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,?)},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 ? induced by the dynamical system ?,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.