The Attractors Of Operator Semigroup On The Topological Space

According to the global attractors and global B-attractors of the operator semigroup defined by Olga Ladyzhenskaya on the complete metric space,which is extended to the Hausdorff topological space to study the global attractors of the operator semigroup.We noticed that the global B-attractors are defined by attracts the set of each bounded set in the space,but the definition of bounded set is not in topological space,so we find the relative compact set and the basically bounded set to replace the bounded set,which defines the global C-attractors and the global S-attractors.According to the basically bounded set defines the semigroup of class basic(?).Then combined with the topological knowledge,the limit set is studied by the method of the Cartesian directed set which defined by Mr.Xia Daoxing,and the concept of net convergence on the Hausdorff topological space.It plays a key role for giving related concepts of attractor of operator semigroup on the topological space.In this paper,first,several conditions are established for the existence of the global attrac-tors,the global C-attractors and the global S-attractors.For example,let {Vt} be a semigroup,suppose that it has a finite global absorbing set B0,then it has a non-empty minimal global attractor,which is compact and invariant,and M=?(B0);if {V} has a relative compact global C-absorbing set C0,then it has a non-empty minimal global C-attractor,which is compact and invariant,and Mc=?(C0);if the semigroup {Vt} belongs to class basic(?),it has a basic bounded global S-absorbing set S0,then it has a non-empty minimal global S-attractor,which is compact and invariant,and Ms=?-(S0).Then the paper also discussed the existence con-dition of the minimal compact of the global attractors,the global C-attractors and the global S-attractors.Finally,proving the theorem of connectivity of the minimal compact attractitive sets on the Hausdorff topological space,the connectivity of these three classes of global attrac-tors are obtained.