Summary
It is now well known that the number of parameters and symmetries of an equation affects the bifurcation structure of that equation. The bifurcation behavior of reaction-diffusion equations on certain domains with certain boundary conditions isnongeneric in the sense that the bifurcation of steady states in these equations is not what would be expected if one considered only the number of parameters in the equations and the type of symmetries of the equations. This point was made previously in work by Fujii, Mimura, and Nishiura [6] and Armbruster and Dangelmayr [1], who considered reaction-diffusion equations on an interval with Neumann boundary conditions.
As was pointed out by Crawford et al. [5], the source of this nongenericity is that reaction-diffusion equations are invariant under translations and reflections of the domain and, depending on boundary conditions, may naturally and uniquely be extended to larger domains withlarger symmetry groups. These extra symmetries are the source of the nongenericity. In this paper we consider in detail the steady-state bifurcations of reaction-diffusion equations defined on the hemisphere with Neumann boundary conditions along the equator. Such equations have a naturalO(2)-symmetry but may be extended to the full sphere where the natural symmetry group isO(3). We also determine a large class of partial differential equations and domains where this kind of extension is possible for both Neumann and Dirichlet boundary conditions.
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References
D. Armbruster and G. Dangelmayr. Coupled stationary bifurcations in nonflux boundary value problems,Math. Proc. Camb. Phil. Soc. 101 (1987) 167–192.
L. Bauer, E. L. Reiss, and H. B. Keller. Axisymmetric buckling of rigidly clamped hemispherical shells,Int. J. Non-Linear Mechanics 8 (1973) 31–39; Axisymmetric buckling of hollow spheres and hemispheres,Commun. Pure Appl. Math. 23 (1970) 529–568.
G. E. Bredon.Introduction to Compact Transformation Groups, Academic Press, New York and London, 1972.
P. Chossat, R. Lauterbach, and I. Melbourne. Steady-state bifurcations with O(3) symmetry,Arch. Rational Mech. and Anal. 113 (4) (1991) 313–376.
J. D. Crawford, M. Golubitsky, M. G. M. Gomes, E. Knobloch, and I. N. Stewart. Boundary conditions as symmetry constraints, inSingularity Theory and Its Applications, Symposium Proceedings Warwick 1989, vol. 2, Springer-Verlag, Heidelberg. To appear 1991.
H. Fujii, M. Mimura, and Y. Nishiura. A picture of the global bifurcation diagram in ecological interacting and diffusing systems,Physica 5D (1982) 1–42.
D. Gilbarg and N. S. Trudinger.Elliptic Partial Differential Equations of Second Order, Grundlehren der math. Wissenschaften 224, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
M. Golubitsky, I. N. Stewart, and D. G. Schaeffer.Singularities and Groups in Bifurcation Theory, Vol. II, Appl. Math. Sci. Ser. 69, Springer-Verlag, New York, 1988.
M. G. M. Gomes. Steady-state Mode Interactions in Rectangular Domains, M.Sc. Thesis, University of Warwick, 1989.
M. G. M. Gomes, R.M. Roberts, and I. N. Stewart. Singularity theory for boundary value problems in rectangular domains. In preparation.
S. Helgason.Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, San Francisco, London, 1978.
E. Ihrig and M. Golubitsky. Pattern selection withO(3) symmetry,Physica 12D (1984) 1–33.
S. Kobayashi and K. Nomizu.Foundations of Differential Geometry, Volume I, Interscience Publishers, New York and London, 1963.
O. A. Ladyzhenskaya and N. N. Ural'tseva.Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, Vol. 46, Academic Press, New York and London, 1968.
S. Mizohara.The Theory of Partial Differential Equations, Cambridge University Press, Cambridge, 1973.
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Communicated by Jerrold Marsden
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Field, M., Golubitsky, M. & Stewart, I. Bifurcations on hemispheres. J Nonlinear Sci 1, 201–223 (1991). https://doi.org/10.1007/BF01209066
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DOI: https://doi.org/10.1007/BF01209066