February  2017, 37(2): 1109-1127. doi: 10.3934/dcds.2017046

Dynamics of spike in a Keller-Segel's minimal chemotaxis model

1. 

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

3. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

4. 

Center for Financial Engineering, Soochow University, Suzhou, Jiangsu 215006, China

* Corresponding author: Yajing Zhang

Received  October 2014 Revised  March 2016 Published  November 2016

Fund Project: This work is partially supported by NSF DMS-1008905, NNSFC (No. 61374089), China Scholarship Council, NSF of Shanxi Province(No. 2014011005-2), Hundred Talent Program of Shanxi and International Cooperation Projects of Shanxi Province (No. 2014081026)

The dynamics are studied for the Keller-Segel's minimal chemotaxis model
$τ u_t=(u_x-kuv_x)_x, \ \ \ \ v_t=v_{xx}-v+u$
on a bounded interval with homogeneous Neumann boundary conditions, where
$\tau\geqslant 0$
and
$k\gg 1$
are parameters and the total mass of
$u$
is scaled to be one. In general, the dynamics can be divided into three stages: the first stage is very short in which
$u$
quickly becomes a delta like function with mass concentrated near the point of global maximum of
$v$
; in the second stage, the point of the global maximum of
$v$
drifts towards the boundary of the domain and reaches it at the end of the second stage; in the third stage, the profile of the solution evolves to a steady state profile. This paper considers a special case in which the relaxation parameter
$\tau$
is set to be zero, so the first stage takes no time. A free boundary problem describing the second stage is presented. Rigorous asymptotic behavior is proven for the third stage evolution.
Citation: Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
References:
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N. AlikakosP. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differ. Eqns., 90 (1991), 81-135.  doi: 10.1016/0022-0396(91)90163-4.  Google Scholar

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L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. Roy. Soc. Lond., 439 (1992), 669-682.  doi: 10.1098/rspa.1992.0176.  Google Scholar

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[7]

X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differ. Eqns., 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017.  Google Scholar

[8]

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

[9]

X. Chen and M. Kowalczyk, Dynamics of an interior sipke in the Gierer-Meinhardt system, Siam J. Math. Anal., 33 (2001), 172-193.  doi: 10.1137/S0036141099364954.  Google Scholar

[10]

X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the Shadow Gierer-Meinhardt system, Adv. Differ. Eqns., 6 (2001), 847-872.   Google Scholar

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S. Childress, Chemotactic collapse in two dimensions, in Lecture Notes in Biomath. , 55, Springer, (1984), 61-66. Google Scholar

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

[13]

P. C. Fife and L. Hsiao, The generation and propagation of internal layers, Nonlinear Anal., 12 (1988), 19-41.  doi: 10.1016/0362-546X(88)90010-7.  Google Scholar

[14]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[15]

G. Fusco, A geometric approach to the dynamics of $u_t =\varepsilon ^2 u_{xx} +f(u)$ for small $\varepsilon $, in Problems Involving Change of Type, Springer, 359 (1990), 53-73. Google Scholar

[16]

G. Fusco and J. K. Hale, Slow motion manifolds, dormant instability and singular perturbations, J. Dyn. Diff. Eqns., 1 (1989), 75-94.  doi: 10.1007/BF01048791.  Google Scholar

[17]

H. GajewskiK. Zacharias and Dr. Konrad Gröger, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.  Google Scholar

[18]

M. Herrero and J. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  doi: 10.1007/BF01445268.  Google Scholar

[19]

M. Herrero and J. Velázquez, Chemotaxis collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.  doi: 10.1007/s002850050049.  Google Scholar

[20]

T. Hillen and K. J. Painter, Global existence far a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[21]

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.  Google Scholar

[22]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Meth. Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569.  Google Scholar

[23]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅰ, Jahresber. DMV, 105 (2003), 103-165.   Google Scholar

[24]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅱ, Jahresber. DMV, 106 (2004), 51-69.   Google Scholar

[25]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modellingchemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.2307/2153966.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[30]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.   Google Scholar

[31]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in twodimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.  Google Scholar

[32]

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.  Google Scholar

[33]

K. Osaki and A. Yagi, Finite dimensional attractors for one dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469.   Google Scholar

[34]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[35]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.  Google Scholar

[36]

X. Sun and M. J. Ward, Dynamics and coarsening of interfaces for the viscous Cahn-Hilliard equation in one spatial dimension, Stud. Appl. Math., 105 (2000), 203-234.  doi: 10.1111/1467-9590.00149.  Google Scholar

[37]

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

[38]

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.  Google Scholar

show all references

References:
[1]

