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Van Duong Dinh

Publications and source records attributed to Van Duong Dinh.

At least 37 records · Page 2Linked to original sources
arXiv2020

Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS) \[ i\partial_t u + Δu + μ|x|^{-b} |u|^αu = 0, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d \] with $b, α>0$. First, we revisit the local well-posedness in $H^1(\mathbb{R}^d)$ for (INLS) of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. $μ=-1$ when $0<α<α^\star$ where $α^\star = \frac{4-2b}{d-2}$ for $d\geq 3$, and $α^\star = \infty$ for $d=1, 2$ by assuming that the initial data belongs to the weighted $L^2$ space $Σ=\{u \in H^1(\mathbb{R}^d) : |x| u \in L^2(\mathbb{R}^d) \}$. Finally, we combine the local theory and the decaying property to show the scattering in $Σ$ for the defocusing (INLS) in the case $α_\star<α<α^\star$, where $α_\star = \frac{4-2b}{d}$.

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arXiv2020

Long time dynamics for the focusing nonlinear Schrödinger equation with exponential nonlinearities

In this paper, we study the focusing nonlinear Schrödinger equation with exponential nonlinearities \[ i \partial_t u + Δu = - \left(e^{4π|u|^2} - 1 - 4πμ|u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] where $μ\in \{0, 1\}$. By using variational arguments, we first derive invariant sets where the global existence and finite time blow-up occur. In particular, we obtain sharp thresholds for global existence and finite time blow-up. In the case $μ=1$, by adapting a recent argument of Arora-Dodson-Murphy \cite{ADM}, we study the long time dynamics of global solutions. It turns out that either there exist $t_n\rightarrow +\infty$ and $R_n \rightarrow \infty$ such that $u(t_n)$ vanishes inside $B(0,R_n)$ for all $n\geq 1$ or the solution scatters in $H^1$.

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arXiv2020

Global dynamics for a class of inhomogeneous nonlinear Schrödinger equations with potential

We consider a class of $L^2$-supercritical inhomogeneous nonlinear Schrödinger equations with potential in three dimensions \[ i\partial_t u + Δu - V u = \pm |x|^{-b} |u|^αu, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3, \] where $0 \frac{4-2b}{3}$. In the focusing case, by adapting an argument of Dodson-Murphy, we first study the energy scattering below the ground state for the equation with radially symmetric initial data. We then establish blow-up criteria for the equation whose proof is based on an argument of Du-Wu-Zhang. In the defocusing case, we also prove the energy scattering for the equation with radially symmetric initial data.

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arXiv2020

Blow-up criteria for linearly damped nonlinear Schrödinger equations

We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations \[ i\partial_t u + Δu + i a u= \pm |u|^αu, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N, \] where $a>0$ and $α>0$. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up $H^1$ solutions to the focusing problem in the mass-critical and mass-supercritical cases.

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arXiv2020

On nonlinear Schrödinger equations with attractive inverse-power potentials

We study the Cauchy problem for nonlinear Schrödinger equations with attractive inverse-power potentials. By using variational arguments, we first determine a sharp threshold of global well-posedness and blow-up for the equation in the mass-supercritical case. We next study the existence and non-existence of minimizers for the energy functional with prescribed mass constraint. In the mass-critical case, we also study the blow-up behavior of minimizers when the mass tends to a critical value.

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arXiv2019

Existence, non-existence and blow-up behavior of minimizers for the mass-critical fractional nonlinear Schrödinger equations with periodic potentials

We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional nonlinear Schrödinger equation with periodic potentials. Using the concentration-compactness principle, we show a complete classification for the existence and non-existence of minimizers for the problem. In the mass-critical case, under a suitable assumption of the potential, we give a detailed description of blow-up behavior of minimizers once the mass tends to a critical value.

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arXiv2019

Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions

We consider a class of $L^2$-supercritical inhomogeneous nonlinear Schrödinger equations in two dimensions \[ i\partial_t u + Δu = \pm |x|^{-b} |u|^αu, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] where $0 2-b$. By adapting a new approach of Arora-Dodson-Murphy \cite{ADM}, we show the energy scattering for the equation with radially symmetric initial data. In the focusing case, our result extends the one of Farah-Guzmán \cite{FG-high} to the whole range of $b$ where the local well-posedness is available. In the defocusing case, our result extends the one in \cite{Dinh-scat} where the energy scattering for non-radial initial data was established in dimensions $N\geq 3$.

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arXiv2019

Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation

We study the Cauchy problem for an inhomogeneous Gross-Pitaevskii equation. We first derive a sharp threshold for global existence and blow up of the solution. Then we construct and classify finite time blow up solutions at the minimal mass threshold. Additionally, using variational techniques, we study the existence, the orbital stability and instability of standing waves.

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arXiv2018

On nonlinear Schrödinger equations with repulsive inverse-power potentials

In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equations with repulsive inverse-power potentials \[ i \partial_t u + Δu - c |x|^{-σ} u = \pm |u|^αu, \quad c>0. \] We study the local and global well-posedness, finite time blow-up and scattering in the energy space $H^1$ for the equation. These results extend a recent work of Miao-Zhang-Zheng [Nonlinear Schrödinger equation with coulomb potential, arXiv:1809.06685] to a general class of inverse-power potentials and higher dimensions.

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arXiv2018

Scattering in the weighted $L^2$-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

We investigate the defocusing inhomogeneous nonlinear Schrödinger equation $$ i \partial_tu + Δu = |x|^{-b} \left({\rm e}^{α|u|^2} - 1- α|u|^2 \right) u, \quad u(0)=u_0, \quad x \in \mathbb{R}^2, $$ with $0<b<1$ and $α=2π(2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weighted space $Σ(\mathbb{R}^2)=\{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\}$. Then we combine the local theory with the decay estimate to obtain scattering in $Σ$ when the Hamiltonian is below the value $\frac{2}{(1+b)(2-b)}$.

