arXiv subjects

JinMyong An

Publications and source records attributed to JinMyong An.

14 recordsLinked to original sources

arXiv2024

Regular Strichartz estimates in Lorentz-type spaces with application to the $H^s$-critical inhomogeneous biharmonic NLS equation

In this paper, we investigate the Cauchy problem for the $H^s$-critical inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t}\pm Δ^{2} u=λ|x|^{-b}|u|^σu,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $λ\in \mathbb C$, $d\ge 3$, $1\le s<\frac{d}{2}$, $0<b<\min \left\{4,2+\frac{d}{2}-s \right\}$ and $σ=\frac{8-2b}{d-2s}$. First, we study the properties of Lorentz-type spaces such as Besov-Lorentz spaces and Triebel-Lizorkin-Lorentz spaces. We then derive the regular Strichartz estimates for the corresponding linear equation in Lorentz-type spaces. Using these estimates, we establish the local well-posedness as well as the small data global well-posedness and scattering in $H^s$ for the $H^s$-critical IBNLS equation under less regularity assumption on the nonlinear term than in the recent work \cite{AKR24}. This result also extends the ones of \cite{SP23,SG24} by extending the validity of $d$, $b$ and $s$. Finally, we give the well-posedness result in the homogeneous Sobolev spaces $\dot{H}^s$.

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arXiv2024

On the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential

In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential \[iu_{t} +Δu-c|x|^{-a}u=\pm |x|^{-b} |u|^{σ} u,\;\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\in \mathbb N$, $c\in \mathbb R$, $a,b>0$ and $σ>0$. First, we establish the local well-posedness in the fractional Sobolev spaces $H^s(\mathbb R^d)$ with $s\ge 0$ by using contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, the global existence and blow-up of $H^1$-solution are investigated. Our results extend the known results in several directions.

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arXiv2023

On stability and instability of the ground states for the focusing inhomogeneous NLS with inverse-square potential

In this paper, we study the stability and instability of the ground states for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential (for short, INLS$_c$ equation): \[iu_{t} +Δu+c|x|^{-2}u+|x|^{-b} |u|^{σ} u=0,\; u(0)=u_{0}(x) \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $0<b<2$, $0<σ<\frac{4-2b}{d-2}$ and $c\neq 0$ be such that $c<c(d):=\left(\frac{d-2}{2}\right)^{2}$. In the mass-subcritical case $0<σ<\frac{4-2b}{d}$, we prove the stability of the set of ground states for the INLS$_{c}$ equation. In the mass-critical case $σ=\frac{4-2b}{d}$, we first prove that the solution of the INLS$_c$ equation with initial data $u_{0}$ satisfying $E(u_0)<0$ blows up in finite or infinite time. Using this fact, we then prove that the ground state standing waves are unstable by blow-up. In the intercritical case $\frac{4-2b}{d}<σ<\frac{4-2b}{d-2}$, we finally show the instability of ground state standing waves for the INLS$_c$ equation.

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arXiv2023

Continuous dependence of the Cauchy problem for the inhomogeneous biharmonic NLS equation in Sobolev spaces

In this paper, we study the continuous dependence of the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] in the standard sense in $H^s$, i.e. in the sense that the local solution flow is continuous $H^s\to H^s$. Here $d\in \mathbb N$, $s>0$, $λ\in \mathbb R$ and $σ>0$. To arrive at this goal, we first obtain the estimates of the term $f(u)-f(v)$ in the fractional Sobolev spaces which generalize the similar results of An-Kim [5](2021) and Dinh [16](2018), where $f(u)$ is a nonlinear function that behaves like $λ|u|^σu$ with $λ\in \mathbb R$. These estimates are then applied to obtain the standard continuous dependence result for IBNLS equation with $0<s <\min \{2+\frac{d}{2},\frac{3}{2}d\}$, $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$ and $0<σ< σ_{c}(s)$, where $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. Our continuous dependence result generalizes that of Liu-Zhang [27](2021) by extending the validity of $s$ and $b$.

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arXiv2022

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

We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,\;u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $λ\in \mathbb R$, $d\in \mathbb N$, $0\le s<\min\left\{2+\frac{d}{2},d\right\}$, $0 0$. 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 $\frac{8-2b}{d}\le σ\le σ_{c}(s)$ with $σ<\infty$.

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arXiv2022

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

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $λ\in \mathbb R$, $d\in \mathbb N$, $0<s<\min \{2+\frac{d}{2},\frac{3}{2}d\}$ and $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in $H^{s}(\mathbb R^{d})$ if $\frac{8-2b}{d}<σ< σ_{c}(s)$ and the initial data is sufficiently small, where $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c}(s)=\infty$ if $s\ge \frac{d}{2}$.

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arXiv2022

Local well-posedness for the inhomogeneous biharmonic nonlinear Schrödinger equation in Sobolev spaces

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $d\in \mathbb N$, $s\ge 0$, $0 0$ and $λ\in \mathbb R$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is locally well-posed in $H^{s}(\mathbb R^{d})$ if $d\in \mathbb N$, $0\le s <\min \{2+\frac{d}{2},\frac{3}{2}d\}$, $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$ and $0<σ< σ_{c}(s)$. Here $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. Our local well-posedness result improves the ones of Guzmán-Pastor [Nonlinear Anal. Real World Appl. 56 (2020) 103174] and Liu-Zhang [J. Differential Equations 296 (2021) 335-368] by extending the validity of $s$ and $b$.

