Blowup of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation

Dinh, Van Duong

doi:10.1016/j.na.2018.04.024

Cited by 71 publications

(35 citation statements)

References 25 publications

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“…This new approach will be very important to carry out the analysis in the present study. For other recent works about the INLS model we refer the reader to Hong (2016), Killip et al (2017), Lu et al (2018), Combet and Genoud (2016), Dinh (2018Dinh ( , 2019.…”

Section: Introductionmentioning

confidence: 99%

Scattering for the Radial Focusing Inhomogeneous NLS Equation in Higher Dimensions

Farah

Guzmán

2019

Bull Braz Math Soc, New Series

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We consider the inhomogeneous nonlinear Schrödinger equation iu t + u + |x| −b |u| α u = 0, x ∈ R N , where N ≥ 2, 4−2b N < α < 4−2b N −2 (4−2b N < α < ∞ if N = 2) and 0 < b < min{N /3, 1}. For a radial initial data u 0 ∈ H 1 (R N), under a certain smallness condition, we prove that the corresponding solution is global and scatters. The smallness condition is related to the ground state solution of −Q + Q + |x| −b |Q| α Q = 0 and the critical Sobolev index s c = N 2 − 2−b α. This is an extension of the recent work (Farah and Guzmán in J Differ Equ 262(8):4175-4231, 2017) by the same authors, where they consider the case N = 3 and α = 2. The proof is inspired by the concentration-compactness/rigidity method developed by Kenig and Merle (Invent

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Section: Introductionmentioning

confidence: 99%

Scattering for the Radial Focusing Inhomogeneous NLS Equation in Higher Dimensions

Farah

Guzmán

2019

Bull Braz Math Soc, New Series

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“…Since t → e −aαt is bounded on [0, T * ), the proof of this result follows by the same lines as in [7,Lemma 3.4].…”

Section: Lemma 22 (Lwp In the Energy-critical Casementioning

confidence: 68%

“…The restriction α ≤ 4 comes from the Young inequality (see [7]). If we consider 4 N ≤ α ≤ α * , then this restriction only effects the validity of α in 2D.…”

Section: Remarkmentioning

confidence: 99%

Blow-up criteria for linearly damped nonlinear Schrödinger equations

Dinh¹

2021

EECT

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We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations i∂tu + ∆u + iau = ±|u| α u, (t, x) ∈ [0, ∞) × R N , where a > 0 and α > 0. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up H 1 solutions to the focusing problem in the mass-critical and mass-supercritical cases.

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“…In this case, it is say that the solution u blows up in a finite time and the time T is called the blow-up time of the solution u. The theoretical study of the phenomenon of blow-up and in particular blow-up solutions for semilinear Schrödinger equations has been the subject of investigations of many authors (see [1,3,9,15,17,18,23], and the references cited therein). This paper is interested by the numerical study of the above problem.…”

Section: Introductionmentioning

confidence: 99%

Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition

N'gohisse¹,

Nabongo²,

Traoré³

2019

IJAMTP

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Theoretical study of the phenomenon of blow-up solutions for semilinear Schrödinger equations has been the subject of investigations of many authors. It is said that the maximal time interval of existence of the solution blows up in a finite time when this time is finite, and the solution develops a singularity in a finite time. In fact, semilinear Schrhödinger equation models a lot of physical phenomenon such as nonlinear optics, energy transfer in molecular systems, quantum mechanics, seismology, plasma physics. In the past, certain authors have used numerical methods to study the phenomenon of blow-up for semilinear Schrödinger equations. They have considered the same problem and one proves that the energy of the system is conserved, and the method used to show blow-up solutions are based on the energy's method. This paper proposes a method based on a modification of the method of Kaplan using eigenvalues and eigenfunctions to show that the semidiscrete solution blows up in a finite time under some assumptions. The semidiscrete blow-up time is also estimate. Similar results are obtain replacing the reaction term by another form to generalise the result. Finally, this paper propose two schemes for some numerical experiments and a graphics is given to illustrate the analysis.

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Blowup ofH1solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation

Cited by 71 publications

References 25 publications

Scattering for the Radial Focusing Inhomogeneous NLS Equation in Higher Dimensions

Scattering for the Radial Focusing Inhomogeneous NLS Equation in Higher Dimensions

Blow-up criteria for linearly damped nonlinear Schrödinger equations

Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition

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