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Homogenization of semilinear parabolic equations in perforated domains. (English) Zbl 1085.35022

Summary: This paper is devoted to the homogenization of a semilinear parabolic equation with rapidly oscillating coefficients in a domain periodically perforated by \(\varepsilon\)-periodic holes of size \(\varepsilon\). A Neumann condition is prescribed on the boundary of the holes. More precisely, we study the following problem \[ \begin{cases} u'_\varepsilon(x,t)- \text{div}(A^\varepsilon(x)\nabla u_\varepsilon(x,t))= f(u_\varepsilon(x,t))+ g_\varepsilon(x,t)\quad &\text{in }\Omega_\varepsilon\times (0,T),\\ u_\varepsilon= 0\quad &\text{on }\partial\Omega\times (0,T),\\ A^\varepsilon\nabla u_\varepsilon.\nu= 0\quad &\text{on }\partial S_\varepsilon\times (0,T),\\ u_\varepsilon(x,0)= u^0_\varepsilon\quad &\text{in }\Omega_\varepsilon,\end{cases}\tag{1} \] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\), \(\Omega_\varepsilon= \Omega\setminus S_\varepsilon\) is a domain perforated by a closed subset \(S_\varepsilon\) of \(\varepsilon\)-periodic holes of the same size as the period, \(g_\varepsilon\in L^{2(0,T,L^2)}(0,T,L^2(\Omega_\varepsilon))\), \(u^0_\varepsilon\in L^2(\Omega_\varepsilon)\) and \(f\) is a continuous function with a linear growth. The matrix \(A^\varepsilon\) is of the form \(A^\varepsilon(x)= A({x\over\varepsilon})\) and \(A\) is a periodic bounded matrix field uniformly positive definite.
When the initial data \((u^0_\varepsilon, g_\varepsilon)\in L^2(\Omega_\varepsilon)\times L^2(0, T,L^2(\Omega_\varepsilon))\) converge in some sense to \((u^0,g)\in L^2(\Omega)\times L^2(0, T,L^2(\Omega))\) and contrary to the case whithout holes, the presence of the holes do not allows to prove a compactness of the solutions of (1) in \(L^2\). To overcome this difficulty, we introduce a suitable auxiliary linear problem to which we apply a corrector results. Then, we describe the asymptotic behaviour of the semilinear problem as \(\varepsilon\to 0\), and we give the limit equation as the following \[ \begin{aligned} \theta u'- \text{div}(A^0\nabla u)= \theta f(u)+\theta g\quad &\text{in }\Omega\times (0,T),\\ u= 0\quad &\text{on }\partial\Omega\times (0,T),\\ u(x,0)= u^0\quad &\text{in }\Omega,\end{aligned} \] with \(A^0\) is the homogenized operator of \(A^\varepsilon\) and \(\theta\) denote the proportion of the material of the perforated domain.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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References:

[5] M. Briane, A. Damlamian and P. Donato, Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar Vol. XIII, H-convergence for perforated domains 391, eds. D. Cioranescu and J. L. Lions (Longman, New York, 1998) pp. 62–100. · Zbl 0943.35005
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