Profile decompositions and blowup phenomena of mass critical fractional Schrödinger equations

We consider the focusing energy-critical inhomogeneous nonlinear Schrödinger equation:3 , and p = 5 − 2b. On the road map of Kenig-Merle [22] we show the global well-posedness and scattering of radial solutions under energy condition, and rigidity condition −bg(x) ≤ x · ∇g(x). We also provide sharp finite time blowup results for non-radial and radial solutions. For this we utilize the localized virial identity.2010 Mathematics Subject Classification. M35Q55, 35Q40.

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“…Proof. One can show (i), (ii), (iii), and (iv) by exactly the same way as in [23,8,7]. We omit the details.…”

Section: Profile Decompositionmentioning

confidence: 62%

On the global well-posedness of focusing energy-critical inhomogeneous NLS

2020

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“…When γ = 0 and α ∈ (0, 1), the problem (1.1) is a nonlocal model known as nonlinear fractional Schrödinger equation which has also attracted much attentions recently [9,10,11,12,19,20,21,22,23,24,15,16]. The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics, which was derived by Laskin [29,30] as a result of extending the Feynman path integral, from the Brownian-like to Levy-like quantum mechanical paths.…”

Section: (T) = M (U(t)) :=mentioning

confidence: 99%

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

Saanouni

2016

Journal of Mathematical Physics

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Abstract. Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrödinger equation are investigated. Consider the initial value problem for an inhomogeneous nonlinear Schrödinger equation Contentswhich models various physical contexts in the description of nonlinear waves such as propagation of a laser beam and plasma waves. For example, when γ = 0, it arises in nonlinear optics, plasma physics and fluid mechanics [2,3]. When γ > 0, it can be thought of as Date: February 16, 2016. 1991 Mathematics Subject Classification. 35Q55.

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“…

We study the existence of ground states for the nonlinear Choquard equation driven by fractional Laplacian:where the nonlinearity satisfies the general Berestycki-Lions-type assumptions.In [21,22], the authors discussed the initial value problem for the boson star equation. Coti Zelati and Nolasco [23,24] obtained the existence of a ground state of pseudorelativistic Hartree equation.

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mentioning

confidence: 99%