Existence Of Time-dependent Attractors For Berger Equations With Nonlinear Damping

In this paper,we study the asymptotic behavior of solutions for the Berger equations with nonlinear damping in the time-dependent space.i)This part study the Berger equations with nonlinear damping:(?)then,the existence of time-dependant global attractors is proved.Due to the existence of nonlinear term and the non-local term(M(‖▽u‖2)△u)of Kirchhoff type,the verification of the compactness solution of the equation becomes difficult and key.In ordr to overcome the difficulties,we establish the cor-responding solution process {U(t,τ)},and using asymptotic priori estimation tech-nique and contraction function method.Finally,the existence of time-dependant global attractors in(H2∩H01)× L2 is obtained.ii)This part study the Berger equations with nonlinear damping and non-autonomous external force:(?)then,the existence of time-dependant pullback attractors is proved.Since the equation contains nonlinear terms,nonlocal term M(‖▽u‖2)△u of Kirchhoff type,and the external force term h(x,t)depended on time t,it is difficult to verify the compactness of the solution of the equation.Therefore,by use of the cocyclic function φ induced by the dynamical system θ,asymptotic priori estimation technique and contraction function method,we overcome the above essential difficul-ties.Finally,the existence of time-dependant pullback attractors in(H2∩H01)×L2 is gained.