Sectorial operators and inertial manifolds for partial functional differential equations in admissible spaces

We prove the existence of inertial manifolds for partial functional differential equation du(t) dt + Au(t) = F (t)ut + g(t, ut) under the conditions that the partial differential operator A is positive such that −A is sectorial with a sufficiently large gap in its spectrum; the operator F (t) is linear, and g is a nonlinear operator satisfying φ-Lipschitz condition for φ belonging to an admissible function space. Our main methods are based on Lyapunov-Perron equations combined with analytic semigroups and admissibility of function spaces.