Dichotomies for Linear Evolutionary Equations in Banach Spaces

In this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a Banach space. The theory we present here applies to general time-varying linear equations in Banach spaces. As a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set. In the case of dissipative systems, our theory applies to the study of the flow in the vicinity of the global attractor. The theory formulated here holds for linear evolutionary systems which are uniformly α-contracting and applies to the study of the linearization of nonlinear equations of the following type: (a) parabolic PDEs, including systems of reaction diffusion equations and the Navier-Stokes equations; (b) hyperbolic PDEs, including the nonlinear wave equation and the nonlinear Schrodinger equation with dissipation; (c) retarded differential equations; and (d) certain neutral differential delay equations.