Bulletin of the
Korean Mathematical Society BKMS

The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$ \left \{ \begin{aligned} \frac{du}{dt}+Au & =F(t,u_t), \quad t\geq s, \medskip \\ u_s (\theta)&=\phi(\theta), \ \ \forall \theta \in ( -\infty, 0], \ \ s \in \R, \quad \end{aligned} \right. $$ where $A$ is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part $F$ may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

Keywords: Admissible inertial manifolds, admissible function spaces, infinite delay, Lyapunov-Perron method, Mackey-Glass, distributed delay

MSC numbers: Primary 34K30, 35B40, 35K58, 37L25