Global attractors for the extensible thermoelastic beam system

This work is focused on the dissipative system $$ \begin{cases} \partial_{tt}u+\partial_{xxxx}u +\partial_{xx}\theta-\big(\beta+\|\partial_x u\|_{L^2(0,1)}^2\big)\partial_{xx}u=f\\ \noalign{\vskip.7mm} \partial_{t} \theta -\partial_{xx}\theta -\partial_{xxt} u= g \end{cases} $$ describing the dynamics of an extensible thermoelastic beam, where the dissipation is entirely contributed by the second equation ruling the evolution of $\theta$. Under natural boundary conditions, we prove the existence of bounded absorbing sets. When the external sources $f$ and $g$ are time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity for all parameters $\beta\in\mathbb{R}$. The same result holds true when the first equation is replaced by $$ \partial_{tt} u-\gamma\partial_{xxtt} u+\partial_{xxxx}u +\partial_{xx}\theta-\big(\beta+\|\partial_x u\|_{L^2(0,1)}^2\big)\partial_{xx}u=f $$ with $\gamma>0$. In both cases, the solutions on the attractor are strong solutions.