The existence and uniqueness of unstable eigenvalues for stripe
      patterns in the Gierer-Meinhardt system

Kota Ikeda; Kota Ikeda

doi:10.3934/nhm.2013.8.291

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 291-325. doi: 10.3934/nhm.2013.8.291

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The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system

Kota Ikeda ^{1
,}

1.
Graduate School of Advanced Mathematical Science, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, Kanagawa 214-8571

Received: 01 March 2012 Revised: 01 January 2013
35K57, 35K50.

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. This system has a stationary solution with a stripe pattern on a rectangular domain, but numerical results suggest that such stripe pattern is unstable. In [8], Kolokolnikov et al. proved the existence of a positive eigenvalue, which is called an unstable eigenvalue, for a stationary solution with a stripe pattern by the NLEP method, which implies the instability of the stripe pattern. In addition, the uniqueness of the unstable eigenvalue was shown under some technical assumptions in [8]. In this paper, we prove the existence and uniqueness of an unstable eigenvalue by using the SLEP method without any extra conditions. We also prove the existence of a single-spike solution in one-dimension.
- Gierer-Meinhardt system,
- eigenvalue problem,
- SLEP method
Citation: Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system[J]. Networks and Heterogeneous Media, 2013, 8(1): 291-325. doi: 10.3934/nhm.2013.8.291

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Abstract

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. This system has a stationary solution with a stripe pattern on a rectangular domain, but numerical results suggest that such stripe pattern is unstable. In [8], Kolokolnikov et al. proved the existence of a positive eigenvalue, which is called an unstable eigenvalue, for a stationary solution with a stripe pattern by the NLEP method, which implies the instability of the stripe pattern. In addition, the uniqueness of the unstable eigenvalue was shown under some technical assumptions in [8]. In this paper, we prove the existence and uniqueness of an unstable eigenvalue by using the SLEP method without any extra conditions. We also prove the existence of a single-spike solution in one-dimension.

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