Long-Ttime Dynamical Behavior Of The Non-autonomous Strongly Damped Wave Equations

In this paper,we consider the long-time behavior of the evolution process generated by the non-autonomous strongly damped wave equations.The long-time behavior of dynamical systems is an important and challenging problem,since it can provide useful information on the future evolution of the systemThe long-time behaviour of dissipative dynamical systems generated by evolution equa-tions of mathematical physics can be described in terms of the so-called global attractors,which is,by definition,a compact invariant subset of the phase space,which attracts the images of all the bounded subsets as time goes to infinity.However,the approach based on the concept of global attractors has two rather essen-tial drawbacks:on the one hand,the rate of convergence can be arbitrarily slow and it is usually very difficult to estimate this rate in terms of the physical parameters of the prob-lem and,on the other hand,the global attractor is,in general,only upper semi-continuous with respect to perturbations,so that the global attractor can change drastically under very small perturbations of the initial dynamical system.These drawbacks obviously lead to very essential difficulties in numerical simulations of global attractors and even make the global attractor,in some sense,unobservableIn order to overcome these drawbacks,the concept of an exponential attractor has been suggested.Like global attractors of semigroups in the autonomous context,global pullback attractors are generally not stable under perturbations and the rate of convergence to the attractor is unknown,which motivates to consider pullback exponential attractorsPullback exponential attractors are families of compact subsets of the phase space whose fractal dimension is uniformly bounded and that pullback attract all bounded sets at an exponential rate.They are,due to the exponential rate of attraction,more stable under perturbations and contain the global pullback attractor.In particular,to show the existence of a pullback exponential attractor is one way of proving the existence and finite dimensionality of the global pullback attractor.In this paper,we prove the existence of pullback exponential attractors for evolution processes generated by non-autonomous strongly damped wave equation with critical expo-nent.To do this end,we show that the generated evolution process satisfies the smoothing property and possesses a semi-invariant family of pullback absorbing sets.We prove that the evolution process can be represented as a sum U=S+C,where the family of operators S satisfies the smoothing property with respect to phase spaceV:=H01(?)×L2(?)and an auxiliary normed space W compactly embedded into V,and C is a family of contractions in the stronger space V.