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

  • Juncheng Wei
  • Matthias Winter
Part of the Applied Mathematical Sciences book series (AMS, volume 189)

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.

Keywords

Assure 

References

  1. 8.
    Bates, P., Shi, J.: Existence and instability of spike layer solutions to singular perturbation problems. J. Funct. Anal. 196, 211–264 (2002) MathSciNetCrossRefMATHGoogle Scholar
  2. 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) MathSciNetMATHGoogle Scholar
  3. 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) MathSciNetCrossRefMATHGoogle Scholar
  4. 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) MathSciNetCrossRefMATHGoogle Scholar
  5. 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) CrossRefMATHGoogle Scholar
  6. 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) MathSciNetCrossRefMATHGoogle Scholar
  7. 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) MathSciNetMATHGoogle Scholar
  8. 68.
    Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986) MathSciNetCrossRefMATHGoogle Scholar
  9. 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) CrossRefMATHGoogle Scholar
  10. 80.
    Gui, C., Wei, J.: Multiple interior peak solutions for some singularly perturbed Neumann problems. J. Differ. Equ. 158, 1–27 (1999) MathSciNetCrossRefMATHGoogle Scholar
  11. 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) MathSciNetCrossRefMATHGoogle Scholar
  12. 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) CrossRefMATHGoogle Scholar
  13. 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) CrossRefMATHGoogle Scholar
  14. 226.
    Takagi, I.: Point-condensation for a reaction-diffusion system. J. Differ. Equ. 61, 208–249 (1986) CrossRefMATHGoogle Scholar
  15. 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) MathSciNetCrossRefMATHGoogle Scholar
  16. 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) CrossRefMATHGoogle Scholar
  17. 256.
    Wei, J., Winter, M.: Stationary solutions for the Cahn-Hilliard equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15, 459–492 (1998) MathSciNetCrossRefMATHGoogle Scholar
  18. 257.
    Wei, J., Winter, M.: On the Cahn-Hilliard equations II: interior spike layer solutions. J. Differ. Equ. 148, 231–267 (1998) MathSciNetCrossRefMATHGoogle Scholar
  19. 272.
    Wei, J., Winter, M.: Symmetric and asymmetric multiple clusters in a reaction-diffusion system. Nonlinear Differ. Equ. Appl. 14, 787–823 (2007) MathSciNetCrossRefMATHGoogle Scholar
  20. 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) MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Juncheng Wei
    • 1
  • Matthias Winter
    • 2
  1. 1.Department of MathematicsThe Chinese University of Hong KongHong KongChina
  2. 2.Department of Mathematical SciencesBrunel UniversityUxbridgeUK

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