We first prove the existence of a compact positively invariant set which exponentially attracts any bounded set for abstract multi-valued semidynamical systems. Then, we apply the abstract theory to handle retarded ordinary differential equations and lattice dynamical systems, as well as reactiondiffusion equations with infinite delays. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.

<p style='text-indent:20px;'>In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in <inline-formula><tex-math id="M2">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \alpha&gt;\frac{1}{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ s&gt;1. $\end{document}</tex-math></inline-formula> We prove the existence of <inline-formula><tex-math id="M5">\begin{document}$ (H^{2\alpha^-+s}(\mathbb{T}^2),H^{2\alpha+s}(\mathbb{T}^2)) $\end{document}</tex-math></inline-formula>-global attractor <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A}, $\end{document}</tex-math></inline-formula> that is, <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> is compact in <inline-formula><tex-math id="M8">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> and attracts all bounded subsets of <inline-formula><tex-math id="M9">\begin{document}$ H^{2\alpha^-+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> with respect to the norm of <inline-formula><tex-math id="M10">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2). $\end{document}</tex-math></inline-formula> The asymptotic compactness of solutions in <inline-formula><tex-math id="M11">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Furthermore, we show the existence of the exponential attractor in <inline-formula><tex-math id="M12">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2), $\end{document}</tex-math></inline-formula> whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of <inline-formula><tex-math id="M13">\begin{document}$ H^{2\alpha^-+s}(\mathbb{T}^2) $\end{document}</tex-math></inline-formula> are all in the topology of <inline-formula><tex-math id="M14">\begin{document}$ H^{2\alpha+s}(\mathbb{T}^2). $\end{document}</tex-math></inline-formula></p>