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For an abstract parabolic equation with initial condition and multidimensional parabolic initial-boundary value problem with absolute terms rapidly oscillating in time, inverse problems of finding these absolute terms from some information about the partial asymptotics of the solutions of the original problems are stated and solved. In this case, the absolute terms are the products of two factors, one of which is described by a rapidly oscillating function (i.e., depends on fast time), while the second term may depend on ordinary time but does not depend on fast time. The following three cases are considered: where, in this pair, only one of the factors is known and where only the mean value of the rapidly oscillating factor is known.

The paper is devoted to the development of the theory of inverse problems for evolution equations with summands rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints (overdetermination conditions) are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.

The averaging method is justified for normal systems of differential equations with rapidly oscillating summands proportional to the square root of the oscillation frequency in the case of the boundary-value problem on a finite interval and for the problem of bounded solutions on the positive semiaxis with boundary condition at its left endpoint.

For a linear second-order parabolic equation with high-frequency coefficients, the complete asymptotics of time-periodic solutions is constructed and justified. The limit averaged stationary problem is assumed degenerate.

Abstract: Two classes of systems of linear ordinary differential equations containing high-frequency terms proportional to certain positive powers of the oscillation frequency are considered. Complete asymptotic expansions of fundamental systems of solutions are constructed and substantiated.

Abstract: Systems of first-order semilinear partial differential equations with terms that oscillate at a frequency ω ≫ 1 in a single variable and are proportional to $$ \sqrt \omega $$ are considered. The Krylov-Bogolyubov-Mitropol’skii averaging method is substantiated for such equations. Based on the two-scale expansion method, an algorithm for constructing complete asymptotics of solutions is proposed and justified.

Without Abstract:

Abstract: The paper deals with averaging problems for parabolic equations. We prove that exponential dichotomy is preserved without any assumption concerning the almost-periodicity of the coefficients.

We obtain a criterion for invertibility in Hölder spaces of linear parabolic operators of arbitrary order with any number of spatial variables and almost-periodic coefficients.

Abstract: An algorithm for constructing the higher approximations of the averaging method for quasilinear parabolic equations with fast oscillating coefficients is suggested and justified.

We justify the averaging method for abstract parabolic equations with stationary principal part that contain non-linearities (subordinate to the principal part) some of whose terms are rapidly oscillating in time with zero mean and are proportional to the square root of the frequency of oscillation. Our interest in the exponent 1/2 is motivated by the fact that terms proportional to lower powers of the frequency have no influence on the average. For linear equations of the same type, we justify an algorithm for the study of the stability of solutions in the case when the stationary averaged problem has eigenvalues on the imaginary axis (the critical case).