Scattering in the Energy Space for Boussinesq Equations

Abstract

In this note we show that all small solutions in the energy space of the generalized 1D Boussinesq equation must decay to zero as time tends to infinity, strongly on slightly proper subsets of the space-time light cone. Our result does not require any assumption on the power of the nonlinearity, working even for the supercritical range of scattering. For the proof, we use two new Virial identities in the spirit of works (Kowalczyk et al. in J Am Math Soc 30:769-798, 2017; Kowalczyk et al. in Lett Math Phys 107(5):921-931, 2017). No parity assumption on the initial data is needed.