Model of stationary diffusion with absorption in domains with fine-grained random boundaries. (English. Russian original) Zbl 1435.35303
Ukr. Math. J. 71, No. 5, 792-807 (2019); translation from Ukr. Mat. Zh. 71, No. 5, 692-705 (2019).
Summary: We consider a boundary-value problem for the equation of stationary diffusion in a porous medium filled with small inclusions in the form of balls with absorbing surfaces. The process of absorption is described by the Robin nonlinear boundary condition. The locations and radii of the inclusions are randomly distributed and described by a set of finite-dimensional distribution functions. We study the asymptotic behavior of solutions to the problem when the number of balls increases and their radii decrease. We deduce a homogenized equation for the main term of the asymptotics and determine sufficient conditions for the distribution functions under which the solutions converge to the solutions of the homogenized problem in probability.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 76S05 | Flows in porous media; filtration; seepage |
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