Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model

Discrete and Continuous Dynamical Systems - Series B

Volumn 4, Issue 4, 2004, Pages 1033-1064

  Kolokolnikov, Theodore ^a   Ward, Michael J ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Gierer Meinhardt model; Green's function; Pattern formation; Reaction diffusion equations

Indexed keywords

EID: 4944224628     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2004.4.1033     Document Type: Article

References (30)

1. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, (1965).
2. (n+2)/(n-2), SIAM Rev., 38 No. 2 (1996), pp. 191-238.
3. P. Bates, N. Fusco, Equilibria with Many Nuclei for the Cahn-Hilliard Equation, J. Diff. Eq., 160, No. 2, (2000), pp. 283-356.
4. L. A. Caffarelli, A. Friedman, Convexity of Solutions of Semilinear Elliptic Equations, Duke Math. J., 52, No. 2, (1985), pp. 431-456.
5. X. Chen, M. Kowalczyk, Slow Dynamics of Interior Spikes in the Shadow Gierer-Meinhardt System, Adv. in Diff. Equat., 8, No. 7, (2001), pp. 847-872.
6. X. Chen, M. Kowalcyzk, Dynamics of an Interior Spike in the Gierer-Meinhardt System, SIAM. J. Math. Anal., 38, No. 1, (2001), pp. 172-193.
7. M. Del Pino, P. Felmer, M. Kowalczyk, Boundary Spikes in the Gierer-Meinhardt System, Commun. Pure Appl. Anal., 1, No. 4, (2002), pp. 437-456.
8. A. Gierer, H. Meinhardt, A Theory of Biological Pattern Formation, Kybernetik, 12, (1972), pp. 30-39.
9. C. Gui, J. Wei, Multiple Interior Peak Solutions for Some Singularly Perturbed Neumann Problems, J. Diff. Eq., 158, No. 1, (1999), pp. 1-27.
10. B. Gustafsson, On the Convexity of a Solution of Liouville's Equation, Duke Math. J., 60, No. 2, (1990), pp. 303-311.
11. L. Harrison, D. Holloway, Order and Localization in Reaction-Diffusion Pattern, Physica A, 222, (1995), pp. 210-233.
12. D. Iron, M. J. Ward, A Metastable Spike Solution for a Non-Local Reaction-Diffusion Model, SIAM J. Appl. Math., 60, No. 3, (2000), pp. 778-802.
13. D. Iron, M. J. Ward, The Dynamics of Boundary Spikes for a Non-local Reaction-Diffusion Model, Europ. J. Appl. Math., 11, No. 5, (2000), pp. 491-514
14. D. Iron, M. J. Ward, J. Wei, The Stability of Spike Solutions to the One-Dimensional Gierer-Meinhardt Model, Physica D, 150, No. 1-2, (2001), pp. 25-62.
15. T. Kolokolnikov, M. J. Ward, Reduced Wave Green's Functions and Their Effect on the Dynamics of a Spike for the Gierer-Meinhardt Model, Europ. J. Appl. Math., 14, No. 5, (2003), pp. 513-545.
16. M. Kowalczyk, Multiple Spike Layers in the Shadow Gierer-Meinhardt System: Existence of Equilibria and Approximate Invariant Manifold, Duke Math. J., 98, No. 1, (1999), pp. 59-111.
17. H. Meinhardt, Models of Biological Pattern Formation, Academic Press, London, (1982).
18. H. Meinhardt, The Algorithmic Beauty of Sea Shells, Springer-Verlag, Berlin, (1995).
19. W. Ni, Diffusion, Cross-Diffusion, and Their Spike-Layer Steady-States, Notices of the AMS, Vol. 45, No. 1, (1998), pp. 9-18.
20. Y. Nishiura, Global Structure of Bifurcating Solutions of Some Reaction-Diffusion Systems, SIAM J. Math. Anal., 13, No. 4, (1982), pp. 555-593.
21. J. E. Pearson, Complex Patterns in a Simple System, Science, 216, (1993), pp. 189-192.
22. Y. Nishiura, D. Ueyama, A Skeleton Structure of Self-Replicating Dynamics, Physica D, 130, (1999), pp. 73-104.
23. M. J. Ward, An Asymptotic Analysis of Localized Solutions for Some Reaction-Diffusion Models in Multi-Dimensional Domains, Stud. Appl. Math., 97, No. 2, (1996), pp. 103-126.
24. M. J. Ward, D. McInerney, P. Houston, D. Gavaghan, P. Maini, The Dynamics and Pinning of a Spike for a Reaction~Diffusion System, SIAM J. Appl. Math., 62, No. 4, (2002), pp. 1297-1328.
25. J. Wei, On Single Interior Spike Solutions for the Gierer-Meinhardt System: Uniqueness and Stability Estimates, Europ. J. Appl. Math., 10, No. 4, (1999), pp. 353-378.
26. J. Wei, On the Boundary Spike Layer Solutions of Singularly Perturbed Semilinear Neumann Problem, J. Diff. Eq., 134, No. 1, (1997), pp. 104-133.
27. J. Wei, On the Interior Spike Layer Solutions to a Singularly Perturbed Neumann Problem, Tohoku Math. J., 50, No. 2, (1998), pp. 159-178.
28. J. Wei, Uniqueness and Critical Spectrum of Boundary Spike Solutions, Proc. Royal Soc. Edinburgh, Section A (Mathematics), 131, No. 6, (2001), pp. 1457-1480.
29. J. Wei, M. Winter, Spikes for the Gierer-Meinhardt System in Two Dimensions: The Strong Coupling Case, J. Diff. Eq., 178, No. 2, (2002), pp. 478-518.
30. R. Wong, Asymptotic Approximations of Integrals, Academic Press, San Diego, CA, (1989).