Synchronization of a Stochastic Reaction-Diffusion System On a Thin Two-Layer Domain

Abstract

A system of semilinear parabolic stochastic partial differential equations with additive space-time noise is considered on the union of thin bounded tubular domains D1,ε := Γ × (0, ε) and D2,ε := Γ × (−ε, 0) joined at the common base Γ ⊂ Rd, where d ≥ 1. The equations are coupled by an interface condition on Γ which involves a reaction intensity k(x , ε), where x = (x , xd+1) ∈ Rd+1 with x ∈ Γ and |xd+1| < ε. Random influences are included through additive space-time Brownian motion, which depend only on the base spatial variable x ∈ Γ and not on the spatial variable xd+1 in the thin direction. Moreover, the noise is the same in both layers D1,ε and D2,ε. Limiting properties of the global random attractor are established as the thinness parameter of the domain ε → 0, i.e., as the initial domain becomes thinner, when the intensity function possesses the property limε→0 ε−1k(x , ε) = +∞. In particular, the limiting dynamics is described by a single stochastic parabolic equation with the averaged diffusion coefficient and a nonlinearity term, which essentially indicates synchronization of the dynamics on both sides of the common base Γ. Moreover, in the case of nondegenerate noise we obtain stronger synchronization phenomena in comparison with analogous results in the deterministic case previously investigated by Chueshov and Rekalo [EQUADIFF-2003, F. Dumortier et al., eds., World Scientific, Hackensack, NJ, 2005, pp. 645–650; Sb. Math., 195 (2004), pp. 103–128].