Admissible inertial manifolds for infinite delay evolution equations

Bull. Korean Math. Soc. Published online January 8, 2021

Le Anh Minh
Hong Duc University

Abstract : 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
$$
\left \{
\begin{aligned}
\frac{du}{dt}+Au & =F(t,u_t),
\quad t\geq s,
\medskip
\\
u(s+\theta) & = u_s (\theta), \ \ \forall \theta \in ( -\infty, 0], \ \ s \in \R,
\quad
\end{aligned}
\right.
$$
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.