Asymptotic expansion of solutions of quasilinear parabolic problems in perforated domains
The author deals with an averaging of the solution of the initial-boundary value problem for a quasilinear parabolic equation in a perforated domain, assuming that the diameters of perforations tend to zero (and the number of perforations eventually tends to infinity). Under some condition on the coefficients, a priori estimates are obtained and the weak convergence of the solutions in the space \(W^{1,1/2}_2\) is proved. After formulation of some local (or model) problems the series of correctors in the form of an asymptotic expansion is constructed. The main result is a theorem about strong convergence of the remainders of this expansion to zero in the space \(W^{1,1/2}_2\). Note that, because of very general assumptions about the sequence of domains, the weak limit of solutions is not identified. However, it is important to point out that the convergence result is obtained without additional assumption about the distribution of perforations.
scientific article; zbMATH DE number 923084