Well-posedness and parabolic smoothing effect for higher order Schr\"odinger type equations with constant coefficients

TL;DR

This paper investigates higher order Schrödinger equations with constant coefficients, establishing well-posedness, smoothing effects, and classifying equations based on their regularity properties.

Contribution

It provides a comprehensive analysis of well-posedness and smoothing effects for a class of higher order Schrödinger equations, including a classification based on smoothing behavior.

Findings

Proves $L^2$ well-posedness using energy inequalities.

Identifies the parabolic smoothing effect for these equations.

Classifies equations into three types based on their smoothing properties.

Abstract

We consider the Cauchy problem of a class of higher order Schr\"odinger type equations with constant coefficients. By employing the energy inequality, we show the $L^2$ well-posedness, the parabolic smoothing and a breakdown of the persistence of regularity. We classify this class of equations into three types on the basis of their smoothing property.