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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 and 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), 575-596. [2] P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433. [3] X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system, Adv. Differential Equations, 6 (2001), 847-872. [4] M. del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Commun. Pure Appl. Anal., 1 (2002), 437-456. doi: 10.3934/cpaa.2002.1.437. [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), 63-79 (electronic). doi: 10.1137/S0036141098332834. [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), 155-177. doi: 10.1080/03605300008821511. [7] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. D. D. E. 14 (2002), 85-137. doi: 10.1023/A:1012980128575. [8] S.-I. Ei, Dynamics and their interaction of spikes on smoothly curved boundaries for reaction-diffusion systems in 2D, preprint. [9] L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, 19, (1998). [10] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., 12 (1972), 30-39. Academic Press, (1981), 369-402. [11] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39. doi: 10.1007/BF00289234. [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), 522-538. doi: 10.4153/CJM-2000-024-x. [13] D. Iron and M. J. Ward, The dynamics of boundary spikes for a nonlocal reaction-diffusion model, European J. Appl. Math., 11 (2000), 491-514. doi: 10.1017/S0956792500004253. [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), 25-62. doi: 10.1016/S0167-2789(00)00206-2. [15] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [16] H. Meinhardt, "Models of Biological Pattern Formation,'' Academic Press, 1982. [17] Y. Miyamoto, Stability of a boundary spike layer for the Gierer-Meinhardt system, European J. Appl. Math., 16 (2005), 467-491. doi: 10.1017/S0956792505006376. [18] Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains, Quarterly of Applied Mathematics, 65 (2007), 357-374. [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), 819-851. doi: 10.1002/cpa.3160440705. [20] W.-M. Ni and I. Takagi, Locating the peaks of least energy solution to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4. [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), 259-272. [22] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593. doi: 10.1137/0513037. [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), 124-127. [24] A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72. doi: 10.1098/rstb.1952.0012. [25] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Diff. Eq., 134 (1997), 104-133. doi: 10.1006/jdeq.1996.3218. [26] J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math., 10 (1999), 353-378. doi: 10.1017/S0956792599003770. [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), 1485-1496. doi: 10.1142/S0218127400000979. [28] J. Wei, Uniqueness and critical spectrum of boundary spike solutions, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1457-1480. doi: 10.1017/S0308210500001487. [29] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458. doi: 10.1007/s00332-001-0380-1.

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), 575-596. [2] P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433. [3] X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system, Adv. Differential Equations, 6 (2001), 847-872. [4] M. del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Commun. Pure Appl. Anal., 1 (2002), 437-456. doi: 10.3934/cpaa.2002.1.437. [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), 63-79 (electronic). doi: 10.1137/S0036141098332834. [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), 155-177. doi: 10.1080/03605300008821511. [7] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. D. D. E. 14 (2002), 85-137. doi: 10.1023/A:1012980128575. [8] S.-I. Ei, Dynamics and their interaction of spikes on smoothly curved boundaries for reaction-diffusion systems in 2D, preprint. [9] L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, 19, (1998). [10] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., 12 (1972), 30-39. Academic Press, (1981), 369-402. [11] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39. doi: 10.1007/BF00289234. [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), 522-538. doi: 10.4153/CJM-2000-024-x. [13] D. Iron and M. J. Ward, The dynamics of boundary spikes for a nonlocal reaction-diffusion model, European J. Appl. Math., 11 (2000), 491-514. doi: 10.1017/S0956792500004253. [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), 25-62. doi: 10.1016/S0167-2789(00)00206-2. [15] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [16] H. Meinhardt, "Models of Biological Pattern Formation,'' Academic Press, 1982. [17] Y. Miyamoto, Stability of a boundary spike layer for the Gierer-Meinhardt system, European J. Appl. Math., 16 (2005), 467-491. doi: 10.1017/S0956792505006376. [18] Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains, Quarterly of Applied Mathematics, 65 (2007), 357-374. [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), 819-851. doi: 10.1002/cpa.3160440705. [20] W.-M. Ni and I. Takagi, Locating the peaks of least energy solution to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4. [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), 259-272. [22] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593. doi: 10.1137/0513037. [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), 124-127. [24] A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72. doi: 10.1098/rstb.1952.0012. [25] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Diff. Eq., 134 (1997), 104-133. doi: 10.1006/jdeq.1996.3218. [26] J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math., 10 (1999), 353-378. doi: 10.1017/S0956792599003770. [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), 1485-1496. doi: 10.1142/S0218127400000979. [28] J. Wei, Uniqueness and critical spectrum of boundary spike solutions, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1457-1480. doi: 10.1017/S0308210500001487. [29] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458. doi: 10.1007/s00332-001-0380-1.
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