Spectral barriers and inertial manifolds for dissipative partial differential equations

In recent years, the theory of inertial manifolds for dissipative partial differential equations has emerged as an active area of research. An inertial manifold is an invariant manifold that is finite dimensional, Lipschitz, and attracts exponentially all trajectories. In this paper, we introduce the notion of a spectral barrier for a nonlinear dissipative partial differential equation. Using this notion, we present a proof of existence of inertial manifolds that requires easily verifiable conditions, namely, the existence of large enough spectral barriers.