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A justification of the averaging method for abstract parabolic equations
I. B. Simonenko
Abstract:
In this paper the method of averaging of N. N. Bogoljubov is applied to abstract parabolic equations of the form
\begin{equation}
\frac{dx}{dt}=Ax+f(x,\omega t),
\end{equation}
where
$A$ is a linear, in general unbounded, operator generating an analytic semigroup, and
$f$ is an operator subordinate to
$A$, in general a nonlinear map, possessing the mean
$$
\lim_{N\to+\infty}\frac1N\int_0^Nf(x,t)\,dt=Fx.
$$
Other conditions on the mapping
$f$ are formulated in terms of the theory of semigroups.
The main results are contained in two theorems.
Theorem 1 relates the initial value problem for equation (1) with the equation
\begin{equation}
\frac{dy}{dt}=Ay+Fy.
\end{equation}
Theorem 2, in the case of periodic dependence of the mapping
$f$ on time, establishes a connection between the stability of the stationary solution to equation (2) and the stability of the corresponding periodic solution of (1).
Bibliography: 5 titles.
UDC:
517.43+517.948+513.881
MSC: 34C29,
47D03,
65L05,
34A12,
47L10 Received: 26.02.1969
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