Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T05:13:39.174Z Has data issue: false hasContentIssue false
  • English
  • Français

A CONLEY INDEX CALCULATION

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  13 August 2009

E. N. DANCER
E. N. DANCER*
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We discuss a Conley index calculation which is of importance in population models with large interaction. In particular, we prove that a certain Conley index is trivial.

Keywords

Conley indexconnecting orbits

MSC classification

Secondary: 35K55: Nonlinear parabolic equations 35K57: Reaction-diffusion equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 3 , December 2009 , pp. 510 - 520
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Alessandrini, G. and Vessella, S., ‘Remark on the strong unique continuation property for parabolic operators’, Proc. Amer. Math. Soc. 132 (2004), 499–501 (electronic).CrossRefGoogle Scholar
[2]Bardos, C. and Tartar, L., ‘Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines’, Arch. Ration. Mech. Anal. 50 (1973), 10–25.CrossRefGoogle Scholar
[3]Caffarelli, L. A. and Friedman, A., ‘Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations’, J. Differential Equations 60 (1985), 420–433.CrossRefGoogle Scholar
[4]Cartan, H., Calcul Différentiel (Hermann, Paris, 1967).Google Scholar
[5]Chen, X.-Y., ‘On the scaling limits at zeros of solutions of parabolic equations’, J. Differential Equations 147 (1998), 355–382.CrossRefGoogle Scholar
[6]Conley, C., Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38 (American Mathematical Society, Providence, RI, 1978).CrossRefGoogle Scholar
[7]Dancer, E. N., ‘On positive solutions of some pairs of differential equations. II’, J. Differential Equations 60 (1985), 236–258.CrossRefGoogle Scholar
[8]Dancer, E. N., ‘On connecting orbits for competing species equations with large interactions’, Topol. Methods Nonlinear Anal. 24 (2004), 1–19.CrossRefGoogle Scholar
[9]Dancer, E. N. and Du, Y. H., ‘Competing species equations with diffusion, large interactions, and jumping nonlinearities’, J. Differential Equations 114 (1994), 434–475.CrossRefGoogle Scholar
[10]Dancer, E. N. and Zhang, Z., ‘Dynamics of Lotka–Volterra competition systems with large interaction’, J. Differential Equations 182 (2002), 470–489.CrossRefGoogle Scholar
[11]Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840 (Springer, Berlin, 1981).CrossRefGoogle Scholar
[12]Lieberman, G. M., Second Order Parabolic Differential Equations (World Scientific, River Edge, NJ, 1996).CrossRefGoogle Scholar
[13]Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16 (Birkhäuser, Basel, 1995).Google Scholar
[14]McCord, C., ‘The connection map for attractor–repeller pairs’, Trans. Amer. Math. Soc. 307 (1988), 195–203.CrossRefGoogle Scholar
[15]Rybakowski, K. P., The Homotopy Index and Partial Differential Equations (Springer, Berlin, 1987).CrossRefGoogle Scholar
[16]Vol’pert, A. and Vol’pert, V., ‘The Fredholm property of elliptic operators in unbounded domains’, Tr. Mosk. Mat. Obs. 67 (2006), 148–227.Google Scholar