Distributed space scales in a semilinear reaction-diffusion system including a parabolic variational inequality : a well-posedness study

This paper treats the solvability of a semilinear reaction-diffusion system, which incorporates transport (diffusion) and reaction effects emerging from two separated spatial scales: x - macro and y - micro. The system’s origin connects to the modeling of concrete corrosion in sewer concrete pipes. It consists of three partial differential equations which are mass-balances of concentrations, as well as, one ordinary differential equation tracking the damage-by-corrosion. The system is semilinear, partially dissipative, and coupled via the solid-water interface at the microstructure (pore) level. The structure of the model equations is obtained in [7] by upscaling of the physical and chemical processes taking place within the microstructure of the concrete. Herein we ensure the positivity and L8-bounds on concentrations, and then prove the global-in-time existence and uniqueness of a suitable class of positive and bounded solutions that are stable with respect to the two-scale data and model parameters. The main ingredient to prove existence include fixed-point arguments and convergent two-scale Galerkin approximations. Keywords: Reaction and diffusion in heterogeneous media, two-scale Galerkin approximations, parabolic variational inequality, well-posedness