WEAK SOLUTIONS FOR COUPLED REACTION-DIFFUSION SYSTEMS WITH PATTERN FORMATION BY A STOCHASTIC FIXED POINT THEOREM

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 systemwhere ri, bi, ci, i > 0, ai R, and gi are linear, i = 1, 2, and the exponent q 1.The operator A = -(-) /2 is a fractional power of the Laplacian, 1 < 2. 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 martingale solution of the coupled system.