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Exponential dichotomy properties are studied for non-autonomous quasilinear partial differential equations that can be written as an ordinary dif- ferential equation du/dt + Au = F (u, t) in a Hilbert space H. It is assumed that the non-linear function F (u, t) is essentially subordinated to the linear operator A; namely, the gap property from the theory of inertial manifolds must hold. Integral manifolds M+ and M− attracting at an exponential rate an arbitrary solution of this equation as t → +∞ and t →− ∞, respectively, are constructed. The general results established are applied to the study of the dichotomy properties of solutions of a one-dimensional reaction-diffusion system and of a dissipative hyperbolic equation of sine-Gordon type. Bibliography: 18 titles.