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Van Duong Dinh

Publications and source records attributed to Van Duong Dinh.

50 records · Page 3Linked to original sources
arXiv2018

On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation

We consider the focusing $L^2$-supercritical fractional nonlinear Schrödinger equation \[ i\partial_t u - (-Δ)^s u = -|u|^αu, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d\geq 2, \frac{d}{2d-1} \leq s <1$ and $\frac{4s}{d}<α<\frac{4s}{d-2s}$. By means of the localized virial estimate, we prove that the ground state standing wave is strongly unstable by blow-up. This result is a complement to a recent result of Peng-Shi [J. Math. Phys. 59 (2018), 011508] where the stability and instability of standing waves were studied in the $L^2$-subcritical and $L^2$-critical cases.

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arXiv2018

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

We consider 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 $d \geq 3$, $c\ne 0$, $c<λ(d)=\left(\frac{d-2}{2}\right)^2$ and $0<α\leq \frac{4}{d}$. Using the profile decomposition obtained recently by the first author \cite{Bensouilah}, we show that in the $L^2$-subcritical case, i.e. $0<α<\frac{4}{d}$, the sets of ground state standing waves are orbitally stable. In the $L^2$-critical case, i.e. $α=\frac{4}{d}$, we show that ground state standing waves are strongly unstable by blow-up.

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arXiv2018

Mass concentration and characterization of finite time blow-up solutions for the nonlinear Schrödinger equation with inverse-square potential

We consider the $L^2$-critical NLS with inverse-square potential $$ i \partial_t u +Δu + c|x|^{-2} u = -|u|^{\frac{4}{d}} u, \quad u(0) = u_0, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, $$ where $d\geq 3$ and $c\ne 0$ satisfies $c<λ(d) := \left(\frac{d-2}{2}\right)^2$. Using a refined compactness lemma, we extend the mass concentration of finite time blow-up solutions established in the attractive case by the first author in [Bensouilah] to $c<λ(d)$. By means of a simple and short limiting profile theorem, we get the same classification result obtained by Csobo and Genoud in [CsoboGenoud] for $0<c<λ(d)$. It also enables us to extend the classification to $c<λ(d)$.

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arXiv2018

Blowup of $H^1$ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation

In this paper, we consider a class of the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b} |u|^αu = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $α_\star\leq α<α^\star$ where $α_\star =\frac{4-2b}{d}$ and $α^\star=\frac{4-2b}{d-2}$ if $d\geq 3$ and $α^\star = \infty$ if $d=1,2$. In the mass-critical case $α=α_\star$, we prove that if $u_0$ has negative energy and satisfies either $xu_0 \in L^2$ with $d\geq 1$ or $u_0$ is radial with $d\geq 2$, then the corresponding solution blows up in finite time. Moreover, when $d=1$, we prove that if the initial data (not necessarily radial) has negative energy, then the corresponding solution blows up in finite time. In the mass and energy intercritical case $α_\star< α<α^\star$, we prove the blowup below ground state for radial initial data with $d\geq 2$. This result extends the one of Farah in \cite{Farah} where the author proved blowup below ground state for data in the virial space $H^1\cap L^2(|x|^2 dx)$ with $d\geq 1$.

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arXiv2018

Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space

In this paper, we consider a class of nonlinear fourth-order Schrödinger equation, namely \[ \left\{ \begin{array}{rcl} i\partial_t u +Δ^2 u &=&-|u|^{ν-1} u, \quad 1+ \frac{8}{d}<ν<1+\frac{8}{d-4},\\ u(0)&=&u_0 \in H^γ(\mathbb{R}^d), \quad 5 \leq d \leq 11. \end{array} \right. \] Using the $I$-method combined with the interaction Morawetz inequality, we establish the global well-posedness and scattering in $H^γ(\mathbb{R}^d)$ with $γ(d,ν)<γ<2$ for some value $γ(d,ν)>0$.

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arXiv2017

On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces

We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs mainly bases on the contraction mapping argument using Strichartz estimate. We also apply the technique of Christ-Colliander- Tao in \cite{ChristCollianderTao} to prove the ill-posedness for (NLHW) in some cases of the super-critical range.

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arXiv2017

On the focusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space

In this paper, we consider the focusing mass-critical nonlinear fourth-order Schrödinger equation. We prove that blowup solutions to this equation with initial data in $H^γ(\mathbb{R}^d), 5\leq d \leq 7, \frac{56-3d+\sqrt{137d^2+1712d+3136}}{2(2d+32)} <γ<2$ concentrate at least the mass of the ground state at the blowup time. This extends the work in \cite{ZhuYangZhang11} where Zhu-Yang-Zhang studied the formation of singularity for the equation with rough initial data in $\mathbb{R}^4$. We also prove that the equation is globally well-posed with initial data $u_0 \in H^γ(\mathbb{R}^d), 5\leq d \leq 7, \frac{8d}{3d+8}<γ<2$ satisfying $\|u_0\|_{L^2(\mathbb{R}^d)} <\|Q\|_{L^2(\mathbb{R}^d)}$, where $Q$ is the solution to the ground state equation.

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arXiv2017

Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu - |x|^{-b} |u|^αu = 0, \quad u(0)=u_0 \in H^1, \] with $b, α>0$. We firstly study the decaying property of global solutions for the equation when $0<α<α^\star$ where $α^\star = \frac{4-2b}{d-2}$ for $d\geq 3$. The proof makes use of an argument of Visciglia. We next use this decay to show the energy scattering for the equation in the case $α_\star<α<α^\star$, where $α_\star = \frac{4-2b}{d}$.

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arXiv2017

Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary

We firstly prove Strichartz estimates for the fractional Schrödinger equations on $\mathbb{R}^d$ endowed with a smooth bounded metric $g$. We then prove Strichartz estimates for the fractional Schrödinger and wave equations on compact Riemannian manifolds without boundary $(M,g)$. We finally give applications of Strichartz estimates for the local well-posedness of the pure power-type nonlinear fractional Schrödinger and wave equations posed on $(M,g)$.

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arXiv2017

Well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation

We prove the local well-posedness for the nonlinear fourth-order Schrödinger equation (NL4S) in Sobolev spaces. We also studied the regularity of solutions in the sub-critical case. A direct consequence of this regularity is the global well-posedness above mass and energy spaces under some assumptions. Finally, we show the ill-posedness for (NL4S) in some cases of the super-critical range.

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arXiv2016

Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces

We prove the well-posed results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrödinger equations on $\mathbb{R}^d$. These results extend the previous ones in \cite{HongSire} for $σ\geq 2$. This covers the well-known result for the Schrödinger equation $σ= 2$ given in \cite{CazenaveWeissler}. In the case $σ\in (0,2)\backslash \{1\}$, we give the local well-posedness in sub-critical case for all exponent $ν> 1$ in contrast of ones in \cite{HongSire}. This also generalizes the ones of \cite{ChoHwangKwonLee} when $d = 1$ and of \cite{GuoHuo} when $d \geq 2$ where the authors considered the cubic fractional Schrödinger equation with $σ\in (1,2)$. We also give the global existence in energy space under some assumptions. We finally prove the local well-posedness in sub-critical and critical cases for the pure power-type nonlinear fractional wave equations.

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