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

Abstract

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.