Article
MSC:
35B32,
35J85,
35K40,
35K57,
35K58,
47H04,
47N20 | MR 1802290 | Zbl 0963.35016 | DOI: 10.21136/MB.2000.126272
Full entry |
PDF
(7.8 MB)
Feedback
PDF
(7.8 MB)
Feedback
Keywords:
bifurcation; spatial patterns; reaction-diffusion system; mollification; inclusions
bifurcation; spatial patterns; reaction-diffusion system; mollification; inclusions
Summary:
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.
Related articles:
References:
[1] E. N. Dancer: On the structure of solutions of non-linear eigenvalue problems. Indiana Univ. Math. J. 23 (1974), 1069-1076. DOI 10.1512/iumj.1974.23.23087 | MR 0348567 | Zbl 0276.47051
[2] P. Drábek M. Kučera M. Míková: Bifurcation points of reaction-diffusion systems with unilateral conditions. Czechoslovak Math. J. 35 (1985), 639-660. MR 0809047
[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. MR 0822872
[4] P. Drábek M. Kučera: Reaction-diffusion systems: Destabilizing effect of unilateral conditions. Nonlinear Anal. 12 (1988), 1173-1192. DOI 10.1016/0362-546X(88)90051-X | MR 0969497
[5] G. Duvaut J. L. Lions: Les Inéquations en Mechanique et en Physique. Dunod, Paris, 1972. MR 0464857
[6] J. Eisner M. Kučera: Spatial patterns for reaction-diffusion systems with conditions described by inclusions. Appl. of Math. 42 (1997), 421-449. DOI 10.1023/A:1022203129542 | MR 1475051
[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. MR 1845578
[8] J. Eisner M. Kučera: Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions. Fields Inst. Commun. 25 (2000), 239-256. MR 1759546
[9] J. Eisner: Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions. Part II, Examples. To appear in Math. Bohem. MR 1826476
[10] S. Fučík A. Kufner: Nonlinear Differential Equations. Elsevier, Amsterdam, 1980. MR 0558764
[11] A. Gierer H. Meinhardt: A theory of biological pattern formation. Kybernetik 12 (1972), 30-39. DOI 10.1007/BF00289234
[12] M. Kučera J. Neustupa: Destabilizing effect of unilateral conditions in reaction-diffusion systems. Comment. Math. Univ. Carol. 27 (1986), 171-187. MR 0843429
[13] M. Kučera: Stability and bifurcation problems for reaction-diffusion system with unilateral conditions. Equadiff 6 (J. Vosmanský, M. Zlámal, eds.). Brno, Universita J. E. Purkyně, 1986, pp. 227-234. DOI 10.1007/BFb0076074 | MR 0877129
[14] M. Kučera: A global continuation theorem for obtaining eigenvalues and bifurcation points. Czechoslovak Math. J. 38 (1988), 120-137. MR 0925946
[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.
[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. MR 1373224
[17] M. Kučera: Reaction-diffusion systems: Bifurcation and stabilizing effect of conditions given by inclusions. Nonlinear Anal. 27 (1996), 249-260. DOI 10.1016/0362-546X(95)00055-Z | MR 1391435
[18] M. Kučera: Reaction-diffusion systems: Stabilizing effect of conditions described by quasivariational inequalities. Czechoslovak Math. J. 47 (1997), 469-486. DOI 10.1023/A:1022411501260 | MR 1461426
[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. MR 1602910
[20] M. Mimura Y. Nishiura M. Yamaguti: Some diffusive prey and predator systems and their bifurcation problems. Ann. N.Y. Acad. Sci. 316 (1979), 490-521. DOI 10.1111/j.1749-6632.1979.tb29492.x | MR 0556853
[21] H. Meinhardt: The algorithmic beauty of sea shells. Springer-Verlag, Berlin, 1996. MR 1325695
[22] J. D. Murray: Mathematical Biology. Springer-Verlag, Berlin, 1993. MR 1239892
[23] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Praha, Academia, 1967. MR 0227584
[24] L. Nirenberg: Topics in Nonlinear Functional Analysis. New York, 1974. MR 0488102 | Zbl 0286.47037
[25] Y. Nishiura: Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Analysis 13 (1982), 555-593. DOI 10.1137/0513037 | MR 0661590 | Zbl 0505.76103
[26] P. Quittner: Bifurcation points and eigenvalues of inequalities of reaction-diffusion type. J. Reine Angew. Math. 380 (1987), 1-13. MR 0916198 | Zbl 0617.35053
[27] A. M. Turing: The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. (1952), 37-72.
Search
Partner of
