This paper deals with the homogenization of the reaction‐diffusion equations in a domain containing periodically distributed holes of size ε, with a dynamical boundary condition of reactive‐diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes

$${\displaystyle \nabla u_{\epsilon }·\nu +\epsilon \frac{\partial u_{\epsilon }}{\partial t}=\epsilon \delta {\mathrm{\Delta }}_{\mathrm{\Gamma }}{u}_{\epsilon }-\epsilon g(u_{\epsilon }),}$$

where ${\mathrm{\Delta }}_{\mathrm{\Gamma }}$ denotes the Laplace–Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, $\delta >0$ plays the role of a surface diffusion coefficient and g is the nonlinear term. We generalize our previous results established in the case of a dynamical boundary condition of pure‐reactive type, i.e., with $\delta =0$. We prove the convergence of the homogenization process to a nonlinear reaction‐diffusion equation whose diffusion matrix takes into account the reactive‐diffusive condition on the surface of the holes.

中文翻译：

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多孔介质中反应扩散型动力边界条件下抛物线问题的均质化

本文研究了一个具有周期性分布的尺寸为ε的孔的区域中具有反应扩散类型的动态边界条件的反应扩散方程的均匀化，即，我们考虑了孔表面的以下非线性边界条件

$${\displaystyle \nabla {ü}_{\epsilon }·\nu +\epsilon \frac{\partial {ü}_{\epsilon }}{\partial Ť}=\epsilon \delta {\mathrm{\Delta }}_{\mathrm{\Gamma }}{ü}_{\epsilon }-\epsilon G（{ü}_{\epsilon }），}$$

哪里 ${\mathrm{\Delta }}_{\mathrm{\Gamma }}$ 表示孔表面上的Laplace-Beltrami算子，ν是边界的向外法线， $\delta >0$扮演表面扩散系数的角色，而g是非线性项。我们推广了我们在纯反应型动态边界条件下建立的先前结果，即$\delta =0$。我们证明了均化过程对非线性反应扩散方程的收敛性，该方程的扩散矩阵考虑了孔表面的反应扩散条件。