Differential and Integral Equations

Nonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical homogeneous nonlinearity

Satoshi Masaki and Hayato Miyazaki

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Abstract

We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u|^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.

Article information

Source
Differential Integral Equations, Volume 32, Number 3/4 (2019), 121-138.

Dates
First available in Project Euclid: 23 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.die/1548212426

Mathematical Reviews number (MathSciNet)
MR3909981

Zentralblatt MATH identifier
07036978

Subjects
Primary: 35B44: Blow-up 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35P25: Scattering theory [See also 47A40]

Citation

Masaki, Satoshi; Miyazaki, Hayato. Nonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical homogeneous nonlinearity. Differential Integral Equations 32 (2019), no. 3/4, 121--138. https://projecteuclid.org/euclid.die/1548212426


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