Coupled Stochastic Allen-Cahn equations: Existence and Uniqueness

TL;DR

This paper establishes the global existence and uniqueness of solutions for a coupled system of stochastic Allen-Cahn equations with interaction, driven by space-time white noise, using methods from Doering (1987).

Contribution

It provides the first rigorous proof of existence and uniqueness for coupled stochastic Allen-Cahn equations with interaction and white noise.

Findings

Proved global existence of solutions in continuous function space.

Established uniqueness of solutions under specified conditions.

Applied and adapted Doering's methods to coupled stochastic PDEs.

Abstract

In this paper, we concern a system of stochastic PDE's. Our system consists of two components. Each component evolves according to the sotchastic Allen-Cahn equation with a symmetric double well potential and with addtional small space-time white noise. The two components interact with each other by an attractive linear force. We aim to give a global existence and uniqueness result in a space of continuous functions on $\mathbb{R} \times \mathbb{R}^+$. We apply methods proposed by Doering \cite{Doering1987}.