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2014 | OriginalPaper | Chapter

2. Existence of Spikes for the Gierer-Meinhardt System in One Dimension

Authors : Juncheng Wei, Matthias Winter

Published in: Mathematical Aspects of Pattern Formation in Biological Systems

Publisher: Springer London

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Abstract

We give a full account of the existence of multiple spikes for the Gierer-Meinhardt system in an interval on the real line. We present a unified rigorous approach based on the Liapunov-Schmidt method which is very flexible and consider the cases of symmetric and asymmetric multi-spike solutions. We also study clustered multiple spikes.

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previous chapter Introduction
next chapter The Nonlocal Eigenvalue Problem (NLEP)
Literature
8.
Bates, P., Shi, J.: Existence and instability of spike layer solutions to singular perturbation problems. J. Funct. Anal. 196, 211–264 (2002) MathSciNetCrossRefMATH
9.
Bates, P., Dancer, E.N., Shi, J.: Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability. Adv. Differ. Equ. 4, 1–69 (1999) MathSciNetMATH
27.
Chen, X., del Pino, M., Kowalczyk, M.: The Gierer and Meinhardt system: the breaking of homoclinics and multi-bump ground states. Commun. Contemp. Math. 3(3), 419–439 (2001) MathSciNetCrossRefMATH
43.
del Pino, M., Kowalczyk, M., Chen, X.: The Gierer-Meinhardt system: the breaking of homoclinics and multi-bump ground states. Commun. Contemp. Math. 3, 419–439 (2001) MathSciNetCrossRefMATH
44.
del Pino, M., Kowalczyk, M., Wei, J.: Multi-bump ground states of the Gierer-Meinhardt system in \(\mathbb{R}^{2}\). Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, 53–85 (2003) CrossRefMATH
49.
Doelman, A., Gardner, R.A., Kaper, T.J.: Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50, 443–507 (2001) MathSciNetCrossRefMATH
50.
Doelman, A., Kaper, T.J., van der Ploeg, H.: Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation. Methods Appl. Anal. 8, 387–414 (2001) MathSciNetMATH
68.
Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986) MathSciNetCrossRefMATH
74.
Gilbarg, D., Trudinger, S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 224. Springer, Berlin (1983) CrossRefMATH
80.
Gui, C., Wei, J.: Multiple interior peak solutions for some singularly perturbed Neumann problems. J. Differ. Equ. 158, 1–27 (1999) MathSciNetCrossRefMATH
81.
Gui, C., Wei, J., Winter, M.: Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17, 47–82 (2000) MathSciNetCrossRefMATH
191.
Oh, Y.G.: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V) a . Commun. Partial Differ. Equ. 13, 1499–1519 (1988) CrossRefMATH
192.
Oh, Y.G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131, 223–253 (1990) CrossRefMATH
226.
Takagi, I.: Point-condensation for a reaction-diffusion system. J. Differ. Equ. 61, 208–249 (1986) CrossRefMATH
240.
Ward, M.J., Wei, J.: Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability. Eur. J. Appl. Math. 13, 283–320 (2002) MathSciNetCrossRefMATH
248.
Wei, J.: On the interior spike layer solutions for some singular perturbation problems. Proc. R. Soc. Edinb., Sect. A, Math. 128, 849–874 (1998) CrossRefMATH
256.
Wei, J., Winter, M.: Stationary solutions for the Cahn-Hilliard equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15, 459–492 (1998) MathSciNetCrossRefMATH
257.
272.
Wei, J., Winter, M.: Symmetric and asymmetric multiple clusters in a reaction-diffusion system. Nonlinear Differ. Equ. Appl. 14, 787–823 (2007) MathSciNetCrossRefMATH
273.
Wei, J., Winter, M.: Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in \({\mathbb {R}}\). Methods Appl. Anal. 14, 119–163 (2007) MathSciNetMATH
Metadata
Title
Existence of Spikes for the Gierer-Meinhardt System in One Dimension
Authors
Juncheng Wei
Matthias Winter
Copyright Year
2014
Publisher
Springer London
Book
Mathematical Aspects of Pattern Formation in Biological Systems

Print ISBN: 978-1-4471-5525-6

Electronic ISBN: 978-1-4471-5526-3

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-1-4471-5526-3_2

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