A HOPF-BIFURCATION IN A PARABOLIC FREE-BOUNDARY PROBLEM

The occurrence of a Hopf bifurcation in a free boundary problem for a parabolic partial differential equation is investigated. The bifurcation parameter tau is contained in the equation which describes the evolution of the free boundary. The problem investigated in this paper arises as the singular limit of a system of reaction-diffusion equations with McKean reaction dynamics. Numerical evidence is examined, which shows the emergence of periodic steady states for small positive values of tau. A regularization of the problem is introduced, making it possible to deal with the Heaviside discontinuity in the reaction term, and well-posedness of the free boundary problem is obtained by application of results from the theory of nonlinear evolution equations to the regularized problem. It is then shown that a pair of complex eigenvalues of the linearized problem crosses the imaginary axis as tau --> 0, and the existence of a Hopf bifurcation is proved, using an implicit function theorem argument.