Averaging of quasilinear parabolic equations with rapidly oscillating principal part. Exponential dichotomy

The first part of this article is an investigation of the question of preservation of an exponential dichotomy for solutions of a linear parabolic equation upon a high-frequency perturbation of its principal terms. The second part contains a justification of the averaging principle on the whole time axis for quasilinear parabolic equations with rapidly oscillating principal part.