A remark on norm inflation for nonlinear Schrödinger equations

Nobu Kishimoto

doi:10.3934/cpaa.2019067

Article Contents

2019, Volume 18, Issue 3: 1375-1402. Doi: 10.3934/cpaa.2019067

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A remark on norm inflation for nonlinear Schrödinger equations

Nobu Kishimoto

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Received: June 30, 2018

Revised: June 30, 2018

Published: May 2019

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Abstract

We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.

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Mathematics Subject Classification: 35Q55, 35B30.

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