Fan, Xiaoming Random attractor for a damped sine-Gordon equation with white noise. (English) Zbl 1065.37057 Pac. J. Math. 216, No. 1, 63-76 (2004). 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 H. Crauel and F. Flandoli [J. Dyn. Differ. Equations 10, 449-474 (1998; Zbl 0927.37031)]. Reviewer: Hans Crauel (Ilmenau) Cited in 47 Documents MSC: 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35R60 PDEs with randomness, stochastic partial differential equations 35B41 Attractors 35Q53 KdV equations (Korteweg-de Vries equations) 37L55 Infinite-dimensional random dynamical systems; stochastic equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) Keywords:random attractor; sine-Gordon equation; Hausdorff dimension Citations:Zbl 0927.37031 PDFBibTeX XMLCite \textit{X. Fan}, Pac. J. Math. 216, No. 1, 63--76 (2004; Zbl 1065.37057) Full Text: DOI