Higher approximations of the homogenization method for quasilinear parabolic equations (Q1595573)

In [\textit{V. B. Levenshtam}, Sov. Math., Dokl. 43, No. 3, 928-931 (1991; Zbl 0782.35034) and Russ. Acad. Sci., Izv. Math. 41, No. 1, 95-132 (1993; Zbl 0795.35041)], the homogenization method [\textit{N. N. Bogolyubov} and \textit{Yu. A. Mitropol'skij}, Asymptotic methods in the theory of nonlinear oscillations (in Russian), Moscow: Fizmatgiz (1963; Zbl 0111.08801)] was justified for the problem of finding solutions to quasilinear parabolic equations of arbitrary order \(2k\) (with rapidly oscillating coefficients of higher derivatives) that are bounded on the whole time axis. This paper deals with justification of the homogenization method and construction of higher approximations for such equations in the case of the Cauchy problem. For ordinary differential equations, similar algorithms for constructing higher approximations are described and justified in [\textit{V. V. Strygin}, in J. Appl. Math. Mech. 48, 767-769 (1984; Zbl 0591.34040) and \textit{Kh. T. Movlyankulov} and \textit{L. V. Sharova}, Differ. Equations 22, 550-554 (1986; Zbl 0609.34011)].