Decay and Scattering of Small Solutions of a Generalized Boussinesq Equation

We study the long-time behavior of small solutions of the initial-value problem for a generalized Boussinesq equation. We obtain a lower bound for the degrees of nonlinearity which allows us to establish a nonlinear scattering result for small perturbations; that is, the small solutions of the nonlinear problem behave asymptotically like the solution of the associated linear problem. Under certain hypotheses, we can construct a scattering operator for the Boussinesq equation which carries a neighborhood of 0 in the energy spaceXintoX.