On the theory of integral manifolds for some delayed partial differential equations with nondense domain - Jendoubi, C. (Ukrains’kyi Matematychnyi Zhurnal, 2020)
UDC 517.9Integral 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 $\|f(t,φ)-f(t,ψ)\|≤ φ(t)\|φ-ψ\|_{\mathcal{C}}$; here, $φ$ belongs to some admissible spaces and $φ,$ $ψ\in\mathcal{C}:=C([-r,0],X)$. 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.