Strong attractors and their robustness for an extensible beam model with energy damping

Abstract

<p style='text-indent:20px;'>This paper investigates the existence of <i>strong</i> global and exponential attractors and their robustness on the perturbed parameter for an extensible beam equation with nonlocal energy damping in <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset{\mathbb R}^N $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M2">\begin{document}$ u_{tt}+\Delta^2 u-\kappa\phi(\|\nabla u\|^2)\Delta u-M(\|\Delta u\|^2+\|u_t\|^2)\Delta u_t+f(u) = h $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \kappa \in \Lambda $\end{document}</tex-math></inline-formula> (index set) is an extensibility parameter, and where the "<i>strong</i>" means that the compactness, the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of the stronger space <inline-formula><tex-math id="M4">\begin{document}$ {\mathcal H}_2 $\end{document}</tex-math></inline-formula> where the attractors lie in. Under the assumptions that either the nonlinearity <inline-formula><tex-math id="M5">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> is of optimal subcritical growth or even <inline-formula><tex-math id="M6">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> is a true source term, we show that (ⅰ) the semi-flow originating from any point in the natural energy space <inline-formula><tex-math id="M7">\begin{document}$ {\mathcal H} $\end{document}</tex-math></inline-formula> lies in the stronger strong solution space <inline-formula><tex-math id="M8">\begin{document}$ {\mathcal H}_2 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M9">\begin{document}$ t&gt;0 $\end{document}</tex-math></inline-formula>; (ⅱ) the related solution semigroup <inline-formula><tex-math id="M10">\begin{document}$ S^\kappa(t) $\end{document}</tex-math></inline-formula> has a strong <inline-formula><tex-math id="M11">\begin{document}$ ({\mathcal H},{\mathcal H}_2) $\end{document}</tex-math></inline-formula>-global attractor <inline-formula><tex-math id="M12">\begin{document}$ {\mathscr A}^\kappa $\end{document}</tex-math></inline-formula> for each <inline-formula><tex-math id="M13">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> and the family of <inline-formula><tex-math id="M14">\begin{document}$ {\mathscr A}^\kappa, \kappa\in \Lambda $\end{document}</tex-math></inline-formula> is upper semicontinuous on <inline-formula><tex-math id="M15">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> in the topology of stronger space <inline-formula><tex-math id="M16">\begin{document}$ {\mathcal H}_2 $\end{document}</tex-math></inline-formula>; (ⅲ) <inline-formula><tex-math id="M17">\begin{document}$ S^\kappa(t) $\end{document}</tex-math></inline-formula> has a strong <inline-formula><tex-math id="M18">\begin{document}$ ({\mathcal H},{\mathcal H}_2) $\end{document}</tex-math></inline-formula>-exponential attractor <inline-formula><tex-math id="M19">\begin{document}$ \mathfrak {A}^\kappa_{exp} $\end{document}</tex-math></inline-formula> for each <inline-formula><tex-math id="M20">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> and it is Hölder continuous on <inline-formula><tex-math id="M21">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> in the topology of <inline-formula><tex-math id="M22">\begin{document}$ {\mathcal H}_2 $\end{document}</tex-math></inline-formula>. These results break through long-standing existed restriction for the attractors of the extensible beam models in energy space and show the optimal topology properties of them in the stronger phase space.</p>