Bifurcation and Asymptotic Behavior of Solutions of a Delay-Differential Equation with Diffusion

A scalar delay-differential equation with diffusion term in one space dimension, where the diffusivity D is a bifurcation parameter, is considered. The center manifold theory and the method of Lyapunov–Schmidt are used to describe two bifurcations from spatially constant solutions as D decreases. By modifying the equation the order of these bifurcations can be reversed. Then the existence of a compact attractor for a class of such equations is shown and the structure of part of the attractor for the modified equation is investigated. It is known that the solutions are globally $L^2 $-bounded; bounds on the solution operator from one intermediate space to another are constructed to obtain an attractor in the $W^{2,2} $ sense.