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Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms. (English) Zbl 1135.35315
The paper deals with a homogenization of attractors for reaction-diffusion systems of the form \[ \partial_t u= a\Delta u- f\Biggl(\varepsilon, x,{x\over \varepsilon}, u\Biggr)+ g\Biggl(\varepsilon, x,{x\over\varepsilon}\Biggr),\tag{1} \] where \(u\in\mathbb R^N\), and \(x\in\Omega\subset \mathbb R^d\), \(d\leq 3\). The authors consider Dirichlet boundary conditions \(u= 0\) for \(x\in\partial\Omega\). The goal of the authors is to compare the global attractor \({\mathcal A}^\varepsilon\) for (1), for \(\varepsilon\searrow 0\) with attractor \({\mathcal A}^0\) of the formally homogenized system \[ \partial_t u= a\Delta u- f^0(x,u)+ g^0(x), \] where \(f^0\) and \(g^0\) corresponding spatial averaging for \(f(\varepsilon, x,{x\over\varepsilon}, u)\) and \(g(\varepsilon, x,{x\over\varepsilon})\), respectively. Under some suitable assumptions on \(f\) and \(g\), they establish a quantitative homogenization estimate \[ \text{dist}_{L^2}({\mathcal A}^\varepsilon, {\mathcal A}^0)\leq C\varepsilon^\chi, \] where \(\chi\) can be calculated explicitly.

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B41 Attractors
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
35K57 Reaction-diffusion equations
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