Admissible integral manifolds for partial neutral functional-differential equations

Abstract

UDC 517.9

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 Banach space $X$  of the form  $$\begin{align*}& \dfrac{\partial}{\partial t}Fu_t= A(t)Fu_t +f(t,u_t),\quad t \ge s,\quad t,s\in\mathbb{R},\\& u_s=\phi\in\mathcal{C}:= C([-r, 0], X)\end{align*}$$  under the conditions that the family of linear partial differential operators $\left(A(t)\right)_{t\in\mathbb{R}}$ generates the evolution family $\left(U(t,s)\right)_{t\geq s}$ with an exponential dichotomy on the whole line $\mathbb{R};$  the difference operator  $F\colon\mathcal{C}\to X$ is bounded and linear, and the nonlinear delay operator $f$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in\mathcal{C},$ where $\varphi(\cdot)$ belongs to an admissible function space defined on $\mathbb{R}.$  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-delayed heat equation for a material with memory.