Scattering for the beam equation

TL;DR

This paper investigates the scattering behavior of solutions to the nonlinear beam equation, extending known results to one dimension in the defocusing case and establishing conditions for scattering in all dimensions in the focusing case.

Contribution

It extends scattering results for the nonlinear beam equation to one dimension and provides new scattering criteria in higher dimensions for the focusing case.

Findings

Scattering holds in 1D for the defocusing case.

Scattering occurs in all dimensions under energy and norm conditions for the focusing case.

Overcomes difficulties due to lack of scaling invariance and Galilean symmetry.

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.