Journal of Dynamics and Differential Equations

, Volume 6, Issue 4, pp 613–629 | Cite as

Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations

  • Pedro Freitas


The stability of stationary solutions of nonlocal reaction-diffusion equations on a bounded intervalJ of the real line with homogeneous Dirichlet boundary conditions is studied. It is shown that it is possible to have stable stationary solutions which change sign once onJ in the case of constant diffusion when the reaction term does not depend explicitly on the space variable. The problem of the possible types of stable solutions that may exist is considered. It is also shown that Matano's result on the lap-number is still true in the case of nonlocal problems.

Key words

nonlocal reaction-diffusion equations stationary solutions bifurcation from simple eigenvalues 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Brunovský, P., and Fiedler, B. (1988). Connecting orbits in scalar reaction-diffusion equations. In Kirchgraber, U., and Walther, H. O. (eds.),Dynamics Reported, Vol. 1, J. Wiley and Sons and B. G. Teubner, Stuttgart.Google Scholar
  2. Casten, R., and Holland, C. (1978). Instability results for reaction-diffusion equations with Neumann boundary conditions.J. Diff. Eqs. 27, 266–273.Google Scholar
  3. Chafee, N. (1981). The electric balast resistor: Homogeneous and nonhomogeneous equilibria. In de Mottoni, P., and Salvadori, L. (eds.),Nonlinear Differential Equations: Invariance, Stability and Bifurcation, AAcademic Press, New York, pp. 97–127.Google Scholar
  4. Chafee, N., and Infante, E. F. (1974). A bifurcation problem for a nonlinear partial differential equation of parabolic type.Appl. Anal. 4, 17–37.Google Scholar
  5. Crandall, M. G., and Rabinowitz, P. H. (1973). Bifurcation, perturbation of simple eigenvalues, and linearized stability.Arch. Rat. Mech. Anal. 52, 161–180.Google Scholar
  6. Fiedler, B., and Poláčik, P. (1990). Complicated dynamics of scalar reaction diffusion equations with a nonlocal term.Proc. Roy. Soc. Edinburgh (A) 115, parts 1/2, 167–192.Google Scholar
  7. Freitas, P. (1993). A nonlocal Sturm-Liouville eigenvalue problem.Proc. Roy. Soc. Edinburgh (A) 124, 169–188.Google Scholar
  8. Furter, J., and Grinfeld, M. (1989). Local vs. non-local interactions in population dynamics.J. Math. Biol. 27, 65–80.Google Scholar
  9. Hale, J. K., and Sakamoto, K. (1987).Shadow Systems and Attractors in Reaction-Diffusion Equations, LCDS Report 87-28, Brown University, Providence, RI.Google Scholar
  10. Kato, T. (1980).Perturbation Theory for Linear Operators, Springer, New York.Google Scholar
  11. Lacey, A. (1993). Thermal runaway due to ohmic heating: General results and a one-dimensional problem (preprint).Google Scholar
  12. Levitan, B. M., and Sargsjan, I. S. (1975).Introduction to Spectral Theory, Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, RI.Google Scholar
  13. Matano, H. (1979).Asymptotic Behaviour and Stability of Solutions of Semilinear Diffusion Equations, Publ. RIMS, Kyoto University, Vol. 15, pp. 401–454.Google Scholar
  14. Matano, H. (1982). Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation.J. Fac. Sci. Univ. Tokyo Sect. 1A 29, 401–441.Google Scholar
  15. Schaaf, R. (1990).Global Solution Branches of Two Point Boundary Value Problems, Lect. Notes Math., Vol. 1458, Springer, New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Pedro Freitas
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityEdinburghScotland

Personalised recommendations