On minimal non-scattering solution for focusing mass-subcritical nonlinear Schrödinger equation

Masaki, Satoshi

doi:10.48550/arxiv.1301.1742

Cited by 7 publications

(20 citation statements)

References 7 publications

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“…The proof is similar to [31,Theorem 1.10]. Then, the rest of Theorem 1.12 is immediate by looking at energy of cQ(x) for 0 < c 1 (see [32,33,34], for instance).…”

Section: Preliminariesmentioning

confidence: 90%

“…We then modify the definition of E 1 slightly; fix t 0 = 0 and Ẽ1 (α, X, ℓ) := inf ℓ(u(0)) u(t) : solution to ( It is known that the weighted spaces are the spaces in which small data scattering holds, i.e., Ẽ1 (α, X) > 0 (see [13,41,34] and references therein). Further, in [33,34], existence of a minimizer to Ẽ1 is shown for suitable size functions. However, it will turn out soon that Ẽ2 is not finite.…”

Section: It Is Obvious By Definition Thatmentioning

confidence: 99%

“…We argue as in the proof of Proposition 4.2 to obtain the result. See [33,34,32] for details. We only comment the following two respects.…”

Section: Proof Of Theorem 113mentioning

confidence: 99%

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Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation

Masaki

2016

Preprint

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In this paper, we introduce two minimization problems on non-scattering solutions to nonlinear Schrödinger equation. One gives us a sharp scattering criterion, the other is concerned with minimal size of blowup profiles. We first reformulate several previous results in terms of these two minimizations. Then, the main result of the paper is existence of minimizers to the both minimization problems for mass-subcritical nonlinear Schrödinger equations. To consider the latter minimization, we consider the equation in a Fourier transform of generalized Morrey space. It turns out that the minimizer to the latter problem possesses a compactness property, which is so-called almost periodicity modulo symmetry.As in E 1 , a similar infimum value for negative time direction has the same value under the assumption (1.3). Intuitively, E 2 is a minimum size of possible "blowup profiles." 1 1 It is known that, in some cases, a solution that does not scatter for positive time direction tends to an orbit of a static profile, a blowup profile, by a group action, say G, as time approaches to the end of maximal time interval (e.g., a standing wave solution u(t, x) = e itω φω(x)). If a size function ℓ is chosen so that it is invariant under the group action G, then the size of such a solution tends to that of a corresponding profile. Of course, another kind of behavior may take place, in general, and so it may not be a definition of E2.

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“…The proof is similar to [31,Theorem 1.10]. Then, the rest of Theorem 1.12 is immediate by looking at energy of cQ(x) for 0 < c 1 (see [32,33,34], for instance).…”

Section: Preliminariesmentioning

confidence: 90%

Section: It Is Obvious By Definition Thatmentioning

confidence: 99%

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Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation

Masaki

2016

Preprint

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“…Proof. The proof follows from the argument similar to the proof of Proposition 3.2 or as in [40]. We omit the detail.…”

Section: 4mentioning

confidence: 93%

“…For the focusing mass-critical KdV equation, Martel-Merle-Raphaël [37,38,39] and Martel-Merle-Nakanishi-Raphaël [36] classified the dynamics of solution into three cases (blow-up, soliton, away from soliton) in the small neighborhood of Q. As for the mass-subcritical nonlinear Schr dinger equation, the first author treated a minimization problem similar to (1.1) in a framework of weighted space and showed existence of a threshold solution which is smaller than ground state solutions (see [40,41]).…”

Section: Introductionmentioning

confidence: 99%

Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

Masaki

Segata

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Please cite this article in press as: S. Masaki, J.-i. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, Ann. I. H. Poincaré -AN (2017), http://dx. AbstractIn this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation in the scale criticalL r space whereL r = {f ∈ S (R)| f Lr = f L r < ∞}. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted tô L r -framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

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Large data mass-subcritical NLS: critical weighted bounds imply scattering

Killip¹,

Masaki

Murphy

et al. 2017

Nonlinear Differ. Equ. Appl.

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On minimal non-scattering solution for focusing mass-subcritical nonlinear Schrödinger equation

Cited by 7 publications

References 7 publications

Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation

Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation

Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

Large data mass-subcritical NLS: critical weighted bounds imply scattering

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