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.

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一些非稠域时滞偏微分方程的积分流形理论

积分流形在研究非线性演化方程的动力学方面非常有用。我们考虑一个非稠密定义的偏微分方程 1 $$ \frac{du}{dt}=\left(A+B(t)\right)u(t)+f\left(t,{u}_t\right) ,\kern0.72em t\in \mathrm{\mathbb{R}}, $$ 其中 (A,D(A)) 满足 Hille-Yosida 条件，(B(t))t∈R 是算子族在 L(D(A),X) 满足一定的可测性和有界条件，且非线性强迫项 f 满足不等式 ‖f(t, ϕ) − f(t, ψ)‖≤ φ(t)‖ϕ − ψ ‖c，其中 𝜑 属于容许空间，而 𝜙,ψ ∈ C ≔ C([−r, 0], X)。我们首先提出了稳定流形和（1）的每个温和解之间的指数收敛结果。然后我们证明这些解存在中心不稳定流形。