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

"Harmonic Analysis and Nonlinear Partial Differential Equations". July 6～8, 2015. edited by Hideo Kubo and Mitsuru Sugimoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.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