The well-posedness and attractor on an extensible beam equation with nonlocal weak damping

Abstract

In this paper, we consider some new results on the well-posedness and the asymptotic behavior of the solutions for a class of extensible beams equation with the nonlocal weak damping and nonlinear source terms. Our contribution is threefold. First, we establish the well-posedness by means of the monotone operator theory with locally Lipschitz perturbation. Then we show that the related solution semigroup \begin{document}$ \{S_{t}\}_{t\geq0} $\end{document} in phase space \begin{document}$ \mathcal{H} $\end{document} has a finite-dimensional global attractor \begin{document}$ \mathcal{A} $\end{document} which has some regularity when the growth exponent of the nonlinearity \begin{document}$ f(u) $\end{document} is up to the subcritical and critical case, respectively. Finally, we obtain the exponential attractor \begin{document}$ \mathcal{A}_{exp} $\end{document} of the dynamical system \begin{document}$ (\mathcal{H}, S_{t}) $\end{document}. These results deepen and extend our previous works([31], [30]), where we only considered the existence of the global attractors in the case of degenerate damping.