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

TL;DR

This paper investigates the local well-posedness and smoothing effects of fifth order semilinear dispersive equations on the torus, revealing conditions under which solutions exist and exhibit parabolic-like behavior due to nonlinear influences.

Contribution

It establishes new conditions for local well-posedness and smoothing effects for fifth order dispersive equations, highlighting the nonlinear term's influence on solution behavior.

Findings

Local well-posedness depends on the nonlinear term satisfying specific conditions.

Smoothing effects occur on one side of the time interval when conditions are met.

Nonexistence results show the nonlinear term's dominant influence prevents perturbative treatment.

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.