Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T18:01:06.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Poincaré–Bendixson theory for certain reaction–diffusion boundary-value problems

Published online by Cambridge University Press:  14 November 2011

Russell A. Smith
Russell A. Smith
Affiliation:
Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper studies orbits in a function space described by solutions of a system of reaction–diffusion equations in a bounded domain with a boundary condition of homogeneous Robin type. The omega-limit set of a bounded semi-orbit is shown to have a simple structure, provided that certain hypotheses are satisfied. For a two-dimensional system of Fitz-Hugh Nagumo type, these hypotheses yield explicit sufficient conditions for the existence of at least one periodic trajectory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 1 , 1994 , pp. 33 - 69
Copyright
Copyright © Royal Society of Edinburgh 1994

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

1Ahlfors, L. V.. Complex Analysis (New York: McGraw-Hill, 1966).Google Scholar
2Alikakos, N. D.. An application of the invariance principle to reaction diffusion equations. J. Differential Equations 33 (1979), 201–225.CrossRefGoogle Scholar
3Angenent, S. and Fiedler, B.. The dynamics of rotating waves in scalar reaction diffusion equations. Trans. Amer. Math. Soc. 307 (1988), 545–568.CrossRefGoogle Scholar
4Chafee, N. and Infante, E. F.. A bifurcation problem for nonlinear partial differential equations of parabolic type. Appl. Anal. 4 (1974), 17–37.CrossRefGoogle Scholar
5Chueh, K., Conley, C. and Smoller, J.. Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26 (1977), 373–392.CrossRefGoogle Scholar
6Conway, E., Hoff, D. and Smoller, J.. Large time behavior of solutions of systems of nonlinear reaction–diffusion equations. SIAM J. Appl. Math. 35 (1978), 1–16.CrossRefGoogle Scholar
7Courant, R. and Hilbert, D.. Methods of Mathematical Physics, Vol. 1 (New York: Wiley, 1953).Google Scholar
8Dautray, R. and Lions, J.-L.. Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3 (Berlin: Springer, 1990).Google Scholar
9Fiedler, B. and Mallet-Paret, J.. A Poincaré–Bendixson theorem for scalar reaction diffusion equations. Arch. Rational Mech. Anal. 107 (1989), 325–345.CrossRefGoogle Scholar
10Fife, P. C.. Pattern formation in reacting and diffusing systems. J. Chem. Phys. 64 (1976), 554–564.CrossRefGoogle Scholar
11Hale, J. K.. Large diffusivity and asymptotic behavior in parabolic systems. J. Math. Anal. Appl. 118 (1986), 455–466.CrossRefGoogle Scholar
12Henry, D.. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840 (New York: Springer, 1981).CrossRefGoogle Scholar
13Hurewicz, W. and Wallman, H.. Dimension Theory (Princeton: Princeton University Press, 1946).Google Scholar
14Ladyzhenskaya, O. A.. A dynamical system generated by the Navier–Stokes equation. J. Soviet Math. 27(1972), 458–479.Google Scholar
15Mallet-Paret, J. and Sell, G.. Inertial manifolds for reaction diffusion equations in higher space dimensions. J. Amer. Math. Soc. 1 (1988), 805–865.CrossRefGoogle Scholar
16Marion, M.. Attractors for reaction–diffusion equations: existence and estimate of their dimension. Appl. Anal. 25(1987), 101–147.CrossRefGoogle Scholar
17Nemytskii, V. V. and Stepanov, V. V.. Qualitative Theory of Differential Equations (Princeton: Princeton University Press, 1960).Google Scholar
18Pliss, V. A.. Nonlocal Problems of the Theory of Oscillations (New York: Academic Press, 1966).Google Scholar
19Romanov, A. V.. Asymptotic finite-dimensionality of semilinear parabolic equations. Soviet Math. Dokl. 40 (1990), 122–125.Google Scholar
20Smith, R. A.. Massera's convergence theorem for periodic nonlinear differential equations. J. Math. Anal. Appl. 120 (1986), 679–708.CrossRefGoogle Scholar
21Smith, R. A.. Convergence theorems for periodic retarded functional differential equations. Proc. London Math. Soc. 60 (1990), 581–608.CrossRefGoogle Scholar
22Smith, R. A.. Poincaré–Bendixson theory for certain retarded functional differential equations. Differential Integral Equations 5 (1992), 213–240.CrossRefGoogle Scholar
23Smoller, J.. Shock Waves and Reaction–Diffusion Equations (New York: Springer, 1983).CrossRefGoogle Scholar
24Temam, R.. Infinite Dimensional Dynamical Systems in Mechanics and Physics (New York: Springer, 1988).CrossRefGoogle Scholar
25Tyson, J. J. and Othmer, H. G.. The dynamics of feedback control circuits in biochemical pathways. Progr. Theoret. Biol. 5 (1978), 1–62.Google Scholar