Weak solutions for coupled reaction-diffusion systems with pattern formation by a stochastic fixed point theorem

TL;DR

This paper proves the existence of weak solutions for stochastic reaction-diffusion systems with pattern formation, using a fixed point theorem applied to the laws of associated Ornstein-Uhlenbeck processes.

Contribution

It introduces a novel approach employing a Schauder-Tychonoff fixed point theorem to establish weak solutions for coupled stochastic reaction-diffusion equations with fractional Laplacian.

Findings

Existence of probabilistic weak solutions for the system.

Application of fixed point theorem to infinite-dimensional Ornstein-Uhlenbeck laws.

Framework applicable to systems with fractional Laplacian operators.

Abstract

Chemical and biochemical reactions can exhibit surprisingly different behaviours, ranging from multiple steady-state solutions to oscillatory solutions and chaotic behaviours. These types of systems are often modelled by a system of reaction-diffusion equations coupled by a nonlinearity. In the article, we study the existence of stochastically perturbed equations of this type. In particular, we show the existence of a probabilitic weak solution of the following stochastic system \begin{align*} \dot {u} & = r_1\,\Delta u+ a_1\, u + b_1 -c_1\, u\cdot v^q+\sigma_1\, g_1(u)\circ \dot W_1, \\ \dot{v} & = r_2 \,A v + a_2\, v + b_2 +c_2\, u\cdot v^q + \sigma_2\, g_2(v)\circ \dot W_2, \end{align*} where $r_i,b_i,c_i, \sigma_i>0$, $a_i\in\mathbb{R}$, and $g_i$ are linear, $i=1,2$, and the exponent $q\geq 1$. The operator $A=-(-\Delta)^{\aleph/2}$ is a fractional power of the Laplacian, $1<\aleph \le2$. The main result is obtained by a Schauder-Tychonoff type fixed point theorem for the controlled versions of the laws of the respective (infinite dimensional) Ornstein-Uhlenbeck system, from which we infer the existence of a weak solution of the coupled system.