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

Masaki, Satoshi; Segata, Jun-ichi

doi:10.48550/arxiv.1602.05331

Cited by 6 publications

(32 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, when α < 2, it seems difficult to derive those properties in the Sobolev or hat-Lebesgue spaces by several reasons. In [24], it turns out that a use of the generalized hat-Morrey space enables us to establish well-posedness theory good enough and to obtain the concentration compactness lemma equipped with a decoupling inequality 1 . Our estimate in Theorem 1.3 removes several technical restrictions made in [24].…”

Section: Introductionmentioning

confidence: 99%

“…In [24], it turns out that a use of the generalized hat-Morrey space enables us to establish well-posedness theory good enough and to obtain the concentration compactness lemma equipped with a decoupling inequality 1 . Our estimate in Theorem 1.3 removes several technical restrictions made in [24]. See Subsection 1.2 below for more details.…”

Section: Introductionmentioning

confidence: 99%

“…In all above studies, the refinements were restricted to the case α = 2 and the diagonal case p = q. In [24] the authors proved the refined Stein-Tomas estimate for (1.1) in the case α = 2 and used it for proving existence of a minimal non-scattering solution for the mass-subcritical generalized KdV equation in the Lα -framework. A similar refinement in the Schrödinger case was done by the first author [22], including its application to existence of a minimal non-scattering solution for the mass-subcritical nonlinear Schrödinger equation in M β γ,δ -framework.…”

Section: Introductionmentioning

confidence: 99%

“…We briefly outline the proofs for Theorems 1.3, the proof of Theorem 1.5 is similar. The diagonal case p = q can be handled by the bilinear technique as in [28,24]. However, this approach does not work well in the non-diagonal case.…”

Section: Introductionmentioning

confidence: 99%

“…As its applications, we prove the small data scattering and existence of a special non-scattering solutions, which are minimal in suitable sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation. Especially, a use of the refined non-diagonal estimate removes several technical restrictions on the previous work [24] about the existence of the special non-scattering solution.2000 Mathematics Subject Classification. Primary 35Q53; Secondary 35B40.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application

Masaki,

Segata

2017

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we give an improvement of the non-diagonal Strichartz estimate for Airy equation by using a Morrey type space. As its applications, we prove the small data scattering and existence of a special non-scattering solutions, which are minimal in suitable sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation. Especially, a use of the refined non-diagonal estimate removes several technical restrictions on the previous work [24] about the existence of the special non-scattering solution.2000 Mathematics Subject Classification. Primary 35Q53; Secondary 35B40. Key words and phrases. generalized Korteweg-de Vries equation, scattering problem.1 Although the decoupling (in)equality is not obtained in Lα with α = 2, it can be established in a wider space M α 2,δ thanks to the local L 2 structure of M α 2,δ .

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application

Masaki,

Segata

2017

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Long Range Scattering for Nonlinear Schrödinger Equations with Critical Homogeneous Nonlinearity

Masaki¹,

Miyazaki²

2018

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

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.2010 Mathematics Subject Classification. 35B44, 35Q55, 35P25.

show abstract

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

Masaki

2016

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 6 publications

References 36 publications

Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application

Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application

Long Range Scattering for Nonlinear Schrödinger Equations with Critical Homogeneous Nonlinearity

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

Contact Info

Product

Resources

About