A Hopf bifurcation in a parabolic free boundary problem

Abstract

The occurrence of a Hopf bifurcation in a free boundary problem for a parabolic partial differential equation is investigated. The bifurcation parameter τ 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 τ. 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 τ → 0, and the existence of a Hopf bifurcation is proved, using an implicit function theorem argument.