Regular global attractors of type III thermoelastic extensible beams

For β ∈ ℝ, the authors consider the evolution system in the unknown variables u and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ \begin{gathered} \partial _{tt} u + \partial _{xxxx} u + \partial _{xxt} \alpha - \left( {\beta + \left\| {\partial _x u} \right\|_{L^2 }^2 } \right)\partial _{xx} u = f, \hfill \\ \partial _{tt} \alpha - \partial _{xx} \alpha - \partial _{xxt} \alpha - \partial _{xxt} u = 0 \hfill \\ \end{gathered} \right. $$\end{document} describing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established.