Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

This paper deals with the asymptotic behavior of a class of second-order evolution equations with dispersive and dissipative terms' critical nonlinearity in locally uniform spaces. First of all, we prove the global well-posedness of solutions to the evolution equations in the locally uniform spaces H-lu(1)(R-N) x H-lu(1)(R-N) and define a strong con-tinuous analytic semigroup. Secondly, the existence of the (H-lu(1)(R-N) x H-lu(1)(R-N ), H-?(1) (R-N) x H-?(1) (R-N))-global attractor is established. Finally, we obtain the asymptotic regularity of solutions which appear to be optimal and the existence of a bounded subset(in H-lu(2)(R-N) xH(lu)(2)(R-N)), which at-tracts exponentially every initial H-lu(1)(R-N) x H-lu(1)(R-N)-bounded set with respect to the H-lu(1)(R-N) x H-lu(1)(R-N)-norm.