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

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 ⅆ u ⅆ t = ( A + B ( t ) ) u ( t ) + f ( t , u t ) , t ∈ R , ( 1 ) where ( A , D ( A ) ) satisfies the Hille – Yosida condition, ( B ( t ) ) t ∈ R is a family of operators in ℒ ( 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 ‖ 𝒞 , here, φ belongs to some 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 such solutions.Our main methods are invoked by the extrapolation theory and the Lyapunov – Perron method based on the admissible functions properties.