Admissible Integral Manifolds for Partial Neutral Functional-Differential Equations

We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in a Banach space X of the form

$\begin{array}{l}\begin{array}{cc}\frac{\partial }{\partial t}F{u}_{t}=A\left(t\right)F{u}_{t}+f\left(t,{u}_{t}\right),& \begin{array}{cc}t\ge s,& t,s\in {\mathbb{R}},\end{array}\end{array}\\ {u}_{s}=\phi \in \mathcal{C}:=C\left(\left[-r,0\right],X\right)\end{array}$

under the conditions that the family of linear partial differential operators (A(t))_t∈ℝ generates the evolution family (U(t, s))_t≥s with exponential dichotomy on the whole line ℝ; the difference operator F : $\mathcal{C}\to X$ is bounded and linear, and the nonlinear delay operator f satisfies the φ-Lipschitz condition, i.e., $\Vert f\left(t,\phi \right)-f\left(t,\psi \right)\Vert \le \varphi \left(t\right){\Vert \phi -\psi \Vert }_{\mathcal{C}}$ for ϕ, ψ ∈ 𝒞, where φ(·) belongs to an admissible function space defined on ℝ. We also prove that an unstable manifold of the admissible class attracts all other solutions with exponential rates. Our main method is based on the Lyapunov – Perron equation combined with the admissibility of function spaces. We apply our results to the finite-delay heat equation for a material with memory.