The delay effect on reaction–diffusion equations

Abstract

In this work we study a reaction–diffusion problem with delay and we make an analysis of the stability of solutions by means of bifurcation theory. We take the delay constant as a parameter. Special conditions on the vector field assure existence of a spatially nonconstant positive equilibrium U_k , which is stable for small values of the delay. An increase of the delay destabilizes the equilibrium of U_k and leads to super or subcritical Hopf bifurcation.

Dedicated to Professor José Geraldo Dos Reis.

E-mail: [email protected]

Keywords:

• Delay model with diffusion effects
• Super critical bifurcation
• Normal form
• Convection process
• AMS 1991 Mathematics Subject Classifications: 35B32
• 34K57
• 35R10

Acknowledgments

We appreciate the generous comments of Professor A.N. Carvalho which greatly improved the readability of the article. Santos was supported by FAPESP, grant # 97/11717-4.

Notes

Dedicated to Professor José Geraldo Dos Reis.

E-mail: [email protected]