Stable periodic solutions of a spatially homogeneous nonlocal reaction-diffusion equation

Nonlocal reaction-diffusion equations of the form u_t = u_xx + F(u, α(u)), where α(u) = ∫_-1¹ u(x) dx, are considered together with Neumann or Dirichlet boundary conditions. One of the main results deals with linearisation at equilibria. It states that, for any given set of complex numbers, one can arrange, choosing the equation properly, that this set is contained in the spectrum of the linearisation. The second main result shows that equations of the above form can undergo a supercritical Hopf bifurcation to an asymptotically stable periodic solution.