Unstable Manifolds for Partial Neutral Differential Equations and Admissibility of Function Spaces

Abstract

We prove the existence and attraction property of an unstable manifold for solutions to the partial neutral functional differential equation of the form

$\left\{\begin{array}{ll} \frac{\partial}{\partial t}Fu_{t}= B(t)Fu_{t} +\varPhi(t,u_{t}),\quad t\ge s;~ t,s\in \mathbb{R},\\ u_{s}=\phi\in \mathcal{C}:=C([-r, 0], X) \end{array}\right.$

under the conditions that the family of linear operators $(B(t))_{t\in \mathbb {R}}$ defined on a Banach space X generates the evolution family (U(t, s)) _t≥s having an exponential dichotomy on the whole line $\mathbb {R}$, the difference operator $F:\mathcal {C}\to X$ is bounded and linear, and the nonlinear delay operator Φ satisfies the ϕ-Lipschitz condition, i.e., $\| \Phi (t,\phi ) -\Phi (t,\psi )\| \le \phi (t)\|\phi -\psi \|_{\mathcal {C}}$ for $\phi ,~ \psi \in \mathcal {C}$, where ϕ(⋅) belongs to an admissible function space defined on $\mathbb {R}$. Our main method is based on Lyapunov-Perron’s equations combined with the admissibility of function spaces and the technique of choosing F-induced trajectories.