Homogenization of parabolic problems with dynamical boundary conditions of reactive‐diffusive type in perforated media. Issue 10 (13th June 2020)

• Record Type: Journal Article
• Authors: Anguiano, María
• Is Part Of: Zeitschrift für angewandte Mathematik und Mechanik. Volume 100:Issue 10(2020)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2020-06-13
• Subjects: dynamical boundary conditions -- homogenization -- perforated media -- reaction‐diffusion systems -- surface diffusion; Mathematics -- Periodicals; Mechanics, Applied -- Periodicals; Engineering -- Periodicals; 519
• DOI: 10.1002/zamm.202000088
• Languages: English
• ISSNs: 0044-2267

Abstract

Abstract: 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 ∇ u ε · ν + ε ∂ u ε ∂ t = ε δ Δ Γ u ε − ε g ( u ε ), where Δ Γ denotes the Laplace–Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, δ > 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 δ = 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. Abstract : 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 ∇ u ε · ν + ε ∂ u ε ∂ t = ε δ Δ Γ u ε − ε g ( u ε ), where Δ Γ denotes the Laplace–Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, δ > 0 plays the role of a surface diffusion coefficient and g is the nonlinear term….