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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

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