On the initial value problem of the semilinear beam equation with weak damping I: Smoothing effect

This paper is devoted to the study of the initial value problem for some semilinear damped beam equation. We shall prove the unique global existence of a small solution for the equation and also obtain a certain smoothing effect of the solution. Our method is based on the expansion formula for the propagator to decompose into the dissipative part and the dispersive part. To control the dispersive part without regularity loss, we apply the stationary phase method developed by Van der Corput. As a result, we observe the smoothing effect of the solution which does not occur on the solution for damped wave equations.