Invariant stable manifolds for  partial neutral functional differential equations in admissible spaces on a half-line

Nguyen Thieu  Huy; Pham Van  Bang

doi:10.3934/dcdsb.2015.20.2993

Article Contents

2015, Volume 20, Issue 9: 2993-3011. Doi: 10.3934/dcdsb.2015.20.2993

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Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line

Nguyen Thieu Huy^1, and
Pham Van Bang^2,

1.
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi
2.
Department of Basic Sciences, University of Economic and Technical Industries, 456-Minh Khai Str., Hai Ba Trung, Hanoi

Received: April 30, 2014

Revised: June 30, 2015

Published: October 2015

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Abstract

In this paper we investigate the existence of invariant stable and center-stable manifolds for solutions to partial neutral functional differential equations of the form $$\begin{cases}\frac{\partial}{\partial t}Fu_t = B(t)Fu_t + \Phi(t,u_t),\quad t\in (0,\infty),\cr u_0 = \phi\in \mathcal{C}: = C([-r, 0], X) \end{cases}$$ when the family of linear partial differential operators $(B(t))_{t\ge 0}$ generates the evolution family $(U(t,s))_{t\ge s\ge 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\| \Phi(t,\phi) -\Phi(t,\psi)\| \le \varphi(t)\|\phi -\psi\|_{\mathcal{C}}$ for $\phi, \psi\in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line.

Keywords:

Exponential dichotomy,
partial neutral functional differential equations,
stable and center-stable manifolds,
admissibility of function spaces.

Mathematics Subject Classification: Primary: 34K19, 37D10; Secondary: 35R10.

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