We consider the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^αu = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2$, $0<b<\min\{\tfrac{N}{2},2\}$, and $\tfrac{4-2b}{N}<α<\tfrac{4-2b}{N-2}$. These constraints make the equation mass-supercritical and energy-subcritical. We extend the results of Farah-Guzmán and Miao-Murphy-Zheng and prove scattering below the ground state with general initial data.
We adapt the argument of Dodson-Murphy to give a simple proof of scattering below the ground state for the intercritical inhomogeneous nonlinear Schrödinger equation. The decaying factor in the nonlinearity obviates the need for a radial assumption.
We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.
In this paper we consider the inhomogeneous nonlinear Schrödinger (INLS) equation \begin{align}\label{inls} i \partial_t u +Δu +|x|^{-b} |u|^{2σ}u = 0, \,\,\, x \in \mathbb{R}^N \end{align} with $N\geq 3$. We focus on the intercritical case, where the scaling invariant Sobolev index $s_c=\frac{N}{2}-\frac{2-b}{2σ}$ satisfies $0<s_c<1$. In a previous work, for radial initial data in $\dot H^{s_c}\cap \dot H^1$, we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up…
We study the focusing intercritical NLS \begin{align}\label{abstract_nls} i\pt_t u+Δ_{x,y}u=-|u|^αu\tag{NLS} \end{align} on the semiperiodic waveguide manifold $\R^d_x\times \T_y$ with $d\geq 5$ and $α\in(\frac{4}{d},\frac{4}{d-1})$. In the case $d\leq 4$, with the aid of the semivirial vanishing theory \cite{Luo_inter}, the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the d…
We consider the focusing inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +Δu + |x|^{-b} |u|^{2σ}u = 0,$$ where $N\geq 2$ and $σ$, $b>0$. We first obtain a small data global result in $H^1$, which, in the two spatial dimensional case, improves the third author result in [22] on the range of $b$. For $N\geq 3$ and $\frac{2-b}{N}<σ<\frac{2-b}{N-2}$, we also study the local well posedness in $\dot H^{s_c}\cap \dot H^1 $, where $s_c=\frac{N}{2}-\frac{2-b}{2σ}$. Sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^1$ are also esta…
We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +Δu +|x|^{-b} |u|^{2σ}u = 0,$$ where $N\geq 3$, $0<b<\min\left\{\frac{N}{2},2\right\}$ and $\frac{2-b}{N}<σ<\frac{2-b}{N-2}$. The scaling invariant Sobolev space is $\dot{H}^{s_c}$ with $s_c=\frac{N}{2}-\frac{2-b}{2σ}$. The restriction on $σ$ implies $0<s_c<1$ and the equation is called intercritical (i.e. mass-supercritical and energy-subcritical). Let $u_0\in \dot H^{s_c}\cap \dot H^1$ be a radial initial data and $u(t)$ the corresponding solution to the INLS equation. We first show that if…
We prove scattering below the mass-energy threshold for the focusing inhomogeneous nonlinear Schrödinger equation \begin{equation} iu_t + Δu + |x|^{-b}|u|^{p-1}u=0, \end{equation} when $b \geq 0$ and $N > 2$ in the intercritical case $0 < s_c <1$. This work generalizes the results of Farah and Guzmán [9], allowing a broader range of values for the parameters $p$ and $b$. We use a modified version of Dodson-Murphy's approach [6], allowing us to deal with the inhomogeneity. The proof is also valid for the classical nonlinear Schrödinger equation ($b = 0$), extending the work in [6] for radial so…
We study the focusing NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^αu\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$ in the intercritical regime $α\in(\frac{4}{d},\frac{4}{d-1})$. By assuming that the \eqref{nls_abstract} is independent of $y$, it reduces to the focusing intercritical NLS on $\mathbb{R}^d$, which is known to have standing wave and finite time blow-up solutions. Naturally, we ask whether these special solutions with non-trivial $y$-dependence exist. In this paper we give an affirmative answer to this question. To that end, we…
In this paper, we study the long-time behavior of global solutions to the Schrödinger-Choquard equation $$i\partial_tu+Δu=-(I_α\ast|\cdot|^b|u|^{p})|\cdot|^b|u|^{p-2}u.$$ Inspired by Murphy, who gave a simple proof of scattering for the non-radial inhomogeneous NLS, we prove scattering theory below the ground state for the intercritical case in energy space without radial assumption.
