Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions

Jan Eisner

Mathematica Bohemica (2000)

  • Volume: 125, Issue: 4, page 385-420
  • ISSN: 0862-7959

Abstract

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Sufficient conditions for destabilizing effects of certain unilateral boundary conditions and for the existence of bifurcation points for spatial patterns to reaction-diffusion systems of the activator-inhibitor type are proved. The conditions are related with the mollification method employed to overcome difficulties connected with empty interiors of appropriate convex cones.

How to cite

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Eisner, Jan. "Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions." Mathematica Bohemica 125.4 (2000): 385-420. <http://eudml.org/doc/29426>.

@article{Eisner2000,
abstract = {Sufficient conditions for destabilizing effects of certain unilateral boundary conditions and for the existence of bifurcation points for spatial patterns to reaction-diffusion systems of the activator-inhibitor type are proved. The conditions are related with the mollification method employed to overcome difficulties connected with empty interiors of appropriate convex cones.},
author = {Eisner, Jan},
journal = {Mathematica Bohemica},
keywords = {bifurcation; spatial patterns; reaction-diffusion system; mollification; inclusions; bifurcation; spatial patterns; reaction-diffusion system; mollification; inclusions},
language = {eng},
number = {4},
pages = {385-420},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions},
url = {http://eudml.org/doc/29426},
volume = {125},
year = {2000},
}

TY - JOUR
AU - Eisner, Jan
TI - Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions
JO - Mathematica Bohemica
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 125
IS - 4
SP - 385
EP - 420
AB - Sufficient conditions for destabilizing effects of certain unilateral boundary conditions and for the existence of bifurcation points for spatial patterns to reaction-diffusion systems of the activator-inhibitor type are proved. The conditions are related with the mollification method employed to overcome difficulties connected with empty interiors of appropriate convex cones.
LA - eng
KW - bifurcation; spatial patterns; reaction-diffusion system; mollification; inclusions; bifurcation; spatial patterns; reaction-diffusion system; mollification; inclusions
UR - http://eudml.org/doc/29426
ER -

References

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  1. E. N. Dancer, 10.1512/iumj.1974.23.23087, Indiana Univ. Math. J. 23 (1974), 1069-1076. (1974) Zbl0276.47051MR0348567DOI10.1512/iumj.1974.23.23087
  2. P. Drábek M. Kučera M. Míková, Bifurcation points of reaction-diffusion systems with unilateral conditions, Czechoslovak Math. J. 35 (1985), 639-660. (1985) MR0809047
  3. P. Drábek M. Kučera, Eigenvalues of inequalities of reaction-diffusion type and destabilizing effect of unilateral conditions, Czechoslovak Math. J. 36 (1986), 116-130. (1986) MR0822872
  4. P. Drábek M. Kučera, 10.1016/0362-546X(88)90051-X, Nonlinear Anal. 12 (1988), 1173-1192. (1988) MR0969497DOI10.1016/0362-546X(88)90051-X
  5. G. Duvaut J. L. Lions, Les Inéquations en Mechanique et en Physique, Dunod, Paris, 1972. (1972) MR0464857
  6. J. Eisner M. Kučera, 10.1023/A:1022203129542, Appl. of Math. 42 (1997), 421-449. (1997) MR1475051DOI10.1023/A:1022203129542
  7. J. Eisner, Critical and bifurcation points of reaction-diffusion systems with conditions given by inclusions, Preprint Math. Inst. Acad. Sci. of the Czech Republic, No. 118, Praha, 1997. To appear in Nonlinear Anal. (1997) MR1845578
  8. J. Eisner M. Kučera, Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions, Fields Inst. Commun. 25 (2000), 239-256. MR1759546
  9. J. Eisner, Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions. Part II, Examples, To appear in Math. Bohem. MR1826476
  10. S. Fučík A. Kufner, Nonlinear Differential Equations, Elsevier, Amsterdam, 1980. (1980) MR0558764
  11. A. Gierer H. Meinhardt, 10.1007/BF00289234, Kybernetik 12 (1972), 30-39. (1972) DOI10.1007/BF00289234
  12. M. Kučera J. Neustupa, Destabilizing effect of unilateral conditions in reaction-diffusion systems, Comment. Math. Univ. Carol. 27 (1986), 171-187. (1986) MR0843429
  13. M. Kučera, 10.1007/BFb0076074, Equadiff 6 (J. Vosmanský, M. Zlámal, eds.). Brno, Universita J. E. Purkyně, 1986, pp. 227-234. (1986) MR0877129DOI10.1007/BFb0076074
  14. M. Kučera, A global continuation theorem for obtaining eigenvalues and bifurcation points, Czechoslovak Math. J. 38 (1988), 120-137. (1988) MR0925946
  15. M. Kučera M. Bosák, Bifurcation for quasi-variational inequalities of reaction diffusion type, Proceedings of EQUAM 92, International Conference on Differential Equations and Mathematical Modelling, Varenna 1992. SAACM 3, 1993, pp. 121-127. (1992) 
  16. M. Kučera, Bifurcation of solutions to reaction-diffusion system with unilateral conditions, Navier-Stokes Equations and Related Topics (A. Sequeira, ed.). Plenum Press, New York, 1995, pp. 307-322. (1995) MR1373224
  17. M. Kučera, 10.1016/0362-546X(95)00055-Z, Nonlinear Anal. 27 (1996), 249-260. (1996) MR1391435DOI10.1016/0362-546X(95)00055-Z
  18. M. Kučera, 10.1023/A:1022411501260, Czechoslovak Math. J. 47 (1997), 469-486. (1997) MR1461426DOI10.1023/A:1022411501260
  19. M. Kučera, Bifurcation of solutions to reaction-diffusion system with conditions described by inequalities and inclusions, Nonlinear Anal. Theory Methods Appl. 30 (1997), 3683-3694. (1997) MR1602910
  20. M. Mimura Y. Nishiura M. Yamaguti, 10.1111/j.1749-6632.1979.tb29492.x, Ann. N.Y. Acad. Sci. 316 (1979), 490-521. (1979) MR0556853DOI10.1111/j.1749-6632.1979.tb29492.x
  21. H. Meinhardt, The algorithmic beauty of sea shells, Springer-Verlag, Berlin, 1996. (1996) MR1325695
  22. J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. (1993) MR1239892
  23. J. Nečas, Les méthodes directes en théorie des équations elliptiques, Praha, Academia, 1967. (1967) MR0227584
  24. L. Nirenberg, Topics in Nonlinear Functional Analysis, New York, 1974. (1974) Zbl0286.47037MR0488102
  25. Y. Nishiura, 10.1137/0513037, SIAM J. Math. Analysis 13 (1982), 555-593. (1982) Zbl0505.76103MR0661590DOI10.1137/0513037
  26. P. Quittner, Bifurcation points and eigenvalues of inequalities of reaction-diffusion type, J. Reine Angew. Math. 380 (1987), 1-13. (1987) Zbl0617.35053MR0916198
  27. A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. (1952), 37-72. (1952) 

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