\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 06, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/06\hfil Well-posedness for nonlinear Schr\"odinger equations] {Global well-posedness for nonlinear Schr\"odinger equations with energy-critical damping} \author[B. Feng, D. Zhao \hfil EJDE-2015/06\hfilneg] {Binhua Feng, Dun Zhao} % in alphabetical order \address{Binhua Feng (corresponding author)\newline Department of Mathematics, Northwest Normal University, Lanzhou 730070, China} \email{binhuaf@163.com} \address{Dun Zhao \newline School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China} \email{zhaod@lzu.edu.cn} \thanks{Submitted January 7, 2014. Published January 5, 2015.} \subjclass[2000]{35J60, 35Q55} \keywords{Nonlinear Schr\"odinger equation; global solution; \hfill\break\indent energy-critical damping} \begin{abstract} We consider the Cauchy problem for the nonlinear Schr\"odinger equations with energy-critical damping. We prove the existence of global in-time solutions for general initial data in the energy space. Our results extend some results from \cite{acs,Anto}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this article we study the Cauchy problem for the nonlinear Schr\"odinger (NLS) equation with energy-critical damping, \begin{equation}\label{1.1} \begin{gathered} iu_{t}+\frac{1}{2}\Delta u= V(x)u+ \lambda |u|^{2\sigma}u -ia|u|^{\alpha}u ,\quad (t,x)\in [0,\infty )\times \mathbb{R}^{N}, \\ u|_{t=0} = u_0,\quad u_0\in \Sigma , \end{gathered} \end{equation} where $N\geq 3$, $\lambda \in \mathbb{R}$, $a>0$, $0<\sigma \leq\frac{2}{N-2}$, $\alpha=\frac{4}{N-2}$ and $ \Sigma $ denotes the energy space associated to the harmonic potential; i.e., \[ \Sigma =\{u\in H^1(\mathbb{R}^{N}),\; xu\in L^2(\mathbb{R}^{N})\}, \] equipped with the norm \[ \|u\|_{\Sigma}:=\|u\|_{L^2}+\|\nabla u\|_{L^2}+\|xu\|_{L^2}. \] The external potential $V$ is supposed to be an anisotropic quadratic confinement, i.e., \begin{equation}\label{1.2} V(x)=\frac{1}{2}\sum_{j=1}^{N}\omega_j^2x_j^2,\quad \omega_j \in \mathbb{R}. \end{equation} Equation \eqref{1.1} appears in different physical contexts. For example, considering the three-body interaction in collapsing Bose-Einstein condensates (BECs), within the realm of Gross-Pitaevskii theory, the emittance of particles from the condensate is described by the dissipative model involving a quintic nonlinear damping term \cite{kms}; in nonlinear optics, equation \eqref{1.1} with $V = 0$ describes the propagation of a laser pulse within an optical fiber under the influence of additional multi-photon absorption processes, see, e.g., \cite{bi,fi}. For $a=0$, equation \eqref{1.1} simplifies to the classical NLS. It arises in various areas of physics, such as nonlinear optics and nonlinear plasmas; for a broader introduction, see \cite{Ca2003,SS}. It also has received a great deal of attention from mathematicians, for instance, see \cite{cabook,Ca2003,SS,Tao2006} and the references therein. For $a>0$, the last term in \eqref{1.1} is dissipative, see \cite{acs,Anto}. Therefore, the energy of \eqref{1.1} is no longer conserved, in contrast to the usual case of Hamiltonian NLS. When $\sigma =\alpha$, and $0 < \sigma \leq 1/N$, the asymptotic behavior in time of the small solution to \eqref{1.1} has been studied in \cite{KS,Shim}. Numerical studies of \eqref{1.1} can be found in \cite{BJ,BJM,fk,PSS}; in particular, the nonlinear-damping continuation of singular solutions for \eqref{1.1} with critical and supercritical nonlinearities has been considered in \cite{fk}. When $V\equiv 0$, under some assumptions, Feng, Zhao and Sun \cite{feng1} have showed that as $a\to 0$ the solution of \eqref{1.1} converges to that of \eqref{1.1} with $a=0$. In \cite{da} the particular case of a mass critical nonlinearity $\sigma=2/N$ and $V = 0$ has been studied. In there, global in-time existence of solutions is established if $\alpha > 4/N$ and it is claimed that finite time blow-up in the log-log regime occurs if $\alpha<4/N$. The global well-posedness for a cubic NLS equation perturbed by higher-order nonlinear damping has been studied in \cite{Anto}, where, in particular, the energy-critical case of a quintic dissipation in three-dimensional space has been treated. Recently, Antonelli, Carles and Sparber \cite{acs} have done a more systematic study for NLS type equations with general energy-subcritical damping. However, equation \eqref{1.1} with an energy-critical damping or nonlinearity do not seem to have been discussed except $N=3$ and $\sigma =1$. The aim of this paper is to establish the global well-posedness for \eqref{1.1} with an energy-subcritical or critical nonlinearity and an energy-critical damping. To solve this problem, we mainly use the idea of \cite{Anto}. This is shown in the following theorem. \begin{theorem}\label{thm1.1} Let $N\geq 3$, $a>0$, $\alpha = \frac{4}{N-2}$ and $u_0 \in \Sigma$. Assume that $V$ satisfy \eqref{1.2} and suppose further that \begin{itemize} \item[(1)] either $\lambda \geq 0$ and $0< \sigma \leq \frac{2}{N-2}$, \item[(2)] or $\lambda < 0$ and $0< \sigma < \frac{2}{N-2}$. \end{itemize} Then, the Cauchy problem \eqref{1.1} has a unique global solution $u \in C([0,\infty),\Sigma)$. \end{theorem} \noindent\textbf{Remark.} In the case of energy-critical, it is well-known (see, e.g. \cite{Ca2003}) that the usual a-priori estimates on the $H^1$-norm is not sufficient to conclude global existence. The reason is that the local existence time of solutions does not only depend on the $H^1$-norm of $u$, but also on its profile. This is an essential difference with \cite{acs}. Enlightened mainly by the work in \cite{Anto,Tao2006,Tao2,zhang}, we will prove this theorem by combining a-priori estimates and a bootstrap argument. We finally state the following estimate for the time-decay of solutions. The proof is the same as that of \cite[Proposition 4.2]{acs}, so we omit it. \begin{corollary} \label{coro1.2} Let $N\geq 