We consider the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in $H^2(\mathbb{R}^N)$, \begin{equation} iu_t + Δ^2 u - |x|^{-b}|u|^αu=0 \end{equation} when $b > 0$ and $N \geq 5$. We first obtain a small data global result in $H^2$, which, in the five-dimensional case, improves a previous result from Pastor and the second author. In the sequel, we show the main result, scattering below the mass-energy threshold in the intercritical case, that is, $\frac{8-2b}{N} < α<\frac{8-2b}{N-4}$, without assuming radiality of the initial data. The proof combines the decay of the nonlinea…
We extend the result of Farah and Guzmán on scattering for the $3d$ cubic inhomogeneous NLS to the non-radial setting. The key new ingredient is a construction of scattering solutions corresponding to initial data living far from the origin.
We prove scattering below the ground state threshold for an energy-critical inhomogeneous nonlinear Schrödinger equation in three space dimensions. In particular, we extend results of Cho, Hong, and Lee from the radial to the non-radial setting.
We study the focusing NLS equation in $\mathbb{R}^N$ in the mass-supercritical and energy-subcritical (or intercritical) regime, with $H^1$ data at the mass-energy threshold $ \mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where $Q$ is the ground state. Previously, Duyckaerts-Merle studied the behavior of threshold solutions in the $H^1$-critical case, in dimensions $N = 3, 4, 5$, later generalized by Li-Zhang for higher dimensions. In the intercritical case, Duyckaerts-Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts-Roudenko fo…
Consider the focusing inhomogeneous nonlinear Schrödinger equation in $H^1(\mathbb{R}^N)$, $$iu_t + Δu + |x|^{-b}|u|^{p-1}u=0,$$ when $b > 0$ and $N \geq 3$ in the intercritical case $0 < s_c <1$. In previous works, the second author, as well as Farah, Guzmán and Murphy, applied the concentration-compactness approach to prove scattering below the mass-energy threshold for radial and non-radial data. Recently, the first author adapted the Dodson-Murphy approach for radial data, followed by Murphy, who proved scattering for non-radial solutions in the 3d cubic case, for $b<1/2$. This work genera…
We establish the classification of minimal mass blow-up solutions of the $L^2$ critical inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^{\frac{4-2b}{N}}u = 0, \] thereby extending the celebrated result of Merle from the classic case $b=0$ to the case $0<b<\min\{2,N\}$, in any dimension $N\ge1$.
This paper is concerned with the global well-posedness and finite time blowup problem for the 3D focusing energy-critical inhomogeneous NLS. In the previous results \cite{chkl2, chkl3} the authors considered the same problems with the spatial inhomogeneity coefficient $g$ such that $g(x) \sim |x|^{-b}$ for $0 \le b < \frac43$. Here we extend the inhomogeneous index $b$ up to $\frac32$. For this purpose, we improve the local theory and develop a new profile decomposition based on weighted space.
We consider the $L^2$ critical inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$ i \partial_t u +Δu +|x|^{-b} |u|^{\frac{4-2b}{N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in H^1(\mathbb{R}^N)$ satisfies $E[u_0]<0$, then the corresponding solution blows-up in finite time. This is in sharp contrast to the classical $L^2$ critical NLS equation where this type of result is only known in the radial case for $N\geq 2$.
In this paper we consider inhomogeneous cubic-quintic NLS in space dimension $d = 3$: $$ iu_t = -Δu + K_1(x)|u|^2u + K_2(x)|u|^4u. $$ We study local well-posedness, finite time blowup, and small data scattering and non-scattering for the ICQNLS when $K_1, K_2 \in C^4(\mathbb R^3 \setminus \{0\})$ satisfy growth condition $|\partial^j K_i(x)| \lesssim |x|^{b_i-j}\, (j = 0, 1, 2, 3, 4)$ for some $b_i \ge 0$ and for $x \neq 0$. To this end we use the Sobolev inequality for the functions $f \in H^n \,(n = 1, 2)$ such that $\||\mathbf L|^\ell f\|_{H^n} < \infty \,(\ell = 1, 2)$, where $\mathbf L$ i…
We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+Δu+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}<M_k$ are global in time while a finite time blow up singularity formation may occur for $\|u\|_{L^2}>M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of t…
We consider the inhomogeneous biharmonic nonlinear Schrödinger equation $$ i u_t +Δ^2 u+λ|x|^{-b}|u|^αu = 0, $$ where $λ=\pm 1$ and $α$, $b>0$. In the subctritical case, we improve the global well-posedness result obtained in \cite{GUZPAS} for dimensions $N=5,6,7$ in the Sobolev space $H^2(\mathbb{R}^N)$. The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, $α=\frac{8-2b}{N-4}$ are also reported. More precisely, we show well-posedness and a st…
