Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-18T20:22:01.541Z Has data issue: false hasContentIssue false
  • English
  • Français

Centre manifolds for partial differential equations with delays*

Published online by Cambridge University Press:  14 November 2011

Xiaodong Lin ,
Joseph W.-H. So  and
Jianhong Wu
Xiaodong Lin
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Joseph W.-H. So
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Jianhong Wu
Affiliation:
Department of Mathematics, York University, North York, Ontario, Canada M3J IP3
Get access Rights & Permissions [Opens in a new window]

Synopsis

A centre manifold theory for reaction-diffusion equations with temporal delays is developed. Besides an existence proof, we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order. As an illustration, this reduction procedure is applied to the Hutchinson equation with diffusion.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 3-4 , 1992 , pp. 237 - 254
Copyright
Copyright © Royal Society of Edinburgh 1992

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

1Chow, S.-N. and Lu, K.. Invariant manifolds for flows in Banach spaces. J. Differential Equations 74 (1988), 285–317.CrossRefGoogle Scholar
2Fitzgibbon, W. E.. Semilinear functional differential equations in Banach space J. Differential Equations 29 (1978), 1–14.CrossRefGoogle Scholar
3Hale, J. K.. Theory of Functional Differential Equations (New York: Springer, 1977).CrossRefGoogle Scholar
4Hale, J. K.. Flows on centre manifolds for scalar functional differential equations. Proc. Roy. Soc. Edinburgh Sect A 101 (1985), 193–201.CrossRefGoogle Scholar
5Hale, J. K. and Meyer, K. R.. A Class of Functional Differential Equations of Neutral Type, Memoir Amer. Math. Soc. 76 (Providence: American Mathematical Society, 1967).Google Scholar
6Lee, E. B. and Markus, L.. Foundations of Optimal Control Systems (New York: John Wiley, 1967).Google Scholar
7Martin, R. and Smith, H. L.. Abstract functional differential equations and reaction-diffusion systems. Trans. Amer. Math. Soc. 321 (1990), 1–44.Google Scholar
8Martin, R. H. and Smith, H. L.. Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence. J. Reine Angew. Math. 413 (1991), 1–35.Google Scholar
9Memory, M. C.. Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion. SIAM J. Math. Anal. 20 (1989), 533–546.CrossRefGoogle Scholar
10Memory, M. C.. Stable and unstable manifolds for partial functional differential equations. Nonlinear Anal. 16 (1991), 131–142.CrossRefGoogle Scholar
11Memory, M. C.. Invariant manifolds for partial functional differential equations. Proc. 2nd Int. Conf. Mathematical Population Dynamics (to appear).Google Scholar
12Morita, Y.. Destabilization of periodic solutions arising in delay-diffusion systems in several space dimension. Japan J. Appl. Math. 1 (1984), 39–65.CrossRefGoogle Scholar
13Travis, C. C. and Webb, G. F.. Existence and stability for partial functional differential equations. Trans. Amer. Math. Soc. 200 (1974) 395–418.CrossRefGoogle Scholar
14Travis, C. C. and Webb, G. F.. Existence, stability, and compactness in the a-norm for partial functional differential equations. Trans. Amer. Math. Soc. 240 (1978), 129–143.Google Scholar
15Yoshida, K.. The Hopf bifurcation and its stability for semilinear differential equations with time delay arising in ecology. Hiroshima Math. J. 12 (1982), 321–348.CrossRefGoogle Scholar