This dissertation is a contribution to the the study of the longtime dynamics of evolutionary equations in unbounded domains. It is of particular interest to prove the existence of global attractors for solutions of such equations. Th this end one need in general some type of asymtotical compactness. In the case the evolutionary PDE is defined on a bounded domain, asymptotical compactness follows from the regularity estimates and the compactnes of Sobolev embeddings and therefore the existence of attractors has been established for most of the disipative equations of mathematocal physics in a bounded domain. The problem is more challenging when the domain is unbounded since the Sobolev embeddings are no longer comapct, so that the usual regularity estimates may not be sufficient.To overcome this obstacle of compactness, A.V. Babin and M.I. Vishik introduced some weighted Sobolev spaces. In their pioneering paper, Proc. Roy. Soc. Edinb.