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Averaging of quasilinear parabolic problems in domains with fine-grained boundary. (English. Russian original) Zbl 0856.35068

Differ. Equations 31, No. 2, 327-339 (1995); translation from Differ. Uravn. 31, No. 2, 350-363 (1995).
The starting point of the paper is the quasilinear initial-boundary value problem for the parabolic equation with Carathéodory coefficients \[ {\partial u\over \partial t}- \sum^n_{j= 1} {d\over dx_j} a_j\Biggl(x, t, u, {\partial u\over \partial x}\Biggr)= a_0\Biggl(x, t, u, {\partial u\over \partial x}\Biggr) \] in the cylinder \(\Omega^{(s)}\times [0, T]\) with a fine-grained lateral surface, so that \(\Omega^{(s)}\) is a bounded domain \(\Omega\subset \mathbb{R}^n\) \((n\geq 3)\) perforated by finitely many nonintersecting closed sets \(F^{(s)}_i\) depending on a positive integer \(s\). After proving an existence theorem and using the weak convergence (as \(s\to \infty\)) of the sequence of solutions \(u_s(x, t)\) to a limit function \(u_0(x, t)\) the author constructs for this function a new initial-boundary value problem in the cylinder \(\overline\Omega\times [0, T]\) that is just the averaged problem in question. The averaging procedure is based on the asymptotic expansion of the solutions \(u_s(x, t)\) and on pointwise estimates of solutions to nonlinear parabolic model equations.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35C20 Asymptotic expansions of solutions to PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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