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arXiv2018

Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction

We study the existence and stability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction in dimensions $d\leq 3$. We also study the characterization of finite time blow-up solutions with minimal mass to the system under mass resonance condition in dimension $d=4$. Finite time blow-up solutions with minimal mass are showed to be (up to symmetries) pseudo-conformal transformations of a ground state standing wave.

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arXiv2018

On instability of radial standing waves for the nonlinear Schrödinger equation with inverse-square potential

We show the strong instability of radial ground state standing waves for the focusing $L^2$-supercritical nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu + c|x|^{-2} u = - |u|^α u, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $u: \mathbb{R} \times \mathbb{R}^d \rightarrow \mathbb{C}$, $c\ne 0$ satisfies $c<λ(d):=\left(\frac{d-2}{2}\right)^2$ and $\frac{4}{d} <α<\frac{4}{d-2}$. This result extends a recent result of Bensouilah-Dinh-Zhu [{\it On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential}, \url{arXiv:1805.01245}] where the stability and instability of standing waves were shown in the $L^2$-subcritical and $L^2$-critical cases.

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arXiv2018

Blow-up criteria for fractional nonlinear Schrödinger equation

We consider the focusing fractional nonlinear Schrödinger equation \[ i\partial_t u - (-Δ)^s u = -|u|^αu, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $s \in (1/2,1)$ and $α>0$. By using localized virial estimates, we establish general blow-up criteria for non-radial solutions to the equation. As consequences, we obtain blow-up criteria in both $L^2$-critical and $L^2$-supercritical cases which extend the results of Boulenger-Himmelsbach-Lenzmann [{\it Blowup for fractional NLS}, J. Funct. Anal. 271 (2016), 2569--2603] for non-radial initial data.

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arXiv2018

Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential

We consider a class of the focusing nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu -c|x|^{-2}u = - |u|^αu, \quad u(0)=u_0 \in H^1, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $\frac{4}{d}\leq α\leq \frac{4}{d-2}$ and $c\ne 0$ satisfies $c>-λ(d):=-\left(\frac{d-2}{2}\right)^2$. In the mass-critical case $α=\frac{4}{d}$, we prove the global existence and blowup below ground states for the equation with $d\geq 3$ and $c>-λ(d)$. In the mass and energy intercritical case $\frac{4}{d}<α<\frac{4}{d-2}$, we prove the global existence and blowup below the ground state threshold for the equation. This extends similar results of \cite{KillipMurphyVisanZheng} and \cite{LuMiaoMurphy} to any dimensions $d\geq 3$ and a full range $c>-λ(d)$. We finally prove the blowup below ground states for the equation in the energy-critical case $α=\frac{4}{d-2}$ with $d\geq 3$ and $c>-\frac{d^2+4d}{(d+2)^2} λ(d)$.

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arXiv2018

Global in time Strichartz estimates for the fractional Schrödinger equations on asymptotically Euclidean manifolds

In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger operators, namely $e^{-itΛ_g^σ}$ with $σ\in (0,\infty)\backslash \{1\}$ and $Λ_g:=\sqrt{-Δ_g}$ where $Δ_g$ is the Laplace-Beltrami operator on asymptotically Euclidean manifolds $(\mathbb{R}^d,g)$. Let $f_0\in C^\infty_0(\mathbb{R})$ be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part $(1-f_0)(P)e^{-itΛ_g^σ}$ satisfies global in time Strichartz estimates as on $\mathbb{R}^d$ of dimension $d\geq 2$ inside a compact set under non-trapping condition. On the other hand, under the moderate trapping assumption, the high frequency part also satisfies the global in time Strichartz estimates outside a compact set. We next prove that the low frequency part $f_0(P)e^{-itΛ_g^σ}$ satisfies global in time Strichartz estimates as on $\mathbb{R}^d$ of dimension $d\geq 3$ without using any geometric assumption on $g$. As a byproduct, we prove global in time Strichartz estimates for the fractional Schrödinger and wave equations on $(\mathbb{R}^d, g), d\geq 3$ under non-trapping condition.

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arXiv2018

On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation

In this paper we study dynamical properties of blowup solutions to the focusing intercritical (mass-supercritical and energy-subcritical) nonlinear fourth-order Schrödinger equation. We firstly establish the profile decomposition of bounded sequences in $\dot{H}^{γ_{\text{c}}} \cap \dot{H}^2$. We also prove a compactness lemma and a variational characterization of ground states related to the equation. As a result, we obtain the $\dot{H}^{γ_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{γ_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{H}^{γ_{\text{c}}}$-norm.

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arXiv2018

On blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation

We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation \[ i\partial_t u -(-Δ)^s u = -|u|^αu, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where $d \geq 2, \frac{d}{2d-1} \leq s <1$, $\frac{4s}{d}<α<\frac{4s}{d-2s}$ and $u_0 \in \dot{H}^{s_{\text{c}}} \cap \dot{H}^s$ is radial with the critical Sobolev exponent $s_{\text{c}}$. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in $\dot{H}^{s_{\text{c}}} \cap \dot{H}^s$. As a result, we obtain the $\dot{H}^{s_{\text{c}}}$-concentration of blowup solutions with bounded $\dot{H}^{s_{\text{c}}}$-norm and the limiting profile of blowup solutions with critical $\dot{H}^{s_{\text{c}}}$-norm.

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