Homogenization of the diffusion equation with a singular potential for a model of a biological cell network
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Homogenization of the diffusion equation with a singular potential for a model of a biological cell network Latifa Ait Mahiout, Grigory Panasenko
and Vitaly Volpert
Abstract. 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. Mathematics Subject Classification. 35B27, 92B05. Keywords. Multicellular structure, Nutrient consumption, Laplacian with Dirac’s potential, High contrast homogeneization.
1. Introduction The present work continues the discussion on the limits of homogenizability of discrete high contrast models in biology started in the PLOS ONE publication [18]. It turns out that the homogenized model gives an adequate macroscopic description for some bounds of values of parameters and is not adequate out of these bounds. The problem of passage from a microscopic discrete description of an array of cells or particles to a macroscopic description in the form of a differential equation of continuum mechanics is considered in [1–3,6,8,10,18,21]. In particular, the work [12] is devoted to a one-dimensional description of the diffusion process with pointwise absorption modeled by a Helmholtz type equation with a potential including a Dirac-function coefficient, i.e., diffusion-discrete absorption. The present paper is devoted to a multidimensional generalization of this result. Note that a direct generalization of the 1D result is not evident because of the well-known difficulties of the definition of the operator Δ − δ(x − x0 ) in the case of multiple dimensions. One ”toy problem” of this type showing difficulties is presented in the Appendix. The self-adjoint extension of the operator Δ − δ(x − x0 ) via the multipliers in Sobolev spaces [20] does not work for Dirac’s potential, while the classical self-adjoint extensions in the whole space Rn pass via asymptotic approximations of the Dirac’s potential (see [4], see also[19] and the bibliography therein for the self-adjoint extensions of a similar problem of the Laplace operator in a domain containing hole in a point). That is why contrary to [12] in order to model small cells consuming the nutrient we use directly an appr
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