Random attractor for a damped sine-Gordon equation with white noise
It is shown that a sine-Gordon equation with additive white noise, formally given by \[ u_{tt}+\alpha u_t-\Delta u+\beta\sin u=q\dot W \] on an open bounded \(\Omega\subset\mathbb R^n\) with smooth boundary, where \(\alpha>0\), \(q\in H^2(\Omega)\cap H_0^1(\Omega)\), \(\dot W\) is the formal derivative of a one-dimensional Wiener process, imposing Dirichlet conditions, has a random attractor. A nonrandom upper bound for the Hausdorff dimension of the random attractor, which decreases as the damping \(\alpha\) grows, is derived. The Hausdorff dimension of random attractors has been shown to be nonrandom almost surely by \textit{H. Crauel} and \textit{F. Flandoli} [J. Dyn. Differ. Equations 10, 449-474 (1998; Zbl 0927.37031)].
scientific article; zbMATH DE number 2138435