The Pullback Attractor And Its Dimension Estimation For A Class Of Evolution Equations On Unbounded Domains

The problem of understanding the long-time behavior of non-autonomous evolution equations is generally considered to be of fundamental importance,due to its relevance to many physical phenomena.In this paper we study the pullback asymptotic behavior for some non-autonomous evolution equations when the underlying Euclidean domain is unbounded.We are devoted to the following non-autonomous evolution equation on a Hilbert space H:(?)Where A is an unbounded positive linear operator on V?H,B and R are nonlinear operators on V.In the first section,we prove the existence of a pullback attractor for our model in an unbounded domain in which the external force needs not to be bounded,neither almost periodic nor translation compact.It is enough that the external force satisfies an appropriate integrability condition.It is known that the theory of global attractors previously developed for autonomous systems is no longer valid to analyse the long-term behaviour of this kind of non-autonomous problems,and it is necessary to use some techniques for non-autonomous systems.In order to prove the existence of the pullback attractor we will exploit an approach which is based on the use of the energy equations which are in direct connection with the concept of pullback asymptotic compactness,and which permits us to obtain information on the pullback asymptotic behaviour of our systems when the injection V?? is not compact(as in our case of unbounded domain).In the second section,we study some examples that fit in that framework.In the third section,we estimate the fractal dimension of pullback attractors associated to the examples mentioned above.