In this paper, we study the scattering theory for the cubic inhomogeneous Schrödinger equations with inverse square potential $iu_t+Δu-\frac{a}{|x|^2}u=λ|x|^{-b}|u|^2u$ with $a>-\frac14$ and $0<b<1$ in dimension three. In the defocusing case (i.e. $λ=1$), we establish the global well-posedness and scattering for any initial data in the energy space $H^1_a(\mathbb R^3)$. While for the focusing case(i.e. $λ=-1$), we obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson-Murphy [Proc. Amer. Math. Soc.,145(2017…
We consider the focussing energy-critical inhomogeneous nonlinear Schrödinger equation: $$ iu_t + Δu + g|u|^2u = 0, u(0)= φ\in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(φ) < E_g(Q),\;\;\mbox{and}\;\; g_s\|φ\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $ΔQ + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\nabla g(x)| \lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \le 1$, and rigidity condition $-g(x) \le x\…
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…
We consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+λ|x|^{-b}|u|^αu = 0,\;\;\mathcal{L}_a=Δ-\frac{a}{|x|^2}, $$ where $λ=\pm1$, $α,b>0$ and $a>-\frac{(N-2)^2}{4}$. We first establish sufficient conditions for global existence and blow-up in $H^1_a(\mathbb{R}^N)$ for $λ=1$, using a Gagliardo-Nirenberg-type estimate. In the sequel, we study local and global well-posedness in $H^1_a(\mathbb{R}^N)$ in the $H^1$-subcritical case, applying the standard Strichartz estimates combined with the fixed point argument. The …
The aim of this paper is to show the small data scattering for 2D ICQNLS: $$iu_t=-Δu + K_1(x)|u|^2u+K_2(x)|u|^4u.$$ Under the assumption that $\left| \partial^j K_l \right| \lesssim |x|^{b_l -j}$ for $j=0, 1, 2, l=1, 2$ and $0 \le b_l \le l - \frac23$, we prove the small data scattering in an angularly regular Sobolev space $H_θ^{1,1}$. We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space $H_θ^{1, 1} \subset H^1$ with angular regularity such that $\|\partial_θf\|_{H^1} < \infty$, and also use the recently developed angularly averaged Stri…
We consider the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation in $\mathbb{R}^N$, $$i \partial_t u +Δ^2 u -|x|^{-b} |u|^{2σ}u = 0,$$ where $σ>0$ and $b>0$. We first study the local well-posedness in $\dot H^{s_c}\cap \dot H^2 $, for $N\geq 5$ and $0<s_c<2$, where $s_c=\frac{N}{2}-\frac{4-b}{2σ}$. Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^2$ with $0\leq s_c<2$. Finally, we study the phenomenon of $L^{σ_c}$-norm concentration for finite time blow up solutions w…
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 $λ\in \mathbb R$, $d\in \mathbb N$, $0\le s<\min\left\{2+\frac{d}{2},d\right\}$, $0<b<\min \left\{4,\; d-s,\; 2+\frac{d}{2}-s \right\}$ and $0<σ\le σ_{c}(s)$ with $σ<\infty$. Here $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. First, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian $(-Δ)^{s/2}$…
In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schrödinger equation with a potential $$ iu_{t}+Δu-Vu+|x|^{-b}|u|^{2}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{3}}, $$ where $0<b<1$. We first establish global well-posedness and scattering for the radial initial data $u_{0}$ in $H^{1}({\bf R}^{3})$ satisfying $M(u_{0})^{1-s_{c}}E(u_{0})^{s_{c}}<\mathcal{E}$ and $\|u_{0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}<\mathcal{K}$ provided that $V$ is repulsive, where $\mathcal{E}$ and $\mathcal{K}$ are the mass-energy and mass-kinetic of the ground states,…
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 …
The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schrödinger equation $$ i u_t +Δu+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+ΔQ+ |x|^{-b}|Q|^{2}Q=0$ and $s_c=(1+b)/2$. We show that if the radial initial data $u_0$ belongs to $H^1(\mathbb{R}^3)$ and satisfies $E(u_0)^{s_c}M(u_0)^{1-s_c}<E(Q)^{s_c}M(Q)^{1-s_c}$ and $\| \nabla u_0 \|_{L^2}^{s_c} \|u_0\|_{L^2}^{1-s_c}<\|\nabla Q \|_{L^2}^{s_c} \|Q\|_{L^2}^{1-s_c}$, then the corresponding solution is global and scatters in $H^1(\mathbb{R}^3)$. Our proof is based in the ideas int…
We consider the magnetic nonlinear inhomogeneous Schrödinger equation $$i\partial_t u -\left(-i\nabla+\fracα{|x|^2}(-x_2,x_1)\right)^2 u =\pm|x|^{-\varrho}|u|^{p-1}u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^2,$$ where $α\in\mathbb{R}\setminus\mathbb{Z},\,\varrho>0,\,p>1$. We prove a dichotomy of global existence and scattering versus blow-up of energy solutions under the ground state threshold in the inter-critical regime. The scattering is obtained by using the new approach of Dodson-Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, {Proc. Am. Ma…
