Stabilization of transverse vibrations of an inhomogeneous Euler- Bernoulli beam with a thermal effect

We consider an inhomogeneous Euler-Bernoulli (EB) beam of length $L$ clamped at both ends and subject to : an external frictional damping and a thermal effect (Fourier law). We prove the well-posedness of the model and analyze the behavior of the solution as $t \rightarrow + \infty.$ The existence is proved using semigroup theory, and the exponential stabilization of solutions is obtained considering multiplier technique. A numerical illustration of the energy decay is given, based on initial data close to a real physical experiment.