Existence of a global weak solution for a reaction-diffusion problem with membrane conditions

TL;DR

This paper proves the existence of weak solutions for a reaction-diffusion system with membrane boundary conditions, extending $L^1$ theory to models with selective permeability and quadratic reaction terms.

Contribution

It adapts existing $L^1$ reaction-diffusion theory to membrane boundary conditions, establishing weak solutions with minimal initial regularity.

Findings

Existence of weak solutions under $L^1$ initial data.

Establishment of $W^{1,1}$ regularity for solutions.

Extension of $L^2$ integrability lemma to membrane conditions.

Abstract

Several problems, issued from physics, biology or the medical science, lead to parabolic equations set in two sub-domains separated by a membrane with selective permeability to specific molecules. The corresponding boundary conditions, describing the flow through the membrane, are compatible with mass conservation and energy dissipation, and are called the Kedem-Katchalsky conditions. Additionally, in these models, written as reaction-diffusion systems, the reaction terms have a quadratic behaviour. M. Pierre and his collaborators have developed a complete $L^1$ theory for reaction-diffusion systems with different diffusions. Here, we adapt this theory to the membrane boundary conditions and prove the existence of weak solutions when the initial data has only $L^1$ regularity using the truncation method for the nonlinearities. In particular, we establish several estimates as the $W^{1,1}$ regularity of the solutions. Also, a crucial step is to adapt the fundamental $L^2$ (space, time) integrability lemma to our situation.