Spatial patterns for reaction-diffusion systems with conditions described by inclusions

Jan Eisner; Milan Kučera

Applications of Mathematics (1997)

  • Volume: 42, Issue: 6, page 421-449
  • ISSN: 0862-7940

Abstract

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We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.

How to cite

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Eisner, Jan, and Kučera, Milan. "Spatial patterns for reaction-diffusion systems with conditions described by inclusions." Applications of Mathematics 42.6 (1997): 421-449. <http://eudml.org/doc/32991>.

@article{Eisner1997,
abstract = {We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.},
author = {Eisner, Jan, Kučera, Milan},
journal = {Applications of Mathematics},
keywords = {reaction-diffusion systems; variational inequalities; inclusions; bifurcation; stationary solutions; spatial patterns; destabilizing effect; variational inequalities inclusions; stationary solutions},
language = {eng},
number = {6},
pages = {421-449},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Spatial patterns for reaction-diffusion systems with conditions described by inclusions},
url = {http://eudml.org/doc/32991},
volume = {42},
year = {1997},
}

TY - JOUR
AU - Eisner, Jan
AU - Kučera, Milan
TI - Spatial patterns for reaction-diffusion systems with conditions described by inclusions
JO - Applications of Mathematics
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 42
IS - 6
SP - 421
EP - 449
AB - We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.
LA - eng
KW - reaction-diffusion systems; variational inequalities; inclusions; bifurcation; stationary solutions; spatial patterns; destabilizing effect; variational inequalities inclusions; stationary solutions
UR - http://eudml.org/doc/32991
ER -

References

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  1. 10.1512/iumj.1974.23.23087, Indiana Univ. Math. J. 23 (1974), 1069–1076. (1974) MR0348567DOI10.1512/iumj.1974.23.23087
  2. Les Inéquations en Mechanique et en Physique, Dunod, Paris, 1972. (1972) MR0464857
  3. Bifurcation points of reaction-diffusion systems with unilateral conditions, Czech. Math. J. 35 (1985), 639–660. (1985) MR0809047
  4. Reaction-diffusion systems: Destabilizing effect of unilateral conditions, Nonlinear Analysis, Theory, Methods, Applications 12 (1988), 1173-1192. (1988) MR0969497
  5. Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions, (to appear). (to appear) Zbl0977.35020MR1802290
  6. Nonlinear Differential Equations, Elsevier, Amsterdam, 1980. (1980) MR0558764
  7. 10.1007/BF00289234, Kybernetik 12 (1972), 30–3. (1972) DOI10.1007/BF00289234
  8. 10.1007/BF00248417, Arch. Rat. Mech. Analysis 57 (1974), 150–165. (1974) MR0442405DOI10.1007/BF00248417
  9. Bifurcation points of variational inequalities, Czechoslovak Math. J. 32 (1982), 208–226. (1982) MR0654057
  10. 10.1023/A:1022411501260, Czechoslovak Math. J. 47 (122) (1997), 469–486. (1997) MR1461426DOI10.1023/A:1022411501260
  11. Bifurcation of solutions to reaction-diffusion system with conditions described by inequalities and inclusions, Proceedings of the Second World Congress of Nonlinear Analysts, Athens, 1996. (1996) MR1602910
  12. Bifurcation for quasi-variational inequalities of reaction-diffusion type, SAACM 3 (1993), 111–127. (1993) 
  13. Bifurcation of solutions to reaction-diffusion systems with unilateral conditions, Navier-Stokes Equations and related Topics, by A. Sequeira (eds.), Plenum Press, New York, 1995, pp. 307–322. (1995) MR1373224
  14. 10.1016/0362-546X(95)00055-Z, Nonlin. Anal., T.M.A. 27 (1996), 249–260. (1996) MR1391435DOI10.1016/0362-546X(95)00055-Z
  15. 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
  16. Mathematical Biology 19, Springer-Verlag, 1993. (1993) MR1239892
  17. 10.1137/0513037, SIAM J. Math. Analysis 13 (1982), 555–593. (1982) Zbl0505.76103MR0661590DOI10.1137/0513037
  18. Topics in Nonlinear Functional Analysis, Academic Press, New York, 1974. (1974) Zbl0286.47037MR0488102
  19. Spectral analysis of variational inequalities, Commentat. Math. Univ. Carol. 27 (1986), 605–629. (1986) MR0873631
  20. Bifurcation points and eigenvalues of inequalities of reaction-diffusion type, J. reine angew. Math. 380 (1987), 1–13. (1987) Zbl0617.35053MR0916198
  21. 10.1016/0022-1236(71)90030-9, J. Funct. Anal. 7 (1987), 487–513. (1987) MR0301587DOI10.1016/0022-1236(71)90030-9
  22. Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics 309, Springer-Verlag, Berlin-Heidelberg-New York, 1973. (1973) Zbl0248.35003MR0463624
  23. The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. (B) (1952), 37–72. (1952) 

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