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arXiv2021

A note on the $H^{s}$-critical inhomogeneous nonlinear Schrödinger equation

In this paper, we consider the Cauchy problem for the $H^{s}$-critical inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=λ|x|^{-b} f(u),\; u(0)=u_{0} \in H^{s} (\mathbb R^{n}),\] where $n\in \mathbb N$, $0\le s<\frac{n}{2}$, $0<b<\min \left\{2,\;n-s,\; 1+\frac{n-2s}{2} \right\}$ and $f(u)$ is a nonlinear function that behaves like $λ|u|^{σ} u$ with $λ\in \mathbb C$ and $σ=\frac{4-2b}{n-2s}$. First, we establish the local well-posedness as well as the small data global well-posedness in $H^{s}(\mathbb R^{n})$ for the $H^{s}$-critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, we obtain some standard continuous dependence results for the $H^{s}$-critical INLS equation. Our results about the well-posedness and standard continuous dependence for the $H^{s}$-critical INLS equation improve the ones of Aloui-Tayachi [Discrete Contin. Dyn. Syst. 41 (11) (2021), 5409-5437] by extending the validity of $s$ and $b$. Based on the local well-posedness in $H^{1}(\mathbb R^{n})$, we finally establish the blow-up criteria for $H^{1}$-solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.

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arXiv2021

The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u=λ|x|^{-b} |u|^{σ} u,\; u(0)=u_{0} \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $λ=\pm1$, $0 -c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first prove the local well-posedness as well as small data global well-posedness and scattering in $H^{1}$ for $c>-\frac{(d+2-2b)^{2}-4}{(d+2-2b)^{2}}c(d)$ and $0<b<\frac{4}{d}$, by using the contraction mapping principle based on the Strichartz estimates. Based on the local well-posedness result, we then establish the blowup criteria for solutions to the equation in the focusing case $λ=-1$. To this end, we derive the sharp Hardy-Sobolev inequality and virial estimates related to this equation.

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arXiv2021

Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential

In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u+|x|^{-b} |u|^{σ} u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $0 -c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation with general data lying below, at, and above the ground state threshold. Our results extend the global existence and blow-up results of Campos-Guzmán (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).

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arXiv2021

Local and global well-posedness in $L^{2}(\mathbb R^{n})$ for the inhomogeneous nonlinear Schrödinger equation

This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schrödinger (INLS) equation $iu_{t} +Δu=λ\left|x\right|^{-b} \left|u\right|^{σ} u, u(0)=u_{0} \in L^{2}(\mathbb R^{n})$, where $λ\in \mathbb C$, $0<b<\min \left\{2,{\rm \; }n\right\}$ and $0<σ\le \frac{4-2b}{n} $. We prove the local well-posedness and small data global well-posedness of the INLS equation in the mass-critical case $σ=\frac{4-2b}{n} $, which have remained open until now. We also obtain some local well-posedness results in the mass-subcritical case $σ<\frac{4-2b}{n} $. In order to obtain the results above, we establish the Strichartz estimates in Lorentz spaces and use the contraction mapping principle based on Strichartz estimates.

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arXiv2021

Small data global well--posedness and scattering for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f\left(u\right), u\left(0\right)=u_{0} \in H^{s} (\mathbb R^{n}),\] where $0 0$. We prove that the Cauchy problem of the INLS equation is globally well--posed in $H^{s} (\mathbb R^{n})$ if the initial data is sufficiently small and $σ_{0} <σ<σ_{s} $, where $σ_{0} =\frac{4-2b}{n} $ and $σ_{s} =\frac{4-2b}{n-2s} $ if $s<\frac{n}{2} $; $σ_{s} =\infty $ if $s\ge \frac{n}{2} $. Our global well--posedness result improves the one of Guzmán in (Nonlinear Anal. Real World Appl. 37: 249--286, 2017) by extending the validity of $s$ and $b$. In addition, we also have the small data scattering result.

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arXiv2021

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

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f(u),\;u(0)\in H^{s} (\mathbb R^{n} ),\] where $n\in \mathbb N$, $0 0$ and $λ\in \mathbb C$. Recently, An--Kim \cite{AK21} proved the local existence of solutions in $H^{s}(\mathbb R^{n} )$ with $0\le s<\min \{ n,\; 1+n/2\}$. However even though the solution is constructed by a fixed point technique, continuous dependence in the standard sense in $H^{s}(\mathbb R^{n} )$ with $0< s<\min \{ n,\; 1+n/2\}$ 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 $H^{s}(\mathbb R^{n} )$, i.e. in the sense that the local solution flow is continuous $H^{s}(\mathbb R^{n} )\to H^{s}(\mathbb R^{n} )$, if $σ$ satisfies certain assumptions.

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arXiv2021

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

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f(u), ~u(0)=u_{0} \in H^{s} (\mathbb R^{n} ),\] where $n\ge3$, $1\le s<\frac{n}{2} $, $0<b<2$ and $f(u)$ is a nonlinear function that behaves like $λ\left|u\right|^{σ} u$ with $λ\in \mathbb C$ and $σ=\frac{4-2b}{n-2s} $. We establish the local well-posedness as well as the small data global well-posedness and scattering in $H^{s} (\mathbb R^{n} )$ with $1\le s<\frac{n}{2}$ for the critical INLS equation under some assumption on $b$. To this end, we first establish various nonlinear estimates by using fractional Hardy inequality and then use the contraction mapping principle based on Strichartz estimates.

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