Smoothness of the inertial manifold via the spatial averaging principle

Abstract

This paper investigates the $C^{n,\epsilon }{|}_{\{n\ge 2,\epsilon \in (0,1\left)\right\}}$-smoothness of the inertial manifold for an abstract semilinear parabolic equation. Compared with the known results, the required spectral gap condition has been relaxed by applying the principle of spatial averaging initially proposed by J. Mallet-Paret and G. Sell in 1988.