# American Institute of Mathematical Sciences

• Previous Article
Approximation of nonlinear parabolic equations using a family of conformal and non-conformal schemes
• CPAA Home
• This Issue
• Next Article
Qualitative analysis and travelling wave solutions for the SI model with vertical transmission
January  2012, 11(1): 115-145. doi: 10.3934/cpaa.2012.11.115

## Dynamics of a boundary spike for the shadow Gierer-Meinhardt system

 1 Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan 2 Graduate School of Advanced Mathematical Science, Meiji University, Kawasaki, 214-8571, Japan 3 Department of Mathematics, Tokyo Institute of Technology, O-Okayama, Meguro-ku, Tokyo 152-8551

Received  December 2009 Revised  October 2010 Published  September 2011

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. The authors of [3] showed that if an initial value is close to a spiky pattern and its peak is far away from the boundary, the solution of the shadow Gierer-Meinhardt system, called a interior spike solution, moves towards a point on boundary which is the closest to the peak. However it has not been studied how a solution close to a spiky pattern with the peak on the boundary, called a boundary spike solution moves along the boundary. In this paper, we consider the shadow Gierer-Meinhardt system and dynamics of a boundary spike solution. Our results state that a boundary spike moves towards a critical point of the curvature of the boundary and approaches a stable stationary solution.
Citation: Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure & Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115
##### References:
 [1] N. D. Alikakos, P. W. Bates, X. Chen and G. Fusco, Mullins-Sekerka motion of small droplets on a fixed boundary,, J. Geom. Anal., 10 (2000), 575.   Google Scholar [2] P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states,, Invent. Math., 174 (2008), 355.   Google Scholar [3] X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system,, Adv. Differential Equations, 6 (2001), 847.   Google Scholar [4] M. del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system,, Commun. Pure Appl. Anal., 1 (2002), 437.  doi: 10.3934/cpaa.2002.1.437.  Google Scholar [5] M. del Pino, P. L. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems,, SIAM J. Math. Anal., 31 (1999), 63.  doi: 10.1137/S0036141098332834.  Google Scholar [6] M. del Pino, P. L. Felmer and J. Wei, On the role of distance function in some singular perturbation problems,, Comm. Partial Differential Equations, 25 (2000), 155.  doi: 10.1080/03605300008821511.  Google Scholar [7] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. D. D. E. \textbf{14} (2002), 14 (2002), 85.  doi: 10.1023/A:1012980128575.  Google Scholar [8] S.-I. Ei, Dynamics and their interaction of spikes on smoothly curved boundaries for reaction-diffusion systems in 2D,, preprint., ().   Google Scholar [9] L. C. Evans, "Partial Differential Equations,'', Graduate Studies in Mathematics, (1998).   Google Scholar [10] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Mathematical analysis and applications, 12 (1972), 30.   Google Scholar [11] A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetic, 12 (1972), 30.  doi: 10.1007/BF00289234.  Google Scholar [12] C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems,, Canad. J. Math., 52 (2000), 522.  doi: 10.4153/CJM-2000-024-x.  Google Scholar [13] D. Iron and M. J. Ward, The dynamics of boundary spikes for a nonlocal reaction-diffusion model,, European J. Appl. Math., 11 (2000), 491.  doi: 10.1017/S0956792500004253.  Google Scholar [14] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model,, Phys. D, 150 (2001), 25.  doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar [15] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [16] H. Meinhardt, "Models of Biological Pattern Formation,'', Academic Press, (1982).   Google Scholar [17] Y. Miyamoto, Stability of a boundary spike layer for the Gierer-Meinhardt system,, European J. Appl. Math., 16 (2005), 467.  doi: 10.1017/S0956792505006376.  Google Scholar [18] Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains,, Quarterly of Applied Mathematics, 65 (2007), 357.   Google Scholar [19] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819.  doi: 10.1002/cpa.3160440705.  Google Scholar [20] W.-M. Ni and I. Takagi, Locating the peaks of least energy solution to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar [21] W.-M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model,, Japan J. Indust. Appl. Math., 18 (2001), 259.   Google Scholar [22] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems,, SIAM J. Math. Anal., 13 (1982), 555.  doi: 10.1137/0513037.  Google Scholar [23] J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix,, Ann. Math. Statistics, 21 (1950), 124.   Google Scholar [24] A. Turing, The chemical basis of morphogenesis,, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37.  doi: 10.1098/rstb.1952.0012.  Google Scholar [25] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem,, J. Diff. Eq., 134 (1997), 104.  doi: 10.1006/jdeq.1996.3218.  Google Scholar [26] J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates,, European J. Appl. Math., 10 (1999), 353.  doi: 10.1017/S0956792599003770.  Google Scholar [27] J. Wei, On a nonlocal eigenvalue problem and its applications to point-condensations in reaction-diffusion systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1485.  doi: 10.1142/S0218127400000979.  Google Scholar [28] J. Wei, Uniqueness and critical spectrum of boundary spike solutions,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1457.  doi: 10.1017/S0308210500001487.  Google Scholar [29] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case,, J. Nonlinear Sci., 11 (2001), 415.  doi: 10.1007/s00332-001-0380-1.  Google Scholar

