Parabolic smoothing effect and local well-posedness of fifth order semilinear dispersive equations on the torus

Abstract: We consider the Cauchy problem of fifth order dispersive equations on the torus. We assume that the initial data is sufficiently smooth and the nonlinear term is a polynomial depending on $\partial_x^3 u, \partial_x^2 u, \partial_x u$ and $u$. We prove that the local well-posedness holds on $[-T,T]$ when the nonlinear term satisfies a condition and otherwise, 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, which means that the nonlinear term can not be treated as a perturbation of the linear part and the equation has a property of parabolic equations by an influence of the nonlinear term. As a corollary, we also have the same results for $(2j+1)$-st order dispersive equations.