On Local Energy Decay for Solutions of the Dispersion Generalized Benjamin-Ono Equation

Abstract We study the long-time behavior of large solutions to the dispersion generalized Benjamin-Ono equation. By means of virial identities, we identify spatial regions around the origin, growing unbounded in time, not containing the soliton region in which every solution belonging in a suitable Sobolev space, necessarily decays to zero along some sequence of times. A similar result is also obtained for solutions of Benjamin equation, a model for gravity-capillary surface waves of the solitary type on deep water.