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Random attractor for a damped sine-Gordon equation with white noise. (English) Zbl 1065.37057

It is shown that a sine-Gordon equation with additive white noise, formally given by

${u}_{tt}+\alpha {u}_{t}-{\Delta }u+\beta sinu=q\stackrel{˙}{W}$

on an open bounded ${\Omega }\subset {ℝ}^{n}$ with smooth boundary, where $\alpha >0$, $q\in {H}^{2}\left({\Omega }\right)\cap {H}_{0}^{1}\left({\Omega }\right)$, $\stackrel{˙}{W}$ 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)].

##### MSC:
 37L30 Attractors and their dimensions, Lyapunov exponents 35R60 PDEs with randomness, stochastic PDE 35B41 Attractors (PDE) 35Q53 KdV-like (Korteweg-de Vries) equations 37L55 Infinite-dimensional random dynamical systems; stochastic equations 60H15 Stochastic partial differential equations