Bilinear Strichartz's type estimates in Besov spaces with application to inhomogeneous nonlinear biharmonic Schrödinger equation

Li, Xuan; Zhang, Ting

doi:10.1016/j.jde.2021.06.007

Cited by 8 publications

(10 citation statements)

References 30 publications

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“…The IBNLS equation (1.1) has attracted a lot of interest in recent years. See, for example, [7,8,15,16,21,23] and the references therein. Guzmán-Pastor [15] proved that (1.1) is locally well-posed in L 2 , if d ∈ N, 0 < b < min {4, d} and 0 < σ < σ c (0).…”

Section: Introductionmentioning

confidence: 99%

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Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

An¹,

Ryu²,

Kim³

2023

DCDS-B

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In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS)<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta^{2} u = \lambda |x|^{-b}|u|^{\sigma}u, u(0) = u_{0} \in H^{s} (\mathbb R^{d}), $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda \in \mathbb R $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ d\in \mathbb N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0<s<\min \{2+\frac{d}{2}, \frac{3}{2}d\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ 0<b<\min\{4, d, \frac{3}{2}d-s, \frac{d}{2}+2-s\} $\end{document}</tex-math></inline-formula>. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in <inline-formula><tex-math id="M5">\begin{document}$ H^{s}(\mathbb R^{d}) $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M6">\begin{document}$ \frac{8-2b}{d}<\sigma< \sigma_{c}(s) $\end{document}</tex-math></inline-formula> and the initial data is sufficiently small, where <inline-formula><tex-math id="M7">\begin{document}$ \sigma_{c}(s) = \frac{8-2b}{d-2s} $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M8">\begin{document}$ s<\frac{d}{2} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ \sigma_{c}(s) = \infty $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M10">\begin{document}$ s\ge \frac{d}{2} $\end{document}</tex-math></inline-formula>.

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Section: Introductionmentioning

confidence: 99%

“…See Theorem 1.5 of [21] for details. This result about the local wellposedness of (1.1) improves the one of [15] by not only extending the validity of d and s but also removing the lower bound σ > 2−2b d .…”

Section: Introductionmentioning

confidence: 99%

Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

An¹,

Ryu²,

Kim³

2023

DCDS-B

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“…The IBNLS equation (1.1) has attracted a lot of interest in recent years. See, for example, [6,7,10,11,22,23,29,35] and the references therein. Guzmán-Pastor [22] proved that (1.1) is locally well-posed in L 2 , if d ∈ N, 0 < b < min {4, d} and 0 < σ < σ c (0).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Liu-Zhang [29] established the local well-posedness in H s with 0 < s < 2 by using the Besov space theory. More precisely, they proved that the IBNLS equation (1.1) is locally well-posed in…”

Section: Introductionmentioning

confidence: 99%

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Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

An¹,

Ryu²,

Kim³

2022

Preprint

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We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equationFirst, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian (−∆) s/2 with s ∈ (0, 1] established by [1] to any s > 0. Applying this estimate and the contraction mapping principle based on Strichartz estimates in Lorentz spaces, we then establish the local well-posedness in H s for the IBNLS equation in both of subcritical case σ < σ c (s) and critical case σ = σ c (s). We also prove that the IBNLS equation is globally well-posed in H s , if the initial data is sufficiently small and 8−2b d≤ σ ≤ σ c (s) with σ < ∞.

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Local Well-Posedness of a Critical Inhomogeneous Bi-harmonic Schrödinger Equation

Saanouni

Peng

2023

Mediterr. J. Math.

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Bilinear Strichartz's type estimates in Besov spaces with application to inhomogeneous nonlinear biharmonic Schrödinger equation

Cited by 8 publications

References 30 publications

Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

Local Well-Posedness of a Critical Inhomogeneous Bi-harmonic Schrödinger Equation

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