Integral manifolds are very useful in studying the dynamics of nonlinear evolution equations. We consider a nondensely defined partial differential equation 1 $$ \frac{du}{dt}=\left(A+B(t)\right)u(t)+f\left(t,{u}_t\right),\kern0.72em t\in \mathrm{\mathbb{R}}, $$ where (A,D(A)) satisfies the Hille–Yosida condition, (B(t))t∈R is a family of operators in L(D(A),X) satisfying certain measurability and boundedness conditions, and the nonlinear forcing term f satisfies the inequality ‖f(t, ϕ) − f(t, ψ)‖≤φ(t)‖ϕ − ψ‖c, where 𝜑 belongs to admissible spaces and 𝜙,ψ ∈ 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 these solutions. Our main methods are invoked by the extrapolation theory and the Lyapunov–Perron method based on the properties of admissible functions.