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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 $$u_{tt}+\alpha u_t-\Delta u+\beta\sin u=q\dot W$$ on an open bounded $\Omega\subset\Bbb R^n$ 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 {\it H. Crauel} and {\it F. Flandoli} [J. Dyn. Differ. Equations 10, 449-474 (1998; Zbl 0927.37031)].

37L30Attractors and their dimensions, Lyapunov exponents
35R60PDEs with randomness, stochastic PDE
35B41Attractors (PDE)
35Q53KdV-like (Korteweg-de Vries) equations
37L55Infinite-dimensional random dynamical systems; stochastic equations
60H15Stochastic partial differential equations
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