Stability for Euler-Bernoulli Beam Equation with a Local Degenerated Kelvin-Voigt Damping

We consider the Euler-Bernoulli beam equation with a local Kelvin-Voigt dissipation type in the interval (-1, 1). The coefficient damping is only effective in (0, 1) and is degenerating near the 0 point with a speed at least equal to x alpha where alpha is an element of (0, 5). We prove that the semigroup corresponding to the system is polynomially stable and the decay rate depends on the degeneracy speed alpha. Here we develop a new method which consists to use a local analysis approach combined with the classical iterative method.