Stability Of Attractors For Processes On Time-dependent Spaces And Applications

In this doctoral dissertation,we study the stability of pullback l-attractors and pullback l-exponential attractors for the perturbed evolution processes on time-dependent phase spaces.Firstly,we establish a criterion on the upper semicontinuity of pullback l-attractors with respect to perturbed parameters ? ? A,in the sense of the Haus-dorff semi-metric.On that basis,we show a criterion on the residual continuity of pullback l-attractors by using the Baire category theorem,that is,there is a residual subset A*of A such that the pullback l-attractor is continuous at each? ? A*with respect to the Hausdorff metric.By using the Dini theorem,we further prove that the continuity of pullback l-attractors with respect to A is equivalent to the pullback equi-attraction.Secondly,the concept of pullback l-exponential attractor in time-dependent phase spaces is proposed,and the first criteria on the existence and Holder con-tinuity of pullback l-exponential attractors of perturbed evolution processes defined on the time-dependent phase spaces,are given by using quasi-stability method.Finally,by using the criteria given above,we study the longtime behavior of two classes of nonlinear evolution equations on three-dimensional smooth bounded domain ?,and show that:(i)we give the first result on the uniqueness and regularity of energy weak solutions for the following quasi-linear damped wave equations with non-standard growth conditions#12 and prove the existence,regularity and stability with respect to perturbed pa-rameters A of pullback l-attractors and pullback l-exponential attractors for the related perturbed evolution processes defined on the time-dependent Orlicz-Sobolev spaces(?)(ii)we also prove the well-posedness of the following viscoelastic model with time-dependent memory kernals#12 and give a new method to obtain the existence,optimal regularity,finite frac-tal dimensions and stability with respect to viscosity parameters A of pullback l-attractors and pullbac l-exponential attractors for the related evolution pro-cesses.