Scattering for the 4D Zakharov system below the ground state

Abstract

For the Zakharov system in four space dimensions, we prove that all solutions inside the potential well of the ground states are global and scattering in the energy space, with no other restriction such as symmetry. The proof has already been reduced by [3] to ruling out the existence of a minimal non-scattering solution that is precompact along some trajectory. This paper carries out the final step in the proof, namely we exclude the possibility of precompact solutions inside the potential well by combining two distinct arguments depending on the motion of trajectory.