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This research studies two systems composed by the Timoshenko beam model fordouble wall carbon nanotubes, coupled with the heat equation governed byFourier's law. For the first system, the coupling is given by the speed therotation of the vertical filament in the beam $\beta\psi_t$ from the first beamof Tymoshenko and the Laplacian of temperature $\delta\theta_{xx}$, where wealso 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 andthe Timoshenko beams of the double wall carbon nanotubes for the second systemis given by the Laplacian of the rotation speed of the vertical filament in thebeam $\beta\psi_{xxt}$ of the first beam of Timoshenko and the Lapacian of thetemperature $\delta\theta_{xx}$. For the second system, we prove theexponential decay of $S_2(t)$ for $(\beta_1, \beta_2, \beta_3) \in [0,1]^3$ andalso 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 wasthe work; Ramos et al. \cite{Ramos2023CNTs}, whose partial results are part ofour results obtained for the first system for $(\tau_1, \tau_2, \tau_3) = (0,0, 0)$.

Stability and Regularity for Double Wall Carbon Nanotubes Modeled as Timoshenko Beams with Thermoelastic Effects and Intermediate Damping