Admissible integral manifolds for partial neutral functional-differential equations

Thieu Huy Nguyen; Vu Thi Ngoc Ha; Trinh Xuan Yen

doi:10.37863/umzh.v74i10.6257

Thieu Huy Nguyen School Appl. Math. and Informatics, Hanoi Univ. Sci. and Technology, Vietnam https://orcid.org/0000-0003-3790-8592
Vu Thi Ngoc Ha School Appl. Math. and Informatics, Hanoi Univ. Sci. and Technology, Vietnam
Trinh Xuan Yen Hung Yen Univ. Technology and Education, Vietnam)

DOI: https://doi.org/10.37863/umzh.v74i10.6257

Keywords: ADMISSIBLE INTEGRAL

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.

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