$H^infty$ well-posedness for a 2-evolution Cauchy problem with complex coefficients

In this paper we deal with a 2-evolution Cauchy problem coming from the Euler–Bernoulli model for vibrating beams and plates. The leading coefficient, corresponding to the modulus of elasticity, is time-dependent and may vanish at t=0 . We prove a well-posedness result in the scale of Sobolev spaces using a C1 -approach, in this way we have H∞ well-posedness with an (at most) finite loss of regularity. We take special interest in the space and time-dependence of a complex coefficient of the extended principle part, related to the shear force, and in the assumptions we pose on that coefficient in order to get H∞ well-posedness.