Exponential Energy Decay of the Euler-Bernoulli Beam with Shear Diffusion/Thermal Dissipation

In this paper, we study the asymptotic behavior of the Euler-Bernoulli elastic beam model coupled a diffusion and/or a dissipation process. We then use a necessary and sufficient condition for a C₀-semigroup being exponential stable to show that the energy associated with these models decays exponentially. A remarkable feature of the proof is that the argument does not need a positive gap of eigenvalues of the related operator as before. Therefore, it throws light on the problems in higher space dimensions.