Well-posedness  and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity

Hiroyuki Hirayama; Mamoru  Okamoto

doi:10.3934/cpaa.2016.15.831

Article Contents

2016, Volume 15, Issue 3: 831-851. Doi: 10.3934/cpaa.2016.15.831

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Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity

Hiroyuki Hirayama^1, and
Mamoru Okamoto^2,

1.
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602
2.
Department of Mathematics, Institute of Engineering, Academic Assembly, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553

Received: April 30, 2015

Revised: November 30, 2015

Published: April 2016

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Abstract

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schrödinger type equations with derivative nonlinearity. In one dimensional case, the small data global well-posedness and scattering for the fourth order nonlinear Schrödinger equation with the nonlinear term $\partial _x (\overline{u}^4)$ are shown in the scaling invariant space $\dot{H}^{-1/2}$. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$, where $d$ denotes the space dimension.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B65.

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