Time-Averaging under Fast Periodic Forcing of Parabolic Partial Differential Equations: Exponential Estimates

The phases of a large class of parabolic partial differential equations with rapid time-periodic forcing can be separated up to exponential small errors. The originally nonautonomous equation is transformed such that the nonautonomous terms are exponentially small in the period h of the forcing. This is a counterpart for partial differential equations of the theorem by A. Neishtadt (1984, J. Appl. Math. Mech.48, 134–139) for ordinary differential equations. In our case the exponential rate depends on time t and the estimates have the form h exp(−c(t) h−1/3).