show all references

##### References:
 [1] N. D. Alikakos, P. W. Bates, X. Chen and G. Fusco, Mullins-Sekerka motion of small droplets on a fixed boundary,, J. Geom. Anal., 10 (2000), 575.   Google Scholar [2] P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states,, Invent. Math., 174 (2008), 355.   Google Scholar [3] X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system,, Adv. Differential Equations, 6 (2001), 847.   Google Scholar [4] M. del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system,, Commun. Pure Appl. Anal., 1 (2002), 437.  doi: 10.3934/cpaa.2002.1.437.  Google Scholar [5] M. del Pino, P. L. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems,, SIAM J. Math. Anal., 31 (1999), 63.  doi: 10.1137/S0036141098332834.  Google Scholar [6] M. del Pino, P. L. Felmer and J. Wei, On the role of distance function in some singular perturbation problems,, Comm. Partial Differential Equations, 25 (2000), 155.  doi: 10.1080/03605300008821511.  Google Scholar [7] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. D. D. E. \textbf{14} (2002), 14 (2002), 85.  doi: 10.1023/A:1012980128575.  Google Scholar [8] S.-I. Ei, Dynamics and their interaction of spikes on smoothly curved boundaries for reaction-diffusion systems in 2D,, preprint., ().   Google Scholar [9] L. C. Evans, "Partial Differential Equations,'', Graduate Studies in Mathematics, (1998).   Google Scholar [10] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Mathematical analysis and applications, 12 (1972), 30.   Google Scholar [11] A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetic, 12 (1972), 30.  doi: 10.1007/BF00289234.  Google Scholar [12] C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems,, Canad. J. Math., 52 (2000), 522.  doi: 10.4153/CJM-2000-024-x.  Google Scholar [13] D. Iron and M. J. Ward, The dynamics of boundary spikes for a nonlocal reaction-diffusion model,, European J. Appl. Math., 11 (2000), 491.  doi: 10.1017/S0956792500004253.  Google Scholar [14] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model,, Phys. D, 150 (2001), 25.  doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar [15] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [16] H. Meinhardt, "Models of Biological Pattern Formation,'', Academic Press, (1982).   Google Scholar [17] Y. Miyamoto, Stability of a boundary spike layer for the Gierer-Meinhardt system,, European J. Appl. Math., 16 (2005), 467.  doi: 10.1017/S0956792505006376.  Google Scholar [18] Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains,, Quarterly of Applied Mathematics, 65 (2007), 357.   Google Scholar [19] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819.  doi: 10.1002/cpa.3160440705.  Google Scholar [20] W.-M. Ni and I. Takagi, Locating the peaks of least energy solution to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar [21] W.-M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model,, Japan J. Indust. Appl. Math., 18 (2001), 259.   Google Scholar [22] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems,, SIAM J. Math. Anal., 13 (1982), 555.  doi: 10.1137/0513037.  Google Scholar [23] J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix,, Ann. Math. Statistics, 21 (1950), 124.   Google Scholar [24] A. Turing, The chemical basis of morphogenesis,, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37.  doi: 10.1098/rstb.1952.0012.  Google Scholar [25] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem,, J. Diff. Eq., 134 (1997), 104.  doi: 10.1006/jdeq.1996.3218.  Google Scholar [26] J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates,, European J. Appl. Math., 10 (1999), 353.  doi: 10.1017/S0956792599003770.  Google Scholar [27] J. Wei, On a nonlocal eigenvalue problem and its applications to point-condensations in reaction-diffusion systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1485.  doi: 10.1142/S0218127400000979.  Google Scholar [28] J. Wei, Uniqueness and critical spectrum of boundary spike solutions,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1457.  doi: 10.1017/S0308210500001487.  Google Scholar [29] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case,, J. Nonlinear Sci., 11 (2001), 415.  doi: 10.1007/s00332-001-0380-1.  Google Scholar
 [1] Manuel del Pino, Patricio Felmer, Michal Kowalczyk. Boundary spikes in the Gierer-Meinhardt system. Communications on Pure & Applied Analysis, 2002, 1 (4) : 437-456. doi: 10.3934/cpaa.2002.1.437 [2] Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651 [3] Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033 [4] Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363 [5] Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 583-591. doi: 10.3934/dcds.2015.35.583 [6] Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885 [7] Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291 [8] Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192 [9] Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443 [10] Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158 [11] Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108 [12] Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941 [13] Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049 [14] Hideo Ikeda, Masayasu Mimura, Tommaso Scotti. Shadow system approach to a plankton model generating harmful algal bloom. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 829-858. doi: 10.3934/dcds.2017034 [15] Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020049 [16] Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023 [17] Hideaki Takaichi, Izumi Takagi, Shoji Yotsutani. Global bifurcation structure on a shadow system with a source term - Representation of all solutions-. Conference Publications, 2011, 2011 (Special) : 1344-1350. doi: 10.3934/proc.2011.2011.1344 [18] Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947 [19] Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665 [20] Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111

2018 Impact Factor: 0.925