| This paper mainly studies the existence and uniqueness of the pullback attrac-tor of the stochastic delay generalized Kuramoto-Sivashinsky equation with periodic initial value boundary conditions.Moreover,the upper semicontinuity of the pull-back attractor when the time delay tends to zero is established.The specific equation is as follows:where τ∈R,α>0,β>0,γ>0,constant ρ>0 is the delay time of the system,h is a tempered time-dependent forcing,f,φ,g are nonlinear terms,and F is a nonlinear term with delay.The first part introduces the research background and current situation of this article.The second part introduces the related concepts and the existence and unique-ness theorem of the pullback random attractor of the dynamical system.The third part proves the existence of the pullback random attractor of the nonautonomous stochastic delay generalized Kuramoto-Sivashinsky equation with periodic initial value conditions.In order to obtain the existence of the pullback random attractor,Both the asymptotically compactness of the dynamical system in the state space and the existence of the closed measurable pullback absorbing set need to be proved.The difficulty here is mainly the asymptotically compactness of the dynamical system in the state space,we need to use the estimation of the solution of the higher order derivative of the generalized Kuramoto-Sivashinsky equation and the Arzela-Ascoli theorem to obtain the asymptotically compactness of the dynamical system in the state space.The last part proves the upper semicontinuity of the pullback random attractor when the delay time tends to zero.The conclusions of the third part are correct when the delay time is any value,and the conclusions do not need to be proved repeatedly.Therefore,the difficulty in this part is to prove the stability of the dynamical system when the delay time tends to zero.This requires transforming some of the conclusions in the third part to apply them. |
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