Scattering for the beam equation

Abstract

In this paper, we study the scattering for the nonlinear beam equation $u_{tt}+\Delta^2u+mu+\mu |u|^{p-1}u=0$. 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 $\|\Delta u_0\|_{L^2}^2+m\|u_0\|_{L^2}^2<\|\Delta R\|_{L^2}^2+m\|R\|_{L^2}^2$ 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.