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Justification of the averaging method for parabolic equations containing rapidly oscillating terms with large amplitudes. (English. Russian original) Zbl 1105.35137
Izv. Math. 70, No. 2, 233-263 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 2, 25-56 (2006).
The author considers the abstract parabolic equation \[ {dy\over dt}= Ay+ g(y,t,\omega t)+ \omega^\alpha\varphi(y, t,\omega\cdot t),\tag{1} \] where \(A\) is a linear operator (generally speaking, unbounded), the nonlinear vector-valued functions \(g(y,t,\tau)\) and \(\varphi(y,t,\tau)\) have mean values with respect to \(\tau\), the mean value of \(\varphi\) is equal to zero, and \(\alpha\) is a positive number. Here the author considers the case \(\alpha={1\over 2}\), which is called the first bifurcation exponent. Under some suitable assumptions on the data of (1), the author establishes that the solutions of (1) are asymptotically close to the corresponding solutions of the averaged problems. Moreover, the author studies the Lyapunov stability and instability of solutions of (1).

35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
35K10 Second-order parabolic equations
35G05 Linear higher-order PDEs
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