3$, $a>0$, $\omega_j \neq 0$ ($j=1,\dots ,N$) and $u_0 \in \Sigma$. In either of the cases mentioned in Theorem \ref{thm1.1}, the solution to \eqref{1.1} satisfies $u\in L^\infty([0,\infty),\Sigma)$ and there exists $C>0$ such that \begin{equation*} \|u(t)\|_{L^2}^2 \leq Ct^{-\frac{N-2}{N+2}}, \quad \forall t \geq 1. \end{equation*} \end{corollary} This article is organized as follows: in Section 2, we collect some lemmas such as Strichartz's estimates, and a-priori estimates for the solutions of \eqref{1.1}. In section 3, we show Theorem \ref{thm1.1}. \smallskip \noindent\textbf{Notation.} In this article, we use the following notation. $C> 0$ will stand for a constant that may be different from line to line when it does not cause any confusion. Since we exclusively deal with $\mathbb{R}^N$, we often use the abbreviation $ L^{r}=L^{r}(\mathbb{R}^{N})$. Given any interval $I\subset \mathbb{R}$, the norms of mixed spaces $L^q(I,L^{r}(\mathbb{R}^{N}))$ are denoted by $\|\cdot\|_{L^q(I,L^{r})}$. We denote by $U(t):=e^{itH}$, the Schr\"odinger group generated by $H=-\frac{1}{2}\Delta +V$. We recall that a pair of exponents $(q,r)$ is Schr\"odinger-admissible if $\frac{2}{q}=N(\frac{1}{2}-\frac{1}{r})$ and $2 \leq r \leq \frac{2N}{N-2}$, ($ 2\leq r\leq \infty$ if $ N=1$; $2\leq r <\infty$ if $N=2$). Then, for any space-time slab $I\times \mathbb{R}^N$, we can define the Strichartz norm \begin{equation*} \|u\|_{S(I)}=\sup_{(q,r)}\|u\|_{L^q(I,L^{r})}, \end{equation*} where the supremum is taken over all admissible pairs of exponents $(q,r)$. \section{Some lemmas} We first recall the following Strichartz's estimates. \begin{lemma}[\cite{Anto,ca1,ca2,KT})] \label{lem2.1} Let $(q,r)$, $(q_1,r _1)$ and $(q_2,r _2)$ be admissible pairs. Assume that $I$ is some finite time interval. Then it follows \begin{equation*} \|U(\cdot)\varphi \|_{L^{q}(I, L^{r})}\leq C(r,N)|I|^{1/q}\|\varphi \|_{L^2}, \end{equation*} and \begin{equation*} \big\|\int_{I\cap \{s \leq t\}}U(t-s)F(s)ds\big\|_{L^{q_1}(I, L^{r_1})}\leq C(r_1,r_2,N)|I|^{1/q_{1}}\|F \|_{L^{q_2^{\prime }}(I,L^{r_2^{\prime }})}. \end{equation*} \end{lemma} Next, we show that \eqref{1.1} is locally well-posed for any $u_0 \in \Sigma$ and we also establish a blow-up alternative. \begin{proposition}[Local solution] \label{prop2.2} Let $N\geq 3$, $0< \sigma \leq \frac{2}{N-2}$, $\alpha = \frac{4}{N-2}$, $\lambda, a \in \mathbb{R}$ and $V$ satisfy \eqref{1.2}. For every $u_0 \in \Sigma$, there exist $T>0$ and a unique strong solution $u$ defined on $[0,T]$. Let $[0,T^*)$ be the maximal time interval on which $u$ is well-defined, then, the following properties hold: \begin{itemize} \item[(i)] $u,\nabla u, xu \in S([0,T])$ for $00$, we define \begin{equation*} X_T=L^\infty((0,T);L^2)\cap L^q((0,T);L^r)\cap L^\gamma((0,T);L^\rho) \end{equation*} where $r=2\sigma+2$, \[ q=\frac{4\sigma+4}{N\sigma}, \quad \gamma=\frac{2N}{N-2}, \quad \rho=\frac{2N^2}{N^2-2N+4}. \] Since $U(\cdot)\nabla u_0 \in X_T$ by Strichartz's