Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus

TL;DR

This paper investigates the initial value problem for fully nonlinear third-order equations on the torus, establishing conditions under which solutions are smooth and well-posed in one or both time directions, resembling parabolic or dispersive behavior.

Contribution

It provides new conditions that determine whether solutions exhibit parabolic smoothing or dispersive properties for these nonlinear equations.

Findings

Solutions are infinitely smooth in one time direction under certain conditions.

The problem is not well-posed in the opposite time direction in some cases.

Conditions are identified that lead to dispersive behavior with well-posed solutions in both directions.

Abstract

We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution in one direction of time that is infinitely smooth and the problem in not well-posed in the other direction. Under other conditions on the nonlinearity and the data, we prove that the equation behaves like a dispersive one: there exists a unique local solution in both directions of time.