Abstract
The existence of random attractor family for a class of nonlinear high-order Kirchhoff equation stochastic dynamical systems with white noise is studied. The Ornstein-Uhlenbeck process and the weak solution of the equation are used to deal with the stochastic terms. The equation is transformed into a general stochastic equation. The bounded stochastic absorption set is obtained by estimating the solution of the equation and the existence of the random attractor family is obtained by isomorphic mapping method. Temper random compact sets of random attractor family are obtained.
Highlights
In this paper, we study the random attractor family of solutions to the strongly damped stochastic Kirchhoff equation with white noise:( ) utt + M Dmu 2 (−∆)m u + β (−∆)m ut + g ( x,u) = q ( x)W, (1.1)with the Dirichlet boundary condition u ( x,=t ) ∂i=u ∂vi 0=, i1, 2,⋅⋅⋅, m −1, x ∈ ∂Ω,t > 0, (1.2)and the initial value conditions u= ( x, 0) u0 ( x),u= t ( x, 0) u1 ( x), x ∈ Ω ⊂ Rn, (1.3)where m > 1 is a positive integer; β > 0 is a constant; Ω is a bounded region with smooth boundary in Rn . ∆ is the Laplacian with respect to the va
The Ornstein-Uhlenbeck process and the weak solution of the equation are used to deal with the stochastic terms
The bounded stochastic absorption set is obtained by estimating the solution of the equation and the existence of the random attractor family is obtained by isomorphic mapping method
Summary
We study the random attractor family of solutions to the strongly damped stochastic Kirchhoff equation with white noise:. Xu et al [5] studied the non-autonomous stochastic wave equation with dispersion and dissipation terms. The existence of random attractors for non-autonomous stochastic wave equations with product white noise is obtained by using the uniform estimation of solutions and the technique of decomposing solutions in a region. Lin et al [6] studied the existence of stochastic attractors for higher order nonlinear strongly damped Kirchhoff equation.
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