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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \{S_{t}\}_{t\geq0} $\end{document}</tex-math></inline-formula> in phase space <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{H} $\end{document}</tex-math></inline-formula> has a finite-dimensional global attractor <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> which has some regularity when the growth exponent of the nonlinearity <inline-formula><tex-math id="M4">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> is up to the subcritical and critical case, respectively. Finally, we obtain the exponential attractor <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{A}_{exp} $\end{document}</tex-math></inline-formula> of the dynamical system <inline-formula><tex-math id="M6">\begin{document}$ (\mathcal{H}, S_{t}) $\end{document}</tex-math></inline-formula>. These results deepen and extend our previous works([<xref ref-type="bibr" rid="b31">31</xref>], [<xref ref-type="bibr" rid="b30">30</xref>]), where we only considered the existence of the global attractors in the case of degenerate damping.</p>