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

Abstract

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 $H^{1}_{\textrm{lu}}({{\mathbb {R}}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$ and define a strong continuous analytic semigroup. Secondly, the existence of the $(H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),H^{1}_{\rho }({\mathbb {R}}^N)\times H^{1}_{\rho }({\mathbb {R}}^N) )$-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 $H^{2}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{2}_{\textrm{lu}}({\mathbb {R}}^N$)), which attracts exponentially every initial $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$-bounded set with respect to the $H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$-norm.