Well-posedness of a reaction-diffusion model with stochastic dynamical boundary conditions

TL;DR

This paper proves the global existence and uniqueness of solutions for a reaction-diffusion PDE on the half-line with a stochastic boundary condition modeled by a Jacobi process, relevant to chemical reactions.

Contribution

It establishes well-posedness results for a nonlinear reaction-diffusion system with stochastic dynamical boundary conditions, a novel combination in this context.

Findings

Proves global existence of solutions.

Establishes pathwise uniqueness.

Handles low regularity boundary conditions.

Abstract

We study the well-posedness of a nonlinear reaction diffusion partial differential equation system on the half-line coupled with a stochastic dynamical boundary condition, a random system arising from the description of the chemical reaction of sulphur dioxide with calcium carbonate stones. The boundary condition is given by a Jacobi process, solution to a stochastic differential equation with a mean-reverting drift and a bounded diffusion coefficient. The main result is the global existence and the pathwise uniqueness of mild solutions. The proof relies on a splitting strategy, which allows to deal with the low regularity of the dynamical boundary condition.