Scattering for the beam equation

Abstract

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

This paper has been withdrawn by author due to a crucial error in Section 6

References in corpus (1)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Cited by in corpus (1)

• Energy Scattering for a Klein-Gordon Equation with a Cubic Convolution