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 \(\Omega\subset\mathbb R^n\) 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)
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) |