  This research studies two systems composed by the Timoshenko beam model for
double wall carbon nanotubes, coupled with the heat equation governed by
Fourier's law. For the first system, the coupling is given by the speed the
rotation of the vertical filament in the beam $\beta\psi_t$ from the first beam
of Tymoshenko and the Laplacian of temperature $\delta\theta_{xx}$, where we
also consider the damping terms fractionals
$\gamma_1(-\partial_{xx})^{\tau_1}\phi_t$, $\gamma_2(-\partial_{xx})^{\tau_2}
y_t$ and $\gamma_3(-\partial_{xx})^{\tau_3} z_t$, where $(\tau_1, \tau_2,
\tau_3) \in [0,1]^3$. For this first system we proved that the semigroup
$S_1(t)$ associated to system decays exponentially for all $(\tau_1 , \tau_2 ,
\tau_3 ) \in [0,1]^3$. The second system also has three fractional damping
$\gamma_1(-\partial_{xx})^{\beta_1}\phi_t$, $\gamma_2(-\partial_{xx})^{\beta_2}
y_t$ and $\gamma_3(-\partial_{xx})^{\beta_3} z_t$, with $(\beta_1, \beta_2,
\beta_3) \in [0,1]^3$. Furthermore, the couplings between the heat equation and
the Timoshenko beams of the double wall carbon nanotubes for the second system
is given by the Laplacian of the rotation speed of the vertical filament in the
beam $\beta\psi_{xxt}$ of the first beam of Timoshenko and the Lapacian of the
temperature $\delta\theta_{xx}$. For the second system, we prove the
exponential decay of $S_2(t)$ for $(\beta_1, \beta_2, \beta_3) \in [0,1]^3$ and
also show that $S_2(t)$ admits Gevrey classes $s>(\phi+1)/(2\phi)$ for
$\phi=\min\{\beta_1,\beta_2,\beta_3\}, \forall (\beta_1,\beta_2,\beta_3)\in
(0,1)^3$, and proving that $S_2(t)$ is analytic when the parameters $(\beta_1,
\beta_2, \beta_3) \in [1/2,1]^3$. One of the motivations for this research was
the work; Ramos et al. \cite{Ramos2023CNTs}, whose partial results are part of
our results obtained for the first system for $(\tau_1, \tau_2, \tau_3) = (0,
0, 0)$.
