Scattering for the beam equation

In this paper, we study the scattering for the nonlinear beam equation $u_{tt}+Δ^2u+mu+μ|u|^{p-1}u=0$. 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 $\R^d (d\ge 1)$ when the energy $E(u_0,u_1)<E(R,0)$ and $\|Δ u_0\|_{L^2}^2+m\|u_0\|_{L^2}^2<\|ΔR\|_{L^2}^2+m\|R\|_{L^2}^2$ 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.

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