Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si18.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>

An, JinMyong; Kim, JinMyong

doi:10.1016/j.nonrwa.2020.103268

Cited by 13 publications

(44 citation statements)

References 27 publications

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“…We refer the reader to [2,3,17,21] for the physical background of the INLS equation (1.1). We also refer the reader to [1,4,5,8,[10][11][12][13][14][15][16]18] for recent work on the INLS equation (1.1). In particular, the author in [18] studied the local and global well-posedness in H s (R n ) with 0 ≤ s ≤ min n 2 , 1 for the INLS equation (1.1) with f (u) = λ|u| σ u by using the contraction mapping principle based on Strichartz estimates.…”

mentioning

confidence: 99%

“…In particular, the author in [18] studied the local and global well-posedness in H s (R n ) with 0 ≤ s ≤ min n 2 , 1 for the INLS equation (1.1) with f (u) = λ|u| σ u by using the contraction mapping principle based on Strichartz estimates. Recently, the authors in [1] improved the local wellposedness result of [18] by extending the validity of s and b. More precisely, they obtained the following existence result (see Theorem 1.4 and Theorem 1.6 in [1]).…”

mentioning

confidence: 99%

“…Recently, the authors in [1] improved the local wellposedness result of [18] by extending the validity of s and b. More precisely, they obtained the following existence result (see Theorem 1.4 and Theorem 1.6 in [1]).…”

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confidence: 99%

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Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$

An¹,

Kim²,

Chae³

2022

DCDS-B

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We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta u = |x|^{-b} f(u), \;u(0)\in H^{s} (\mathbb R^{n} ), $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ n\in \mathbb N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0<s<\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0<b<\min \{ 2, \;n-s, \;1+\frac{n-2s}{2} \} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> is a nonlinear function that behaves like <inline-formula><tex-math id="M5">\begin{document}$ \lambda |u|^{\sigma } u $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ \sigma>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \lambda \in \mathbb C $\end{document}</tex-math></inline-formula>. Recently, the authors in [<xref ref-type="bibr" rid="b1">1</xref>] proved the local existence of solutions in <inline-formula><tex-math id="M8">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ 0\le s<\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula>. However even though the solution is constructed by a fixed point technique, continuous dependence in the standard sense in <inline-formula><tex-math id="M10">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ 0< s<\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula> doesn't follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial data in the standard sense in <inline-formula><tex-math id="M12">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula>, i.e. in the sense that the local solution flow is continuous <inline-formula><tex-math id="M13">\begin{document}$ H^{s}(\mathbb R^{n} )\to H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula>, if <inline-formula><tex-math id="M14">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> satisfies certain assumptions.

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confidence: 99%

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Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$

An¹,

Kim²,

Chae³

2022

DCDS-B

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“…2 Let a, b be non-negative real numbers and p, q be positive real numbers satisfying 1 p + 1 q = 1. Then for any ε, we have ab εa p + ε − q p b q .…”

Section: Virial Estimatesmentioning

confidence: 99%

“…Let us recall the known results for the INLS c equation (1.1). Using the energy method, Suzuki [24] showed that if 1…”

Section: Introductionmentioning

confidence: 99%

Dynamics of the energy critical nonlinear Schrödinger equation with inverse square potential

Yang¹

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We consider the focusing energy critical NLS with inverse square potential in dimension d = 3, 4, 5 with the details given in d = 3 and remarks on results in other dimensions. Solutions on the energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifolds of the ground state. In the latter case they converge to the ground state exponentially in the energy space as t → ∞ or t → −∞. (In 3-dim without radial assumption, this holds under the compactness assumption of non-scattering solutions on the energy surface.) When the kinetic energy is greater than that of the ground state, we show that all radial H 1 solutions blow up in finite time, with the only two exceptions in the case of 5-dim which belong to the stable/unstable manifold of the ground state. The proof relies on the detailed spectral analysis, local invariant manifold theory, and a global Virial analysis.

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

Zhang

2021

Journal of Differential Equations

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Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in Hs(Rn)

Cited by 13 publications

References 27 publications

Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$

Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$

Dynamics of the energy critical nonlinear Schrödinger equation with inverse square potential

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

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