On the well-posedness of the generalized Korteweg–de Vries equation in scale-critical Lr-space

Masaki, Satoshi; Segata, Jun Ichi

doi:10.2140/apde.2016.9.699

Cited by 18 publications

(22 citation statements)

References 33 publications

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“…Remark 1.8. The proof of [47,Theorem 1.7] shows that there exists a constant δ independent of u 0 Lα such that if…”

Section: Resultsmentioning

confidence: 99%

“…A pair (s L (α), α) is acceptable and conjugate-acceptable if 5/3 α < 20/9. Remark that there exists an acceptable and conjugate-acceptable pair under a weaker assumption 10/7 < α < 10/3 (see [47,Remark 4.1]). …”

Section: Generalized Strichartz' Estimatesmentioning

confidence: 99%

“…The well-posedness of (gKdV) and small data scattering inL α is established by the authors as long as 8/5 < α < 10/3 by introducing a generalized version of Stichartz' estimates adopted to theL r -framework, see [47]. The mass…”

Section: Introductionmentioning

confidence: 99%

“…By small data scattering result in [47], we know that d is bounded by a positive constant from below. Remark that there are several choices on notion of minimality of non-scattering solutions since u(t) Lα is not a conserved quantity.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Remark 1.8. The proof of [47,Theorem 1.7] shows that there exists a constant δ independent of u 0 Lα such that if…”

Section: Resultsmentioning

confidence: 99%

Section: Generalized Strichartz' Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…Proof. The proof follows from an argument similar to that in [19,Lemma 3.4]. Hence we omit the detail.…”

Section: Nonlinear Estimatesmentioning

confidence: 88%

The generalized Korteweg-de Vries equation with time oscillating nonlinearity in scale critical Sobolev space

Segata

Watanabe

2016

Nonlinear Differ. Equ. Appl.

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We consider the generalized Korteweg-de Vries (gKdV) equation with the time oscillating nonlinearity: ∂tu + ∂ 3 x u + g(ωt)∂x(|u| p−1 u) = 0, (t, x) ∈ R × R. Under the suitable assumption on g, we show that if the nonlinear term is mass critical or supercritical i.e., p ≥ 5 and u(0) ∈Ḣ sp , where sp = 1/2 − 2/(p − 1) is a scale critical exponent, then there exists a unique global solution to (gKdV) provided that |ω| is sufficiently large. We also obtain the behavior of the solution to (gKdV) as |ω| → ∞.

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Multilinear estimates with applications to nonlinear Schrödinger and Hartree equations in $$\widehat{L^p}$$ L p ^ -spaces

Hyakuna

2018

J. Evol. Equ.

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On the well-posedness of the generalized Korteweg–de Vries equation in scale-critical Lr-space

Cited by 18 publications

References 33 publications

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

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

The generalized Korteweg-de Vries equation with time oscillating nonlinearity in scale critical Sobolev space

Multilinear estimates with applications to nonlinear Schrödinger and Hartree equations in $$\widehat{L^p}$$ L p ^ -spaces

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