Exponential stabilization for carbon nanotubes modeled as Timoshenko beams with thermoelastic effects

In this article we consider the problem of heat conduction in carbon nanotubes modeled like Timoshenko beams, inspired by the work of J. Yoon \textit{et al.} (Composites Part B: Engineering, \textbf{35}(2), 87--93. 2004). Using the theory of semigroups of linear operators, we prove the well-posedness of the problem and the exponential stabilization of the total energy of the system of differential equations, partially damped, without assuming the known relationship of equality of wave velocities. Furthermore, we analyze the fully discrete problem using a finite difference scheme, introduced by a spatiotemporal discretization that combines explicit and implicit integration methods. We show the construction of numerical energy and simulations that prove our theoretical exponential decay results and display the convergence rates.