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.