Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-16T12:09:24.237Z Has data issue: false hasContentIssue false
  • English
  • Français

Generic properties of equilibria of reaction-diffusion equations with variable diffusion

Published online by Cambridge University Press:  14 November 2011

Carlos Rocha
Carlos Rocha
Affiliation:
Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

It is shown that, generically, scalar one-dimensional parabolic equations ut = (a2(x)ux)x + f(u), x ∈ [0, 1], with Neumann boundary conditions, have all the equilibrium solutions hyperbolic.

Moreover, the bifurcations of these equilibria are generically of the saddle-node type.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 101 , Issue 1-2 , 1985 , pp. 45 - 55
Copyright
Copyright © Royal Society of Edinburgh 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Abraham, R. and Robbin, J.. Transversal Mappings and Flows (New York: Benjamin, 1967).Google Scholar
2Brunovsky, P. and Chow, S.-N.. Generic properties of stationary solutions of reaction-diffusion equations. J. Differential Equations 53 (1984), 1–23.CrossRefGoogle Scholar
3Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
4Fusco, G. and Hale, J. K.. Stable equilibria in a scalar parabolic equation with variable diffusion. SIAM J. Math. Anal, to appear.Google Scholar
5Hale, J. K. and Massat, R.. Asymptotic behaviour of gradient-like systems. (Univ. Fla. Symp. Dyn. Sys. If) (New York: Academic Press, 1982).Google Scholar
6Hale, J. K. and Rocha, C.. Bifurcations in a parabolic equation with variable diffusion. J. Nonlinear Anal. 9 (1985), 479–494.CrossRefGoogle Scholar
7Henry, D.. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).Google Scholar
8Henry, D.. Some infinite-dimensional Morse-Smale systems denned by parabolic partial differential equations, preprint.Google Scholar
9Matano, H.. Convergence of solutions of one-dimensional semilinear parabolic equation. J. Math. Kyoto Univ. 18 (1978), 163–183.Google Scholar
10Peixoto, M. M.. On a generic theory of end point boundary value problems. An. Acad. Brasil Ciênc. 41 (1969), 1–6.Google Scholar
11Peixoto, M. M.. On end-point boundary value problems. J. Differential Equations 44 (1982), 273–280.CrossRefGoogle Scholar
12Smoller, J. and Wasserman, A.. Generic bifurcation of steady-state solutions. J. Differential Equations 52 (1984), 432–438.CrossRefGoogle Scholar