Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise

Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

“…In the stochastic case, pathwise random attractors of lattice systems have been examined in [6,7,15,20,44,45,47] for the first-order systems, and in [17,48] for second-order systems. For random attractors of stochastic PDEs, the reader is referred to [2,3,8,9,11,25,26,27,28,31,33,40,43,50,46] for the first-order equations, and to [41,42,51] for the second-order equations.…”

Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise

“…Recently, random dynamic systems have drawn much attention from researchers due to their wide range of applications. The random attractor is one of most important tools which are used to determine long-time behavior of solutions for autonomous or non-autonomous random dynamic systems, see [1,12,4,3,10,9,2,15,16]. However, random attractors attract the orbit slowly compared to exponential attractors, see [13], which are not conducive to numerical estimation and application.…”

Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

“…The random attractor, was first studied by Ruelle [45,46], is one of most important concepts to describe long-term behavior of solutions for a given random system to capture the essential dynamics with possibly extremely wide fluctuations. Later, Crauel, Debussche, Flandoli, Imkeller, Langa, Schmalfuss, Robinson, Bates, Lu, Caraballo, Kloeden, Wang etc., developed some general theories of random attractors (mainly on existence, semi-continuity and bound of Hausdorff/fractal dimensions) and applications to stochastic evolution equations (such as Navier-Stokes equation, reaction-diffusion equations, wave equations and lattice systems driven by random perturbation or noises), see [4,6,10,11,12,18,19,28,29,30,31,33,34,44,46,49,53,54,65,66] and the references wherein.…”

“…The random attractor and the bounds of its Hausdorff and fractal dimensions for the stochastic wave equations with additive noise (i.e., the random term in (1) is "adW (t)" independent of u) have been studied by many authors, see [12,13,8,18,21,38,54,57,63,65]. For the stochastic system (1) with linear multiplicative noise "au • dW (t)" (depending on the state variable u) and sufficient small coefficient |a| of random term, when the nonlinear function f has a subcubic growth exponent (i.e., f 1 ≡ 0 in (A1)), the existence and the boundedness of fractal dimension of random attractor were studied, see [22,36,52,66], of those, Zhou and Zhao in [66] gave some sufficient conditions to bound the fractal dimension of a random invariant set for a cocycle and applied these conditions to get an upper bound of fractal dimension of the random attractor of system (1).…”