Asymptotic analysis of a coupled ODE‐PDE system arising from heterogeneous diffusion‐reaction kinetics
Michal Beneš, Michael Eden, Adrian Muntean · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · 2024
Abstract This contribution is concerned with the well‐posedness and homogenization of an ordinary differential equation (ODE) of Arrhenius‐type coupled with a doubly nonlinear parabolic partial differential equation (PDE) with rapidly oscillating coefficients and taking into account disparate diffusion‐reaction time scales, including regularly as well as singularly perturbed problems. The ODE‐PDE system is spatially dependent and is subjected to Robin‐type boundary conditions. Such problems are used to model a variety of processes and phenomena such as combustion and exothermal chemical reactions. We will have a special look at the questions of the existence, uniqueness, boundedness, and the asymptotic limit of the microscale problem by applying the two‐scale convergence and unfolding method. A numerical example illustrates both the expected behavior of the approximated solutions as well as the capability of the proposed upscaled models.