Pullback Exponential Attractors For A Semilinear Dissipative Wave Equation On R^N

One of the cenral problems in the theory of dynamical systems is to study the asymptotic behavior of evolutionary processes related to real-world modeling.At present,the study of the asymptotic behavior of autonomous differential equation is quite mature,but there is still a lot of work to be done on the asymptotic behavior of the model of non-autonomous differential equation,especially that in the unbounded domain.This thesis studies the long time dynamic behavior of the following non-autonomous semilinear dissipative wave equation:utt+δut-φ（x）Δu+λf（u）=η（x,t）,x∈RN,t≥0.WhereN≥3,（φx）)-1=:g（x）∈LN/2（RN）∩L∞（RN）.In the theory of infinite dimensional dynamical systems,the long time dynamic behavior of a dissipative partial differential equation can be described by means of a global attractor.But,there are two drawbacks in the method based on the study of the concept of the global attractor:on the one hand,the convergence speed of attractors can be arbitrarily slow,and often hard to estimate;on the other hand,the global attractor under perturbation is usually only upper semi-continuous,so even in very small perturbation,the global attractor may also lead to radical changes.In order to overcome these drawbacks,some scholars put forward the concept of an exponential attractor.However,an exponential attractor is not unique,so it is very important to choose a suitable exponential attractor.Recently,some scholars proposed the concept of a pullback exponential attractor and its construction method.Pullback exponential attractors attract any bounded subsets of the phase space at an exponential rate,and it contains the pullback global attractor.Therefore,proving the existence of the pullback global attractor and its finite dimensionality can be accomplished by proving the existence of the pullback exponential attractor.In this thesis,we prove the existence of the pullback exponential attractor for the evolution process generated by the above non-autonomous semi-linear dissipative wave equation in RN.For this end,it is proved the pullback strongly bounded dissipativity of the evolution process generated by the equation,and also proved that the evolution process can be decomposed into U=S+C,where the operator family S satisfies the smoothing property with respect to phase space V:=D1,2（RN）×Lg2（RN）and the auxiliary normed space W compactly embedded into V,and the family of operators C is a contraction operator on the space V.In order to overcome the difficulties caused by the non-compactness of the operator A=-φΔin an unbounded domain,this thesis has proved the above result in the energy space V=D1,2（RN）×L2g2（RN）.