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We consider the Cauchy problem of fifth order dispersive equations on thetorus. We assume that the initial data is sufficiently smooth and the nonlinearterm is a polynomial depending on $\partial_x^3 u, \partial_x^2 u, \partial_xu$ and $u$. We prove that the local well-posedness holds on $[-T,T]$ when thenonlinear term satisfies a condition and otherwise, the local well-posednessholds with a smoothing effect only on either $[0,T]$ or $[-T,0]$ andnonexistence result holds on the other time interval, which means that thenonlinear term can not be treated as a perturbation of the linear part and theequation has a property of parabolic equations by an influence of the nonlinearterm. As a corollary, we also have the same results for $(2j+1)$-st orderdispersive equations.

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