estimates, we have $\|U(\cdot)\nabla u_0\|_{X_T}\to 0$ as $T\to 0$. Next, we claim that there exists $\eta>0$ such that if $u_0\in \Sigma$ satisfies \begin{equation}\label{l} \|U(\cdot)\nabla u_0\|_{X_T}\leq \eta \end{equation} for some $T>0$, then there exists a unique solution $u\in S([0,T])$ of \eqref{1.1}. Notice that \eqref{l} is satisfied for $T$ small enough. Indeed, fix $\eta>0$, to be chosen later. Duhamel's formulation for \eqref{1.1} reads \begin{equation}\label{l1} u(t)=U(t)u_0-i \lambda \int_{0}^{t}U(t-s)(|u|^{2\sigma} u)(s)ds -a\int_{0}^{t}U(t-s)(|u|^{\frac{4}{N-2}} u)(s)ds. \end{equation} Denote the right hand side by $\Phi(u)(t)$. By Lemma \ref{lem2.1} and H\"older's inequality, we have \begin{equation} \label{l2} \begin{aligned} \|\Phi(u)\|_{X_T} &\leq C\|u_0\|_{L^2} +C\||u|^{2\sigma} u\|_{L^{q'}((0,T);L^{r'})} +C\||u|^{\frac{4}{N-2}} u\|_{L^{\gamma'}((0,T);L^{\rho'})} \\& \leq C\|u_0\|_{L^2} +CT^{2\sigma/\theta}\|u\|_{ L^\infty((0,T);H^1)}^{2\sigma} \|u\|_{L^q((0,T);L^r)}\\ &\quad +C\|u\|_{ L^\gamma((0,T);L^\rho)}\|\nabla u\|_{ L^\gamma((0,T);L^\rho)}^{\frac{4}{N-2}}, \end{aligned} \end{equation} where $\theta=\frac{2\sigma(2\sigma+2)}{2-(N-2)\sigma}$. Next, to estimate $\nabla u$ and $xu$, we notice that \begin{equation*} [\partial_j,H]=\partial_j V(x),\quad [x_j,H]=\partial_j,\quad j=1,\ldots,N. \end{equation*} where $[A,B]=AB-BA$ denotes the usual commutator. Therefore, \begin{equation} \label{l1b} \begin{aligned} \nabla \Phi (u)(t) &= U(t) \nabla u_0-i \lambda \int_{0}^{t}U(t-s)\nabla(|u|^{2\sigma} u)(s)ds \\ &\quad -a\int_{0}^{t}U(t-s)\nabla(|u|^{\frac{4}{N-2}} u)(s)ds\\ &\quad -i \lambda \int_{0}^{t}U(t-s)\Phi(u)(s)\nabla V ds. \end{aligned} \end{equation} Now we estimate the second term of the right-hand side as above. Since $\nabla V$ is sublinear by assumption, \begin{equation} \label{l3} \begin{aligned} \|\nabla \Phi(u)\|_{X_T} &\leq C\|U(\cdot) \nabla u_0\|_{X_T}+CT^{2\sigma/\theta} \|u\|_{ L^\infty((0,T);H^1)}^{2\sigma}\|\nabla u\|_{L^q((0,T);L^r)} \\ &\quad +C\|\nabla u\|_{ L^\gamma((0,T);L^\rho)}^{\frac{N+2}{N-2}}+CT\|x \Phi(u)\|_{L^\infty((0,T);L^2)}\\ &\quad +CT\|\Phi(u)\|_{L^\infty((0,T);L^2)}. \end{aligned} \end{equation} Similarly, we have \begin{equation} \label{l4} \begin{aligned} \|x \Phi(u)\|_{X_T} &\leq C\|xu_0\|_{L^2}+CT^{2\sigma/\theta}\|u\|_{ L^\infty((0,T);H^1)}^{2\sigma} \|x u\|_{L^q((0,T);L^r)} \\ &\quad +C\|x u\|_{ L^\gamma((0,T);L^\rho)}\|\nabla u\|_{ L^\gamma((0,T);L^\rho)}^{\frac{4}{N-2}}\\ &\quad +CT\|\nabla \Phi(u)\|_{L^\infty((0,T);L^2)}. \end{aligned} \end{equation} It is thus easy to see that $\Phi$ maps the set \begin{align*} \mathcal{B} =\Big\{&u;\|\nabla u\|_{ L^\gamma((0,T);L^\rho)}\leq 2\eta,\; \|\nabla u\|_{L^\infty((0,T);L^2)\cap L^q((0,T);L^r)}\leq 2C\|xu_0\|_{L^2}, \\ &\|x u\|_{X_T}\leq 2C\|xu_0\|_{L^2},\|u\|_{X_T}\leq 2C\|u_0\|_{L^2}\Big\} \end{align*} to itself and is a contraction in the $X_T$ norm, provided $\eta$ and $T$ are chosen sufficiently small. The contraction mapping theorem then implies the existence of a unique solution to \eqref{1.1} on $[0,T]$. Finally, by some standard arguments, (i) and (ii) follow. \end{proof} \noindent\textbf{Remark.