Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations

Abstract

<p style='text-indent:20px;'>In this paper we consider the nonlinear beam equations accounting for rotational inertial forces. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact global attractor and an exponential attractor. The main purpose is to trace the behavior of solutions of the nonlinear beam equations when the effect of the rotational inertia fades away gradually. A natural question is whether there are qualitative differences would appear or not. To answer the question, we deal with the rotational inertia with a parameter <inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and consider the difference of behavior between the case <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;\alpha\le1 $\end{document}</tex-math></inline-formula> and the case <inline-formula><tex-math id="M3">\begin{document}$ \alpha = 0 $\end{document}</tex-math></inline-formula>. The main novel contribution of this paper is to show the continuity of global attractors and exponential attractors with respect to <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> in some sense.