Stability and regularity for double wall carbon nanotubes modeled as Timoshenko beams with thermoelastic effects and intermediate damping

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 rotation speed of the vertical filament in the beam from the first beam of Timoshenko and the Laplacian of temperature , where we also consider the damping terms fractionals , , and , where . For this first system, we proved that the semigroup associated to system decays exponentially for all . The second system also has three fractional dampings , , and , with . Furthermore, the couplings between the heat equation and the Timoshenko beams of the double wall carbon nanotubes for the second system are given by the Laplacian of the rotation speed of the vertical filament in the beam of the first beam of Timoshenko and the Lapacian of temperature . For the second system, we prove the exponential decay of the associated semigroup for and also show that this semigroup admits Gevrey classes for , and we finish our investigation proving that is analytic when the parameters .