In this paper, we prove the upper semicontinuity of the strong global attractors on the dissipative index in the topology of the stronger space for the Kirchhoff wave model with structural nonlinear damping:, with ∈ [1∕2, 1). It is continuation of the research in recent literatures 1,2 where the upper semicontinuity of the strong attractor on in the topology of natural energy space is obtained. This result improves and deepens those in recent literatures (Li and Yang in
(1) Background: Aging is a global phenomenon, and China’s aging is extensive and rapid and already at the middle to upper level worldwide. Promoting social interaction and increasing positive psychological qualities in individuals are key components in helping people adapt to the physical and mental changes of the aging process. Among them, how middle-aged and older adults improve their physical and mental health through physical activity is of great concern. (2) Methods: This study measured the physical activity of 2721 middle-aged and elderly square dance participants across China, and structural equation modeling was applied to explore the relationship between square dance exercise and group cohesion as well as the role of perceived social support and psychological capital. (3) Results: The results showed that (a) square dance exercise positively predicts group cohesion among middle-aged and older adults. (b) Perceived social support and psychological capital mediate the relationship between square dance exercise and group cohesion, and the mediating role consists of three pathways: perceived social support alone, psychological capital alone, and perceived social support-psychological capital chain mediation. (c) The mediating effect of psychological capital alone is greater than the mediating effect of perceived social support alone and the mediating effect of the perceived social support-psychological capital chain. (4) Conclusions: This study provides support for the theory and practice of square dance exercise and intervention guidance for increasing positive psychological qualities and group dynamic levels in middle-aged and older adults.
<p style='text-indent:20px;'>In this paper, we investigate the global well-posedness and the existence of strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation in <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset{\mathbb R}^N $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}+\Delta ^2u+\Delta ^2u_t+\Delta \phi (\Delta u) = g(x), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with the hinged boundary condition. We show that (i) when the nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> is quasi-monotone and is of at most the critical growth: <inline-formula><tex-math id="M3">\begin{document}$ 1\leq p\leq p^{*}: = \frac{N+2}{(N-2)^{+}} \ (N\geq 2) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ g = 0 $\end{document}</tex-math></inline-formula>, the model has in phase space <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal H} = V_3\times L^2 $\end{document}</tex-math></inline-formula> a trivial global and exponential attractor, respectively. (ii) In particular when <inline-formula><tex-math id="M6">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>, without any polynomial growth restriction for <inline-formula><tex-math id="M7">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>, the model has a strong global and a strong exponential attractor, respectively. These results deepen and extend the related researches on this topic in recent literature [<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b22">22</xref>]. The method developed here allows us to establish the existence of the strong global and exponential attractor for this nonlinear model.</p>
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