The paper is devoted to a reaction-diffusion problem describing diffusion and consumption of nutrients in a biological tissue consisting of small cells periodically arranged in an extracellular matrix. Cells consume nutrients with a rate proportional to cell area and to nutrient concentration. The dependence on the nutrient concentration can be linear or nonlinear. The cells are modeled by a potential approximating the Dirac’s delta-function. The potential has a periodically distributed support of small measure. The problem contains two small parameters: the diameter of a cell and the distance between the cells (in comparison with the characteristic macroscopic size). In the multi-dimensional formulation assuming some restriction on the relation of parameters, we prove convergence of solution of this problem to the solution of a limiting homogenized problem. We show that the problem is non-homogenizable in classical sense if this restriction fails.

该论文致力于反应扩散问题，描述了营养物质在生物组织中的扩散和消耗，该组织由周期性排列在细胞外基质中的小细胞组成。细胞以与细胞面积和营养物浓度成比例的速率消耗营养物。对营养物浓度的依赖性可以是线性或非线性的。通过近似狄拉克三角函数的电势对单元建模。潜力具有周期性的小规模支持。该问题包含两个小参数：一个单元的直径和单元之间的距离（与特征宏观尺寸相比）。在多维表述中，假设对参数之间的关系有一些限制，我们证明了该问题的解与有限同质化问题的解的收敛性。我们证明，如果此限制失败，则该问题在经典意义上是不可均匀化的。