Admissible integral manifolds for partial neutral functional-differential equations

Abstract

UDC 517.9 We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in Banach space X   of the form  & ∂ ∂ t F u t = A ( t ) F u t + f ( t , u t ) , t ≥ s , t , s ∈ ℝ , & u s = ϕ ∈ 𝒞 : = C ( [ - r ,0 ] , X )  under the conditions that the family of linear partial differential operators ( A ( t ) ) t ∈ ℝ generates the evolution family ( U ( t , s ) ) t ≥ s with an exponential dichotomy on the whole line ℝ ;   the difference operator  F : 𝒞 → X is bounded and linear, and the nonlinear delay operator f satisfies the φ -Lipschitz condition, i.e.,  ‖ f ( t , ϕ ) - f ( t , ψ ) ‖ ≤ φ ( t ) ‖ ϕ - ψ ‖ 𝒞 for ϕ , ψ ∈ 𝒞 , where φ ( ⋅ ) belongs to an admissible function space defined on ℝ .   We also prove that an unstable manifold of the admissible class attracts all other solutions with exponential rates.  Our main method is based on the Lyapunov – Perron equation combined with the admissibility of function spaces.  We  apply our results to the finite-delayed heat equation for a material with memory.