Local well-posedness and parabolic smoothing effect of fifth order dispersive equations on the torus (Harmonic Analysis and Nonlinear Partial Differential Equations)

Abstract

The paper is announcement of the result obtained in [23]. We consider the Cauchy problem of fifth order dispersive equations with polynomial type nonlinearities depending on u; @xu; @2xu and @3xu under the periodic boundary condition. We show the following results. When the non- linear term is non-parabolic resonance type, we have the local well-posedness on (-T; T). On the other hand, when the nonlinear term is parabolic resonance type, the local well-posedness holds with a smoothing effect only on either [0; T) or (-T; 0] and nonexistence result holds on the other time interval.