dc.description.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]. | |