The global Cauchy problem for a vibrating beam equation

We consider the global Cauchy problem for an evolution equation which models an Euler–Bernoulli vibrating beam with time dependent elastic modulus under a force linear function of the displacement u, of the slope ∂xu, of and . These two last derivatives are proportional to the bending moment and to the shear respectively. We show results of well-posedness in Sobolev spaces assuming that the coefficient of the shear term has a decay rate x−σ, σ 1, for the position x→±∞ and that all the coefficients of , 1 k 3, satisfy suitable Levi conditions since we allow the elastic modulus to vanish at some time t=t0.