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

TL;DR

This paper establishes the well-posedness of a semilinear reaction-diffusion system with two spatial scales, modeling concrete corrosion, by proving existence, uniqueness, and stability of solutions using fixed-point and Galerkin methods.

Contribution

It provides a rigorous mathematical analysis ensuring the solvability and stability of a complex two-scale reaction-diffusion model for concrete corrosion.

Findings

Proved global-in-time existence and uniqueness of solutions.

Established positivity and boundedness of concentrations.

Demonstrated stability with respect to data and parameters.

Abstract

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 \cite{tasnim1} by upscaling of the physical and chemical processes taking place within the microstructure of the concrete. Herein we ensure the positivity and $L^\infty-$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.