Reaction-diffusion transport into core-shell geometry: Well-posedness and stability of stationary solutions

TL;DR

This paper analyzes a reaction-diffusion model with discontinuous coefficients in a core-shell geometry, establishing well-posedness, uniqueness, and stability of stationary solutions using monotone operator theory.

Contribution

It introduces a rigorous mathematical framework for reaction-diffusion equations with discontinuous diffusion coefficients in complex geometries, proving well-posedness and stability.

Findings

Well-posedness of the nonlinear PDE established

Unique and asymptotically stable stationary solutions proven

Application of monotone operator theory to complex geometries

Abstract

We investigate a nonlinear parabolic reaction-diffusion equation describing the oxygen concentration in encapsulated pancreatic cells with a general core-shell geometry. This geometry introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. We apply monotone operator theory to show well-posedness of the problem in the strong form. Furthermore, the stationary solutions are unique and asymptotically stable. These results rely on the gradient structure of the underlying PDE.