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>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>

Well-posedness and strong attractors for a beam model with degenerated nonlocal strong damping

“…The global attractor theory of beam equations with nonlocal damping has received much more attention in recent years (cf. [6,7,15,21,22,23,25,26,31,32] and reference therein). Motivated by model (1.6), the authors in [26] studied a more general case…”

“…[6,7,15,21,22,23,25,26,31,32] and reference therein). Motivated by model (1.6), the authors in [26] studied a more general case…”

Strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation

“…Recently, there have been more and more concerns for this issue for the hyperbolic system (cf. [18,20,19,21,25] and references therein). For instance, Sun and Yang [25] have recently investigated the existence of strong (V 2 × L 2 , V 4 × V 2 )global and exponential attractors and their robustness on the perturbed parameter κ in V 4 ×V 2 -topology for an extensible beam equation with nonlocal energy damping in Ω ⊂ R N :…”

“…It worth mentioning that the similarity of this paper with [25] lies in that we define the same operator A = ∆ 2 , with the hinged boundary condition, and rewrite the original model equation as an abstract operator equation by using the fact that A However, in terms of technology, the method used here is completely different from that in [25] because (i) the nonlinearity ∆φ(∆u) in Eq. ( 1) is more complex local one rather than nonlocal ones in Eq.…”

“…(ii) the weak solutions here only possess partial regularity (regularity on the velocity component u t ) as t > 0 instead of complete one (regularity on both u and u t ) as t > 0 for problem (8), (2)-(3), which ensures that the weak solutions of the latter are exactly the strong ones, it is just based on this that we proceeded with further studies in [25] for the strong global and exponential attractors. Here, the loss of higher complete regularity as t > 0 causes that the approach used in [25] fails. Therefore, we use the new strategy as shown above.…”