N. AlikakosP. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differ. Eqns., 90 (1991), 81-135.  doi: 10.1016/0022-0396(91)90163-4.  Google Scholar

[2]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation, Part Ⅰ, J. Differ. Eqns., 111 (1994), 421-457.  doi: 10.1006/jdeq.1994.1089.  Google Scholar

[3]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation, Part Ⅱ, J. Differ. Eqns., 117 (1995), 165-216.  doi: 10.1006/jdeq.1995.1052.  Google Scholar

[4]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.   Google Scholar

[5]

L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. Roy. Soc. Lond., 439 (1992), 669-682.  doi: 10.1098/rspa.1992.0176.  Google Scholar

[6]

J. Carr and R. Pego, Metastable patterns in solutions of $u_t =\varepsilon ^2 u_{xx}- f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576.  doi: 10.1002/cpa.3160420502.  Google Scholar

[7]

X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differ. Eqns., 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017.  Google Scholar

[8]

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

[9]

X. Chen and M. Kowalczyk, Dynamics of an interior sipke in the Gierer-Meinhardt system, Siam J. Math. Anal., 33 (2001), 172-193.  doi: 10.1137/S0036141099364954.  Google Scholar

[10]

X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the Shadow Gierer-Meinhardt system, Adv. Differ. Eqns., 6 (2001), 847-872.   Google Scholar

[11]

S. Childress, Chemotactic collapse in two dimensions, in Lecture Notes in Biomath. , 55, Springer, (1984), 61-66. Google Scholar

[12]

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

[13]

P. C. Fife and L. Hsiao, The generation and propagation of internal layers, Nonlinear Anal., 12 (1988), 19-41.  doi: 10.1016/0362-546X(88)90010-7.  Google Scholar

[14]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[15]

G. Fusco, A geometric approach to the dynamics of $u_t =\varepsilon ^2 u_{xx} +f(u)$ for small $\varepsilon $, in Problems Involving Change of Type, Springer, 359 (1990), 53-73. Google Scholar

[16]

G. Fusco and J. K. Hale, Slow motion manifolds, dormant instability and singular perturbations, J. Dyn. Diff. Eqns., 1 (1989), 75-94.  doi: 10.1007/BF01048791.  Google Scholar

[17]

H. GajewskiK. Zacharias and Dr. Konrad Gröger, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.  Google Scholar

[18]

M. Herrero and J. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  doi: 10.1007/BF01445268.  Google Scholar

[19]

M. Herrero and J. Velázquez, Chemotaxis collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.  doi: 10.1007/s002850050049.  Google Scholar

[20]

T. Hillen and K. J. Painter, Global existence far a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[21]

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.  Google Scholar

[22]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Meth. Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569.  Google Scholar

[23]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅰ, Jahresber. DMV, 105 (2003), 103-165.   Google Scholar

[24]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅱ, Jahresber. DMV, 106 (2004), 51-69.   Google Scholar

[25]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modellingchemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.2307/2153966.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[30]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.   Google Scholar

[31]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in twodimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.  Google Scholar

[32]

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.  Google Scholar

[33]

K. Osaki and A. Yagi, Finite dimensional attractors for one dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469.   Google Scholar

[34]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[35]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.  Google Scholar

[36]

X. Sun and M. J. Ward, Dynamics and coarsening of interfaces for the viscous Cahn-Hilliard equation in one spatial dimension, Stud. Appl. Math., 105 (2000), 203-234.  doi: 10.1111/1467-9590.00149.  Google Scholar

[37]

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

[38]

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.  Google Scholar

Figure 1.  A numerical solution of (1) with $\tau=0$, $k=200$, $\ell=1$, and mesh sizes $\Delta x=1/400$, $\Delta t=3\times 10^{-6}$. Left: snapshots of $v(k,\cdot,t)$ with constant frequency; middle: snapshots of $u$; right: location of the point of maximum of $v(k,\cdot,t)$, which reaches the boundary at $T\approx 0.13$.
Figure 2.  First Row: a numerical solution of (6) with $k=100$, $\ell=4$, and $v_0(x)=\frac{\ell}{\pi}\big|\cos\frac{\pi(x-2.5)}{\ell}\big|$; left is location of maximum of $v$; middle is snapshots of $v(k,\cdot,t)$ with non-uniform time elapses; right is snapshots of $v_x(k,\cdot,t)$. Second Row: the solution of the free boundary problem (7) for $t\in[0,T^-]$ ($T\approx 5.2$), combined with the solution of the boundary value problem (8) for $t\geqslant T$; left is the location of free boundary; middle is snapshots of $w$; right is snapshots of $w_x$.
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