A sharp scattering condition for focusing mass-subcritical nonlinearSchrödinger equation

Satoshi Masaki

doi:10.3934/cpaa.2015.14.1481

Article Contents

2015, Volume 14, Issue 4: 1481-1531. Doi: 10.3934/cpaa.2015.14.1481

This issue Previous Article On the pointwise decay estimate for the wave equation with compactly supported forcing term Next Article Remarks on global solutions of dissipative wave equations with exponential nonlinear terms

A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation

Satoshi Masaki^1,

1.
Laboratory of Mathematics, Institute of Engineering, Hiroshima university, Higashihiroshima Hirhosima, 739-8527

Received: August 31, 2013

Revised: April 30, 2014

Published: June 2015

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Abstract

This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schrödinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for scattering by proving existence of a threshold solution which does not scatter at least for one time direction and of which initial data attains minimum value of a norm of the weighted $L^2$ space in all initial value of non-scattering solution. Unlike in the mass-critical or -supercritical case, ground state is not a threshold. This is an extension of previous author's result to the case where the exponent of nonlinearity is below so-called Strauss number. A main new ingredient is a stability estimate in a Lorenz-modified-Bezov type spacetime norm.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35P25, 35B44.

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