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Mathematische Zeitschrift

, Volume 254, Issue 2, pp 359–383 | Cite as

Standing waves in the FitzHugh-Nagumo system and a problem in combinatorial geometry

  • Juncheng Wei
  • Matthias Winter
Article

Abstract

We show that there is a close relation between standing-wave solutions for the FitzHugh-Nagumo system

Open image in new window

where 0<a<1/2 and δ γ=β 2 ∈ (0,a), and the following combinatorial problem:

(*) Given K points Q 1 , . . . , Q K R N with minimum distance 1, find out the maximum number of times that the minimum distance 1 can occur.

More precisely, we show that for any given positive integer K, there is a δ K >0 such that for 0<δ<δ K , there exists a standing-wave solution (u δ ,ν δ ) to the FitzHugh-Nagumo system with the property that u δ has K spikes Q δ 1,. . .,Q δ K and Open image in new window approaches an optimal configuration in (*), where Open image in new window .

Keywords

FitzHugh-Nagumo system Standing waves Optimal configuration Localized energy method 

Mathematics Subject Classification

Primary 35B40 35B45 Secondary 35J55 92C15 92C40 

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References

  1. 1.
    Alikakos, N., Fusco, G., Kowalczyk, M.: Finite dimensional dynamics and interfaces intersecting the boundary I. Indiana Univ. Math. J. 45, 1119–1155 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alikakos, N., Kowalczyk, M.: Critical points of a singular perturbation problem via reduced energy and local linking. J. Diff. Eqs. 159, 403–426 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bates, P., Dancer, E.N., Shi, J.: Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability. Adv. Diff. Eqs. 4, 1–69 (1999)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bates, P., Fusco, G.: Equilibria with many nuclei for the Cahn-Hilliard equation. J. Diff. Eqs. 160, 283–356 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bates, P., Shi, J.: Existence and instability of spike layer solutions to singular perturbation problems. J. Funct. Anal. 196, 211–264 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bender, C.: Bestimmung der größten Zahl gleich großer Kugeln, welche sich auf eine Kugel von demselben Radius wie die übrigen auflegen lassen. Grunert Arch. 56, 302–313 (1874)zbMATHGoogle Scholar
  7. 7.
    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, 419–439 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Clement, P., Sweers, G.: Existence and multiplicity results for a semilinear eigenvalue problem. Ann. Scuola Norm. Sup. Pisa 14, 97–121 (1987)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Conway, J.H., Sloan, N.J.A.: Sphere packings, lattices and groups, 3rd ed. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 290, Springer, New York, Heidelberg, 1999Google Scholar
  10. 10.
    Coxeter, H.S.M.: An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size. Convexity, Proc. Symp. Pure Math. 7, 53–71 (1963)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Dancer, E.N.: A note on the asymptotic uniqueness for some nonlinearities which change sign. Bull. Austral. Math. Soc. 61, 305–312 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    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)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Dancer, E.N., Yan, S.: Multipeak solutions for a singular perturbed Neumann problem. Pacific J. Math. 189, 241-262 (1999)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Dancer, E.N., Yan, S.: Interior and boundary peak solutions for a mixed boundary value problem. Indiana Univ. Math. J. 48, 1177–1212 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Dancer, E.N., Yan, S.: A minimization problem associated with elliptic systems of FitzHugh-Nagumo type. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 237–253 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    deFigueiredo, D.G., Mitidieri, E.: A Maximum Principle for an elliptic system and applications to semilinear problems. SIAM J. Math. Anal. 17, 836–849 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    del Pino, M., Kowalczyk, M., Wei, J.: Multi-bump ground states for the Gierer-Meinhardt system in R 2. Ann. Inst. H. Poincaré Anal. Non Linéaire 20, 53–85 (2003)zbMATHCrossRefGoogle Scholar
  18. 18.
    Fejes-Toth, L.: New results in the theory of packing and covering. In: Gruber, P.M., Wills, J.M. (eds.) Convexity and its Applications. Birkhäuser, Basel, 1983, pp. 318–359Google Scholar
  19. 19.
    FitzHugh, R.: Impulse and physiological states in models of nerve membrans. Biophysics J. 1, 445–466 (1961)CrossRefGoogle Scholar
  20. 20.
    Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Gui, C, Wei, J.: Multiple interior peak solutions for some singular perturbation problems. J. Diff. Eqs. 158, 1–27 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Gui, C., Wei, J.: On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Canad. J. Math. 52, 522–538 (2000)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Gui, C., Wei, J., Winter, M.: Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 47–82 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Erdös, P., Purdy, G.: Extremal problems in combinatorial geometry. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics. Elsevier Science, B.V., 1995, pp. 809–871Google Scholar
  25. 25.
    Harborth, H.: Solution to problem 664a. Elemente Math. 29, 14–15 (1974)MathSciNetGoogle Scholar
  26. 26.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)Google Scholar
  27. 27.
    Klaasen, G.A., Mitidieri, E.: Standing wave solutions for a system derived from the FitzHugh-Nagumo equations for nerve conduction. SIAM J. Math. Anal. 17, 74–83 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Klaasen, G.A., Mitidieri, E.: Standing wave solutions of a system of reaction-diffusion equations derived from the FitzHugh-Nagumo equations. SIAM J. Appl. Math. 44, 74–83 (1986)Google Scholar
  29. 29.
    Kowalczyk, M.: Multiple spike layers in the shadow Gierer-Meinhardt system: existence of equilibria and approximate invariant manifold. Duke Math. J. 98, 59–111 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Lazer, A.C., McKenna, P.J.: On steady state solutions of a system of reaction-diffusion equations from biology. Nonlinear Anal. 6, 523–530 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Leech, J.: The problem of the thirteen spheres. Math. Gaz. 40, 22–23 (1956)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2071 (1962)Google Scholar
  33. 33.
    Ni, W.-M., Wei, J.: On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure. Appl. Math. 48, 731–768 (1995)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Nishiura, Y.: Coexistence of infinitely many stable solutions to reaction-diffusion system in the singular limit, Dynamics Reported: Expositions in Dynamical Systems, Edited by C.R.K.T. Jones, U. Kirchgraber and H.O. Walther, Vol. 3, Springer-Verlag, New York, 1994Google Scholar
  35. 35.
    Nishiura, Y., Fujii, H.: Stability of singularly perturbed solutions to systems of reaction diffusion euations. SIAM J. Math. Anal. 18, 1726–1770 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Ouyang, T., Shi, J.: Exact multiplicity of positive solutions for a class of semilinear problems. J. Diff. Eqs. 146, 121–156 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Oshita, Y.: On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions. J. Diff. Eqs. 188, 110–134 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Oshita, Y., Ohnishi, I.: Standing pulse solutions for the FitzHugh-Nagumo euations. Jap. J. Ind. Appl. Math. 20, 101–115 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Sweers, G., Troy, W.C.: On the bifurcation curve for an elliptic system of FitzHugh-Nagumo type. Phys. D 177, 1–22 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Reinecke, C., Sweers, G.: Solutions with internal jump for an autonomous elliptic system of FitzHugh-Nagumo type. Math. Nachr. 254, 64–87 (2003)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Reinecke, C., Sweers, G.: Existence and uniqueness of solutions on bounded domains to a FitzHugh-Nagumo type elliptic system. Pacific J. Math. 197, 183–211 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Reinecke, C., Sweers, G.: A boundary layer solution to a semilinear elliptic system of FitzHugh-Nagumo type. C. R. Acad. Sci. Paris Ser. I Math. 329, 27–32 (1999)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Reinecke, C., Sweers, G.: Positive solution on R N to a system of elliptic equations of FitzHugh-Nagumo type. J. Diff. Eqs. 153, 292–312 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Wei, J., Winter, M.: Stationary solutions for the Cahn-Hilliard equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 459–492 (1998)zbMATHCrossRefGoogle Scholar
  45. 45.
    Wei, J., Winter, M.: Multiple boundary spike solutions for a wide class of singular perturbation problems. J. London Math. Soc. 59, 585–606 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Wei, J., Winter, M.: Symmetric and asymmetric multiple clusters in a reaction-diffusion system. NoDEA Nonlinear Differential Equations Appl. (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.School of Information Systems, Computing and MathematicsBrunel UniversityUxbridgeUnited Kingdom
  2. 2.Department of MathematicsThe Chinese University of Hong KongHong Kong

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