The Non-autonomous Dynamical System Pulls Back The Time Dependence Of The Attractor

This thesis studies the time-dependent compactness and asymptotic behavior of pullback attractors for non-autonomous dynamical systems.First,we establish a theoretical criterion on the existence of backward compact pullback attractors for non-autonomous dynamic systems with the backward compact-decay decom-position.That is,when a non-autonomous process S has a backward compact-decay decomposition,There is a backward compact attractor if and only if there exists an increasing,bounded and pullback absorbing set.We apply these ab-stract criteria and consider the backward compact dynamics of space-periodic so-lutions for the non-autonomous complex-valued Schr(?)dinger equation with time dependent force in R:where a>0,u(t,x)is an unknown complex-valued function.An increasing,bounded and pullback absorbing set is obtained if the forcing and its time-derivative are backward uniformly integrable.Split the solution u(t)of equation into low-frequency part and high-frequency part,The method of energy,high-low frequency decomposition,Sobolev embedding and interpolation are quite involved in calculating a priori pullback or forward bound,we show that the non-autonomous Schr(?)dinger equation has a pullback attractor which is compact in the past.Then,we study the robustness of a pullback attractor as the time tends to infinity.We prove that the forward(resp.backward)compactness is a necessary and sufficient condition such that a pullback attractor is upper semi-continuous to a compact set at positive(resp.negative)infinity,and also obtain the minimal limit-set.We further prove the lower semi-continuity of the pullback attractor and get the maximal limit-set at infinity.Some criteria for such robustness are estab-lished when the evolution process is forward or backward omega-limit compact.As an application,We consider the following non-autonomous Ginzburg-Landau equation:where s ? R,?,?,?>0,? is a bounded smooth domain in Rn with n = 1,2 and the unknown a is a complex-valued function.Finally,we study the asymptotically autonomous dynamics for parabolic e-quations,it is shown that the components of the pullback attractor converge to the global attractor if and only if the pullback attractor is forward compact,this result reduces two uniformness conditions in the theoretical result given by Kloe-den and Simsen(J.Math.Anal.Appl.,2015,2017).Some constructions from the forward limit-set of a pullback attractor are established.Some applications are given for two classes of non-autonomous quasi-linear parabolic equations:1?The Laplace equation with spatially variable exponents on a bounded domain ?(?)Rn ut-div(D(t,x)|?u|p(x)-2?u)+ |u|p(x)-2u = B(u);2?The p-Laplace equation on an unbounded domain with dissipative non-linearity and non-autonomous forcing:where ?>0,p>2,and the p-Laplace operator A:W1,P(Rn)? W-1,P'(Rn)is defined by where the nonlinearity is weakly dissipative if p>q.By new methods of induc-tion absorption and varying attraction basins,using cut-off function,we obtain the tail-estimate of solutions,so the pullback attractors is forward compact.We show that the pullback attractor converge to the global attractor.