On the theory of integral manifolds for some delayed partial differential equations with nondense domain

• Date: 2020
• Main Author: Jendoubi, C.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2020
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/6020
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.9 Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation $$\frac{du}{dt}=(A+B(t))u(t)+f(t,u_t),\quad t\in\mathbb{R},\tag{1}$$ where $(A,D(A))$ satisfies the Hille–Yosida condition, $(B(t))_{t\in\mathbb{R}}$ is a family of operators in $\mathcal{L}(\overline{D(A)},X)$ satisfying some measurability and boundedness conditions, and the nonlinear forcing term $f$ satisfies $\|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$;  here, $\varphi$ belongs to some admissible spaces and $\phi,$ $\psi\in\mathcal{C}:=C([-r,0],X)$. We first present an exponential convergence result between the stable manifold and every mild solution of (1).  Then we prove the existence of center-unstable manifolds for such solutions. Our main methods are invoked by the extrapolation theory and the Lyapunov–Perron method based on the admissible functions properties.