Peak solutions for an elliptic system of FitzHugh-Nagumo type

Edward Norman Dancer; Shusen Yan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 4, page 679-709
  • ISSN: 0391-173X

Abstract

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The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.

How to cite

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Dancer, Edward Norman, and Yan, Shusen. "Peak solutions for an elliptic system of FitzHugh-Nagumo type." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.4 (2003): 679-709. <http://eudml.org/doc/84516>.

@article{Dancer2003,
abstract = {The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.},
author = {Dancer, Edward Norman, Yan, Shusen},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {679-709},
publisher = {Scuola normale superiore},
title = {Peak solutions for an elliptic system of FitzHugh-Nagumo type},
url = {http://eudml.org/doc/84516},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Dancer, Edward Norman
AU - Yan, Shusen
TI - Peak solutions for an elliptic system of FitzHugh-Nagumo type
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 4
SP - 679
EP - 709
AB - The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.
LA - eng
UR - http://eudml.org/doc/84516
ER -

References

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  1. [1] A. Bahri, “Critical points at infinity in some variational problems”, Research Notes in Mathematics 182, Longman-Pitman, 1989. Zbl0676.58021MR1019828
  2. [2] P. Clément – G. Sweers, Existence and multiplicity results for a semilinear eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 97-121. Zbl0662.35045MR937538
  3. [3] E. N. Dancer, A note on asymptotic uniqueness for some nonlinearities which change sign, Bull. Austral. Math. Soc. 61 (2000), 305-312. Zbl0945.35031MR1748710
  4. [4] E. N. Dancer – S. Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 (1999), 1177-1212. Zbl0948.35055MR1757072
  5. [5] E. N. Dancer – S. Yan, Singularly perturbed elliptic problem in exterior domains, J. Differential Integral Equations 13 (2000), 747-777. Zbl1038.35008MR1750049
  6. [6] E. N. Dancer – S. Yan, A minimization problem associated with elliptic system of FitzHugh-Nagumo type, Ann. Inst. H. Poincaré, Analyse Non Linéaire, to appear. Zbl1110.35019MR2047356
  7. [7] D. G. deFigueiredo – E. Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal. 17 (1986), 836-849. Zbl0608.35022MR846392
  8. [8] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophy. J. 1 (1961), 445-466. 
  9. [9] R. Gardner – L. A. Peletier, The set of positive solutions of semilinear equations in large ball, Proc. Royal Soc. Edinburgh 104A (1986), 53-72. Zbl0625.35030MR877892
  10. [10] A. Hodgkin – A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952), 500-544. 
  11. [11] G. A. Klaasen – E. Mitidieri, Standing wave solutions for a system derived from the FitzHugh-Nagumo equation for nerve conduction, SIAM J. Math. Anal. 17 (1986), 74-83. Zbl0593.35043MR819214
  12. [12] G. A. Klaasen – W. C. Troy, Stationary wave solutions of a system of reaction-diffusion equations derived from the FitzHugh-Nagumo equations, SIAM J. Appl. Math. 44 (1984), 96-110. Zbl0543.35051MR730003
  13. [13] J. S. Nagumo – S. Arimoto – Y. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio. Engineers 50 (1962), 2061-2070. 
  14. [14] W. M. Ni – I. Takagi – J. Wei, On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem, intermediate solution, Duke Math. J. 94 (1998), 597-618. Zbl0946.35007MR1639546
  15. [15] W. M. Ni – J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problem, Comm. Pure Appl. Math. 48 (1995), 731-768. Zbl0838.35009MR1342381
  16. [16] L. A. Peletier – J. Serrin, Uniqueness of positive solutions of semilinear equations in ℝ n , Arch. Rat. Mech. Anal. 81 (1983), 181-197. Zbl0516.35031MR682268
  17. [17] C. Reinecke – G. Sweers, A boundary layer solution to a semilinear elliptic system of FitzHugh-Nagumo type, C. R. Acad. Sci. Paris Sér. I. Math. 329 (1999), 27-32. Zbl0931.35046MR1703287
  18. [18] C. Reinecke – G. Sweers, A positive solution on ℝ N to a system of elliptic equations of FitzHugh-Nagumo type, J. Differential Equations 153 (1999), 292-312. Zbl0929.35042MR1683624
  19. [19] C. Reinecke – G. Sweers, Existence and uniqueness of solutions on bounded domains to a FitzHugh-Nagumo type elliptic system, Pacific J. Math. 197 (2001), 183-211. Zbl1066.35069MR1810215
  20. [20] C. Reinecke – G. Sweers, Solutions with internal jump for an autonomous elliptic system of FitzHugh-Nagumo type, Math. Nachr. 251 (2003), 64-87. Zbl1118.35009MR1960805
  21. [21] O. Rey, The role of the Green’s function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1-52. Zbl0786.35059MR1040954
  22. [22] G. Sweers – W. Troy, On the bifurcation curve for an elliptic system of FitzHugh-Nagumo type, Physica D: Nonlinear Phenomena 177 (2003), 1-22. Zbl1082.35068MR1965323

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