Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities

Türker Özsarı

doi:10.3934/cpaa.2019027

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2019, Volume 18, Issue 1: 539-558. Doi: 10.3934/cpaa.2019027

This issue Previous Article Semiclassical states for fractional Choquard equations with critical growth Next Article

Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities

Türker Özsarı

Department of Mathematics, Izmir Institute of Technology, Urla, Izmir 35430, Turkey

Received: November 30, 2017

Revised: February 28, 2018

Published: August 2018

The author is supported by Izmir Institute of Technology's BAP Grant 2015IYTE43.

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Abstract

The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly supported or rapidly decaying weight function in virial identities - an idea borrowed from [18]. At the end of the paper, a numerical example satisfying the theory is provided.

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Mathematics Subject Classification: Primary: 35B44, 35Q41, 35Q55; Secondary: 35B45.

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Figure 1. The oscillating coefficient $A_\Omega$ over an extended interval

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Figure 2. Mollifiers $l$ and $r$

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Figure 3. Weight function $\varphi$

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Figure 4. The graph of $m(x)$

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Figure 5. The graph of $m'(x)$

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Figure 6. The graph of initial datum $\text{Re}[u_0(x)]$ ($\rho = 0.1$) and $m(x)$

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