Homogenization of the diffusion equation with a singular potential for a model of a biological cell network

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. © 2020, Springer Nature Switzerland AG.

Авторы
Ait Mahiout L. , Panasenko G.2, 3, 4 , Volpert V. 5, 6, 7
Издательство
Birkhauser Verlag AG
Номер выпуска
6
Язык
Английский
Статус
Опубликовано
Номер
181
Том
71
Год
2020
Организации
  • 1 Laboratoire d’équations aux dérivées partielles non linéaires et histoire des mathématiques, École Normale Supérieure, B.P. 92, Vieux Kouba, Algiers, 16050, Algeria
  • 2 Institut Camille Jordan, UMR 5585 CNRS, University of Lyon, UJM-Saint-Etienne, SFR MODMAD FED 4169, Saint-Etienne, 42023, France
  • 3 National Research University Moscow Power Engineering Institute, Krasnoznamennaya 14, Moscow, 111250, Russian Federation
  • 4 Institute of Applied Mathematics, Vilnius University, Nagarduko str., 24, Vilnius, 03225, Lithuania
  • 5 Institut Camille Jordan, UMR 5585 CNRS, University Lyon 1, Villeurbanne, 69622, France
  • 6 INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, Villeurbanne Cedex, 69200, France
  • 7 Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
Ключевые слова
High contrast homogeneization; Laplacian with Dirac’s potential; Multicellular structure; Nutrient consumption
Дата создания
02.11.2020
Дата изменения
02.11.2020
Постоянная ссылка
https://repository.rudn.ru/ru/records/article/record/64316/
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