Scattering for the beam equation

In this paper, we study the scattering for the nonlinear beam equation $u_{tt}+{\mathrm{\Delta }}^{2}u+mu+\mu \mathrm{\mid }u{\mathrm{\mid }}^{p-1}u=0$. Our results include two aspects. In the defocusing case we show that the scattering holds for $d=1$, which extends the result in \cite{Pau-Beam} to one dimension. In the focusing case, we show that the scattering holds in ${\mathbb{R}}^d(d\ge 1)$ when the energy $E(u_0,u_1)<E(R,0)$ and $\mathrm{\parallel }\mathrm{\Delta }{u}_0{\mathrm{\parallel }}_{L^2}^2+m\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}^2<\mathrm{\parallel }\mathrm{\Delta }R{\mathrm{\parallel }}_{L^2}^2+m\mathrm{\parallel }R{\mathrm{\parallel }}_{L^2}^2$ for ground state $R$. The difficulties lie the absence of the scaling invariance and a Galilean transformation for the equation to control the Momentum vector.