(?)-Pullback Attractor For The Strongly Damped Wave Equations

The understanding of the asymptotic behaviour of dynamical systems is one of the most important problems of modern mathematical physics.One way to treat this problem for a system generated by an autonomous partial differential equation is to analyse the existence and structure of its global attractor.However,non-autonomous systems are also of great importance and interest as they appear in many applications to natural sciences.In recent years more attention was paid to more general non-autonomous differential equations and the processes generated by them.Different approaches were made to find the counterpart of the global attractor in this case.One of the most suitable ones defines the notion of the pullback global attractor as a minimal family of compact invariant sets under the process and pullback attracting each bounded subset of the phase space.The theory of pullback attractors has shown to be very useful in the understanding of the dynamics of non-autonomous dynamical systems.A necessary condition for the existence of pullback attractors for non-autonomous dynamical systems is pullback asymptotic compactness.A technical method for verifying pullback asymptotic compactness,via contractive functions,is developed by C.Y.Sun in[21].In this paper,firstly,we give the equivalence condition of(?)-Pullback asymptotically compactness for a cocycle generated by a non-autonomous system.Secondly,we propose a method for verifying(?)-pullback asymptotic compactness,via cont.ractive functions.Finally,we apply our theory to prove the existence of(?)-pullback attractor for a non-autonomous strongly damped wave equations with super critical nonlinearity which the external force needs not be bounded.It is enough that this term satisfies an appropriate integrability condition.