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Homogenization of parabolic equations—an alternative approach and some corrector-type results. (English) Zbl 0898.35008

The paper deals with the homogenization problem for linear parabolic equations of the type \[ {\partial u\over \partial t} + \sum _{i,j}{\partial \over \partial x_i} \left (a_{ij}\left (x ,{x\over \varepsilon },t ,{t\over \varepsilon ^r} \right){\partial u\over \partial x_j}\right)=f \] with coefficients \(a_{ij}(x,y,t,s)\) periodic in \(y\) and \(s\). Considering a sequence of parameters \(\varepsilon \to 0\), a sequence of problems and the corresponding sequence of solutions \(u^\varepsilon \) is given. The solutions \(u^\varepsilon \) converge to a solution \(u^0\) of the so-called homogenized equation with coefficients \(\bar a_{ij}(x,t)\) independent of \(\varepsilon \).
The homogenization problem can be studied by different methods. The same problem was studied in [A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic analysis for periodic Structures, North-Holland, Amsterdam (1978; Zbl 0404.35001)] by asymptotic expansion methods. The two-scale convergence method [G. Allaire, SIAM J. Math. Anal. 23, No. 6, 1482-1518 (1992; Zbl 0770.35005)] introduced by G. Allaire and Nguetseng in 1989 for elliptic equations seems to be the most efficient. In the present paper the two-scale method is adapted to parabolic problems. It yields formulae for coefficients of the limit homogenized equation and a simple proof of convergence \(u^\varepsilon \to u^0\) including convergence of “corrected” solutions. According to the value of the parameter \(r\) describing ratio of the space period \(\varepsilon \) and the time period \(\varepsilon ^r\), three cases occur: \(r<2\), \(r=2\) and \(r>2\) with different formulae for the homogenized coefficients.
Reviewer: J.Franců(Brno)

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
74E05 Inhomogeneity in solid mechanics
74E30 Composite and mixture properties
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