The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

Abstract

In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher-order Kirchhoff equation with variable coefficients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of Dk{{\mathcal{D}}}_{k}-absorbing sets of the stochastic dynamical system Φk{\Phi }_{k} is obtained, and the asymptotic compactness of Φk{\Phi }_{k} is proved by the decomposition method. Finally, the Dk{{\mathcal{D}}}_{k}-stochastic attractor family of the stochastic dynamical system Φk{\Phi }_{k} in Vm+k(Ω)×Vk(Ω){V}_{m+k}\left(\Omega )\times {V}_{k}\left(\Omega ) is obtained.

Keywords

• nonautonomous higher-order kirchhoff equation
• partial differential equations
• variable coefficient
• the family of random attractors
• additive noise
• 37b55
• 35b41
• 35g31
• 60h15