## 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

### Citations:

Zbl 0404.35001; Zbl 0770.35005
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### References:

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