Journal of Integral Equations and Applications

Stable and center-stable manifolds of admissible classes for partial functional differential equations

Trinh Viet Duoc and Nguyen Thieu Huy

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Abstract

In this paper, we investigate the existence of stable and center-stable manifolds of admissible classes for mild solutions to partial functional differential equations of the form $\dot {u}(t)=A(t)u(t)+f(t,u_t)$, $t\ge 0$. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like $L_p$-spaces and many other function spaces occurring in interpolation theory such as the Lorentz spaces $L_{p,q}$. Results in this paper are the generalization and development for our results in \cite {HD1}. The existence of these manifolds obtained in the case that the family of operators $(A(t))_{t\ge 0}$ generate the evolution family $(U(t,s))_{t\ge s\ge 0}$ having an exponential dichotomy or trichotomy on the half-line and the nonlinear forcing term $f$ satisfies the $\varphi $-Lipschitz condition, i.e., $\| f(t,u_t) -f(t,v_t)\| \le \varphi (t)\|u_t -v_t\|_{\mathcal {C}}$, where $u_t,\ v_t \in \mathcal{C} :=C([-r, 0], X)$, and $\varphi (t)$ belongs to some admissible Banach function space and satisfies certain conditions.

Article information

Source
J. Integral Equations Applications, Volume 30, Number 4 (2018), 543-575.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1543482176

Digital Object Identifier
doi:10.1216/JIE-2018-30-4-543

Mathematical Reviews number (MathSciNet)
MR3881216

Zentralblatt MATH identifier
06989832

Subjects
Primary: 34K19: Invariant manifolds 34K25: Asymptotic theory 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx] 35B40: Asymptotic behavior of solutions

Keywords
Exponential dichotomy and trichotomy partial functional differential equations admissibility of function spaces stable and center-stable manifolds of admissible classes

Citation

Duoc, Trinh Viet; Huy, Nguyen Thieu. Stable and center-stable manifolds of admissible classes for partial functional differential equations. J. Integral Equations Applications 30 (2018), no. 4, 543--575. doi:10.1216/JIE-2018-30-4-543. https://projecteuclid.org/euclid.jiea/1543482176


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