u_0 $ slightly tilted to the left end. Top: Very quickly the cellular density develops the double-boundary spike tilted to the left. This asymmetric profile preserves for an extremely long time to $ t\approx 280 $, when a transition occurs and the small spike on the right disappears within a very short period. Similar dynamics are observed in the chemical concentration $ v(x,t) $ on the right. Bottom: The contour plot of cellular density on the left and the decay of free energy (1.9) along the solution trajectory on the right. In particular, one observes that the energy decays extremely slowly for a very long time and then a phase transition occurs at time $ t\approx 280 $ as described earlier
Figure 7.  This figure demonstrates the instability of the interior spike $ (u^i,v^i) $. Here we choose $ \kappa = 20 $ and the initial data $ (u_0,v_0) = (u^i,v^i) + (0.2e^{-2x^2},0) $ to be small perturbations from the single interior spike, with the initial cell density $ u_0 $ slightly tilted to the left end. Top: Initially the cell density spike decays with a linear rate, and it stabilizes to a single interior spike at time $ t\approx 1 $. Similar to the double-boundary spike in Figure 6, this single interior spike is asymmetric and slightly tilted to the left. Very slowly it shifts to the left end until time $ t\approx 11 $ when a transition occurs quickly and this interior spike eventually stabilizes at the left endpoint. Snapshots capture similar dynamics observed in the chemical concentration $ v(x,t) $ on the top right. Bottom: The energy evolution captures the meta-stability of the interior spike and the occurrence of a phase transition at time $ t\approx11 $. Needless to say, the convergence to the single boundary spike is much faster than that of the double-boundary spike in Figure 6
Figure 8.  Exponential stability of the double-boundary spike $ u^d $ and the interior spike $ u^i $ in the symmetric class. Here we choose the initial data to be symmetric perturbations of these spikes in both simulations. In strong contrast to the dynamics in Figure 6 and Figure 7 that take exponentially long to converge to an equilibrium, we observe exponential decay to the steady states $ u^d $ and $ u^i $ in the plots, at least for $ t\in(0.5,4.5) $, and the exponential convergence rates are $ \lambda^d\approx -4.8786 $ and $ \lambda^i = \approx -4.8771 $ for the double-boundary spike and the single interior spike
Figure 9.  Left: Dynamics and free energy evolution out of $ (\mathbb U^i_4,\mathbb V^i_4) $ with an odd perturbation tilted to the right end. The two interior spikes sense the chemical gradient, attract and move towards each other, and merge into a single interior spike at time $ t\approx 12 $. This interior spike is tilted and shifts to the right end very slowly until it stabilizes on the boundary at time $ t\approx 45 $ through a phase transition. Two phase transitions are observed in the energy evolution at the bottom. Right: Dynamics and free energy evolution out of $ (\mathbb U^i_4,\mathbb V^i_4) $ with an even perturbation. They attract and merge each other very quickly, stay symmetric, and finally stabilize into the single interior spike
Figure 10.  Instability of multi-spikes given in (1.7) and (1.8) with $ n = 5,6 $ and 7 presented on the left, the middle, and the right. Top. Local dynamics of (1.1) out of these multi-spikes are presented here and they demonstrate the dissipation and merging of these multi-spikes which occur in a relatively short period. Middle. Global dynamics out of these multi-spikes readily demonstrate their instability, as well as the stability of the single and double-boundary spike. Bottom. Energy evolution here showcases several phase transitions of (1.1). They suggest the rich spatial-temporal dynamics of this relatively simple system
Figure 11.  Spatial-temporal dynamics of (1.1) out of initial data randomly perturbed from $ (\bar u,\bar v) = (\frac{2}{\pi},\frac{2}{\pi}) $ with $ \chi = 20 $. In each column, we report the local, global dynamics of (1.1), and the evolution of the free energy. These results suggest that the single boundary spike the global attractor in these settings. The center of initial cellular density from the left to the right is $ \bar x_1\approx 1.5630 $. $ \bar x_2\approx 1.5531 $, $ \bar x_3\approx 1.5794 $, respectively. These simulations, together with others we have conducted but not reported here, tend to suggest the asymmetry of the initial data is preserved: initial data tilted to the left will converge to the decreasing single boundary, and initial data tilted to the right will converge to the increasing single boundary
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