Title:
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Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions (English) |
Author:
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Eisner, Jan |
Language:
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English |
Journal:
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Mathematica Bohemica |
ISSN:
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0862-7959 (print) |
ISSN:
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2464-7136 (online) |
Volume:
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125 |
Issue:
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4 |
Year:
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2000 |
Pages:
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385-420 |
Summary lang:
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English |
. |
Category:
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math |
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Summary:
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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. (English) |
Keyword:
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bifurcation |
Keyword:
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spatial patterns |
Keyword:
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reaction-diffusion system |
Keyword:
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mollification |
Keyword:
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inclusions |
MSC:
|
35B32 |
MSC:
|
35J85 |
MSC:
|
35K40 |
MSC:
|
35K57 |
MSC:
|
35K58 |
MSC:
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47H04 |
MSC:
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47N20 |
idZBL:
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Zbl 0963.35016 |
idMR:
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MR1802290 |
DOI:
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10.21136/MB.2000.126272 |
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Date available:
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2009-09-24T21:45:00Z |
Last updated:
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2020-07-29 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/126272 |
. |
Reference:
|
[1] E. N. Dancer: On the structure of solutions of non-linear eigenvalue problems.Indiana Univ. Math. J. 23 (1974), 1069-1076. Zbl 0276.47051, MR 0348567, 10.1512/iumj.1974.23.23087 |
Reference:
|
[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 |
Reference:
|
[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 |
Reference:
|
[4] P. Drábek M. Kučera: Reaction-diffusion systems: Destabilizing effect of unilateral conditions.Nonlinear Anal. 12 (1988), 1173-1192. MR 0969497, 10.1016/0362-546X(88)90051-X |
Reference:
|
[5] G. Duvaut J. L. Lions: Les Inéquations en Mechanique et en Physique.Dunod, Paris, 1972. MR 0464857 |
Reference:
|
[6] J. Eisner M. Kučera: Spatial patterns for reaction-diffusion systems with conditions described by inclusions.Appl. of Math. 42 (1997), 421-449. MR 1475051, 10.1023/A:1022203129542 |
Reference:
|
[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 |
Reference:
|
[8] J. Eisner M. Kučera: Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions.Fields Inst. Commun. 25 (2000), 239-256. MR 1759546 |
Reference:
|
[9] J. Eisner: Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions. Part II, Examples.To appear in Math. Bohem. MR 1826476 |
Reference:
|
[10] S. Fučík A. Kufner: Nonlinear Differential Equations.Elsevier, Amsterdam, 1980. MR 0558764 |
Reference:
|
[11] A. Gierer H. Meinhardt: A theory of biological pattern formation.Kybernetik 12 (1972), 30-39. 10.1007/BF00289234 |
Reference:
|
[12] M. Kučera J. Neustupa: Destabilizing effect of unilateral conditions in reaction-diffusion systems.Comment. Math. Univ. Carol. 27 (1986), 171-187. MR 0843429 |
Reference:
|
[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. MR 0877129, 10.1007/BFb0076074 |
Reference:
|
[14] M. Kučera: A global continuation theorem for obtaining eigenvalues and bifurcation points.Czechoslovak Math. J. 38 (1988), 120-137. MR 0925946 |
Reference:
|
[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. |
Reference:
|
[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 |
Reference:
|
[17] M. Kučera: Reaction-diffusion systems: Bifurcation and stabilizing effect of conditions given by inclusions.Nonlinear Anal. 27 (1996), 249-260. MR 1391435, 10.1016/0362-546X(95)00055-Z |
Reference:
|
[18] M. Kučera: Reaction-diffusion systems: Stabilizing effect of conditions described by quasivariational inequalities.Czechoslovak Math. J. 47 (1997), 469-486. MR 1461426, 10.1023/A:1022411501260 |
Reference:
|
[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 |
Reference:
|
[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. MR 0556853, 10.1111/j.1749-6632.1979.tb29492.x |
Reference:
|
[21] H. Meinhardt: The algorithmic beauty of sea shells.Springer-Verlag, Berlin, 1996. MR 1325695 |
Reference:
|
[22] J. D. Murray: Mathematical Biology.Springer-Verlag, Berlin, 1993. MR 1239892 |
Reference:
|
[23] J. Nečas: Les méthodes directes en théorie des équations elliptiques.Praha, Academia, 1967. MR 0227584 |
Reference:
|
[24] L. Nirenberg: Topics in Nonlinear Functional Analysis.New York, 1974. Zbl 0286.47037, MR 0488102 |
Reference:
|
[25] Y. Nishiura: Global structure of bifurcating solutions of some reaction-diffusion systems.SIAM J. Math. Analysis 13 (1982), 555-593. Zbl 0505.76103, MR 0661590, 10.1137/0513037 |
Reference:
|
[26] P. Quittner: Bifurcation points and eigenvalues of inequalities of reaction-diffusion type.J. Reine Angew. Math. 380 (1987), 1-13. Zbl 0617.35053, MR 0916198 |
Reference:
|
[27] A. M. Turing: The chemical basis of morphogenesis.Phil. Trans. Roy. Soc. (1952), 37-72. |
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