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The asymptotic behaviour of the third boundary-value problem solutions in domains with fine-grained boundaries. (English) Zbl 0892.35018
Cioranescu, Doina (ed.) et al., Homogenization and applications to material sciences. Proceedings of the international conference, Nice, France, June 6–10, 1995. Tokyo: Gakkotosho. GAKUTO Int. Ser., Math. Sci. Appl. 9, 203-213 (1995).
Summary: We consider the initial boundary-value problem ${\partial u^\varepsilon \over \partial t}- \Delta u^\varepsilon= f^\varepsilon (x,t),\;x\in \Omega^\varepsilon,\;t>0;\quad {\partial u^\varepsilon \over\partial n}+ \sigma^\varepsilon (x,u^\varepsilon) =0,\;x\in \partial F^\varepsilon ,\;t>0;$ $u^\varepsilon(x,t)=0,\;x\in\partial \Omega^\varepsilon,\;t>0;\;u^\varepsilon (x,0)= \varphi (x),\;x\in\Omega^\varepsilon;$ where $$\Omega^\varepsilon = \Omega \setminus \cup^{N (\varepsilon)}_{j=1}$$; $$\{F^\varepsilon_j$$, $$j=1,2, \dots, N (\varepsilon)\}$$ is a system of balls of radius $$\varepsilon^\alpha$$, $$2<\alpha\leq 3$$. The centers of the balls form a periodical lattice with period $$\varepsilon$$. A homogenized model of diffusion is obtained, and a corrector is constructed.
For the entire collection see [Zbl 0873.00028].

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35K20 Initial-boundary value problems for second-order parabolic equations
##### Keywords:
periodical lattice; homogenized model