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

TL;DR

This paper analyzes the stability and regularity of double wall carbon nanotubes modeled as Timoshenko beams with thermoelastic effects and fractional damping, proving exponential decay and regularity properties of the associated semigroups.

Contribution

It introduces two coupled Timoshenko beam systems with fractional damping and heat effects, establishing exponential decay and regularity results, extending prior partial findings.

Findings

Exponential decay of semigroups for all fractional damping parameters in [0,1]^3.

Gevrey class regularity for the second system depending on damping parameters.

Analyticity of the semigroup when damping parameters are in [1/2,1]^3.

Abstract

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 $(\tau_1, \tau_2, \tau_3) = (0, 0, 0)$.