# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
##### References:
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Chen, Generation, propagation, and annihilation of metastable patterns, J. Differ. Eqns., 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017. [8] X. Chen, J. Hao, X. Wang, Y. 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. [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. [10] X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the Shadow Gierer-Meinhardt system, Adv. Differ. Eqns., 6 (2001), 847-872. [11] S. Childress, Chemotactic collapse in two dimensions, in Lecture Notes in Biomath. , 55, Springer, (1984), 61-66. [12] S. Childress and J. Perkus, Nonlinear aspects of chemotaxis, Math. Bios., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9. [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. [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. [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. [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. [17] H. Gajewski, K. 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. [18] M. Herrero and J. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  doi: 10.1007/BF01445268. [19] M. Herrero and J. Velázquez, Chemotaxis collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.  doi: 10.1007/s002850050049. [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. [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. [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. [23] D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅰ, Jahresber. DMV, 105 (2003), 103-165. [24] D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅱ, Jahresber. DMV, 106 (2004), 51-69. [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. [26] K. Kang, T. 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. [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. [28] C.-S. Lin, W.-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. [29] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. [30] 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. [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. [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. [33] K. Osaki and A. Yagi, Finite dimensional attractors for one dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469. [34] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407. [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. [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. [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. [38] Y. Zhang, X. Chen, J. Hao, X. 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.

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##### References:
 [1] N. Alikakos, P. 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. [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. [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. [4] P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. [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. [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. [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. [8] X. Chen, J. Hao, X. Wang, Y. 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. [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. [10] X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the Shadow Gierer-Meinhardt system, Adv. Differ. Eqns., 6 (2001), 847-872. [11] S. Childress, Chemotactic collapse in two dimensions, in Lecture Notes in Biomath. , 55, Springer, (1984), 61-66. [12] S. Childress and J. Perkus, Nonlinear aspects of chemotaxis, Math. Bios., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9. [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. [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. [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. [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. [17] H. Gajewski, K. 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. [18] M. Herrero and J. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  doi: 10.1007/BF01445268. [19] M. Herrero and J. Velázquez, Chemotaxis collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.  doi: 10.1007/s002850050049. [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. [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. [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. [23] D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅰ, Jahresber. DMV, 105 (2003), 103-165. [24] D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅱ, Jahresber. DMV, 106 (2004), 51-69. [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. [26] K. Kang, T. 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. [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. [28] C.-S. Lin, W.-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. [29] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. [30] 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. [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. [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. [33] K. Osaki and A. Yagi, Finite dimensional attractors for one dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469. [34] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407. [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. [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. [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. [38] Y. Zhang, X. Chen, J. Hao, X. 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.
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 $T\approx 0.13$.
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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