Dichotomy Gap Conditions, Admissible Spaces, and Inertial Manifolds.

We study the existence of an inertial manifold for the fully non-autonomous evolution equation of the form du dt + A (t) u (t) = f (t , u) , t ∈ R , in certain admissible spaces. We prove the existence of such an inertial manifold in the cases that the family of linear partial differential operators (A (t)) t ∈ R generates an evolution family (U (t , s)) t ≥ s satisfying certain dichotomy estimates, and the nonlinear forcing term f(t, x) satisfies the φ -Lipschitz condition, i.e., f (t , x 1) - f (t , x 2) ⩽ φ (t) A (t) θ (x 1 - x 2) , where φ (·) belongs to some admissible function space such that certain dichotomy gap condition holds. This dichotomy gap condition, on the one hand, extends the spectral gap condition known in the case of autonomous equations, on the other hand, provides a chance to come over the restricted spectral gap condition. [ABSTRACT FROM AUTHOR]