Inertial Manifolds for Partial Neutral Functional Differential Equations in Admissible Spaces

Abstract

We prove the existence of an inertial manifold for the partial neutral functional differential equations of the form \(\frac {\partial }{\partial t}(Fu_{t}) +A (Fu_{t}) = {\Phi }(t,u_{t})\), where the partial differential operator A is positive definite and self-adjoint with a discrete spectrum having a sufficiently large gap; the difference operator \(F: \mathcal {C}_{\beta } \rightarrow X\) is a bounded linear operator, and the nonlinear delay operator Φ satisfies the φ-Lipschitz condition, i.e., \(\|{\Phi }(t,\phi )\| \leq \varphi (t)(1+|\phi |_{\mathcal {C}_{\beta }})\) and \(\|{\Phi }(t,\phi )-{\Phi }(t,\psi )\| \leq \varphi (t)|\phi - \psi |_{\mathcal {C}_{\beta }}\), where φ belongs to an admissible function space defined on \(\mathbb {R}\). Our main method is based on Lyapunov–Perron’s equations combined with the admissibility of function spaces and the technique of choosing F-induced trajectories.