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 Hlu1(RN)×Hlu1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}_{\textrm{lu}}({{\mathbb {R}}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$$\end{document} and define a strong continuous analytic semigroup. Secondly, the existence of the (Hlu1(RN)×Hlu1(RN),Hρ1(RN)×Hρ1(RN))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N),H^{1}_{\rho }({\mathbb {R}}^N)\times H^{1}_{\rho }({\mathbb {R}}^N) )$$\end{document}-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 Hlu2(RN)×Hlu2(RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{2}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{2}_{\textrm{lu}}({\mathbb {R}}^N$$\end{document})), which attracts exponentially every initial Hlu1(RN)×Hlu1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$$\end{document}-bounded set with respect to the Hlu1(RN)×Hlu1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}_{\textrm{lu}}({\mathbb {R}}^N)\times H^{1}_{\textrm{lu}}({\mathbb {R}}^N)$$\end{document}-norm.