In this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations, in particular for ordinary and partial differential equations. These manifolds, which are finite dimensional invariant Lipschitz manifolds, seem to be an appropriate tool for the study of questions related to the long-time behavior of solutions of the evolutionary equations. The inertial manifolds contain the global attractor, they attract exponentially all solutions, and they are stable with respect to perturbations. Furthermore, in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation.
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* This research was supported in part by grants from the National Science Foundation and from the USDOE Oftice of Basic Energy Sciences under Contract DE.AC02.82ER12049. 309