The Cauchy problem for the quadratic beam equation in d ≥ 2

Abstract

The purpose of this paper is to study the Cauchy problem of the beam equation with quadratic nonlinearity. We establish the global well-posedness and scattering of small solutions in $2\le d\le 8$ with the strategy of Strichartz estimates, dispersive estimates and the method of space-time resonance. The main difficulties come from the weak dispersive estimates of beam semi-group and possible resonance. We shall utilize the observation of null structure for space-time resonance, which enables us to obtain enough decay but avoiding the degeneracies.