} For more general potentials, as suggested in the proof, Proposition \ref{prop2.2} remains valid if we assume more generally that $V (x)$ is smooth, and at most quadratic, i.e., $\partial^\alpha V \in L^\infty(\mathbb{R}^N)$ for all $|\alpha|\geq 2$. In the following, we shall derive several a-priori estimates on the solutions of \eqref{1.1}. By the analogous arguments to those of \cite[Lemma 2.7]{acs} and \cite[Lemma 3.1]{Anto}, we obtain the following lemma. \begin{lemma} \label{lem2.3} Let $u(t)\in \Sigma$ be a solution of \eqref{1.1} defined on the maximal interval $[0,T^*)$, and V (x) satisfy \eqref{1.2}. Then it follows \begin{gather}\label{2011} \|u(t)\|_{L^2} \leq \|u_0\|_{L^2},\quad \forall~t\in [0,T^*), \\ \label{202} \int_0^{T^*} \int_{\mathbb{R}^N}|u(t,x)|^{\frac{2N}{N-2}}dxdt\leq C(\|u_0\|_{L^2}). \end{gather} \end{lemma} The a-priori estimates in Lemma \ref{lem2.1} are not sufficient to conclude global well-posedness for \eqref{1.1}. We consequently follow the idea in \cite{acs} and \cite{Anto} and consider the modified energy functional \begin{equation} \label{2.2} \begin{aligned} E(t)&=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u(t,x)|^2dx +\int_{\mathbb{R}^N}V(x)|u(t,x)|^2dx \\ &\quad +\frac{\lambda}{\sigma+1 }\int_{\mathbb{R}^N}|u(t,x)|^{2\sigma +2}dx +\kappa \int_{\mathbb{R}^N}|u(t,x)|^{\frac{2N}{N-2}}dx. \end{aligned} \end{equation} \begin{lemma} \label{lem2.4} Let $u(t)\in \Sigma$ be a solution of \eqref{1.1} defined on the maximal interval $[0,T^*)$, and V (x) satisfy \eqref{1.2}. Moreover, let $0<\kappa <\frac{a(N-2)^2}{2N}$, and assume that \begin{itemize} \item[(1)] either $\lambda \geq 0$ and $0< \sigma \leq \frac{2}{N-2}$, \item[(2)] or $\lambda < 0$ and $0< \sigma < \frac{2}{N-2}$. \end{itemize} Then \begin{gather}\label{203} E(t)\leq E(0)+C(\|u_0\|_{L^2}),\quad \forall ~t\in [0,T^*), \\ \label{204} \int_0^{T^*} \int_{\mathbb{R}^N}|u(x,t)|^{\frac{2(N+2)}{N-2}}dx dt \leq C(E(0),\|u_0\|_{L^2}). \end{gather} \end{lemma} \begin{proof} This is done along the lines of \cite[Proposition 3.1]{acs}. By their, we obtain \begin{align*} \frac{d}{dt}E(t) &=-\Big(a-\kappa\Big(\frac{4}{(N-2)^2}+\frac{2}{N-2}\Big)\Big) \int_{\mathbb{R}^N}|u|^{\frac{4}{N-2}}|\nabla u|^2dx\\ &\quad -2a\frac{2}{N-2}\int_{\mathbb{R}^N}|u|^{\frac{4}{N-2}}|\nabla |u||^2dx \\ &\quad -\kappa\Big(\frac{4}{(N-2)^2}+\frac{2}{N-2}\Big) \int_{\mathbb{R}^N}|u|^{\frac{4}{N-2}}|Re(\bar{\phi}\nabla u) -\operatorname{Im}(\bar{\phi}\nabla u)|^2dx \\ &\quad -2a\int_{\mathbb{R}^N}V(x)|u|^{\frac{2N}{N-2}}dx -2a\lambda \int_{\mathbb{R}^N}|u|^{\frac{4}{N-2} +2\sigma }dx\\ &\quad -2a\kappa \frac{N}{N-2}\int_{\mathbb{R}^N}|u|^{\frac{8}{N-2}+2}dx, \end{align*} where \begin{equation*} \phi (t,x):=\begin{cases} |u(t,x)|^{-1}u(t,x) &\text{if }u(t,x)\neq 0,\\ 0 &\text{if } u(t,x)=0. \end{cases} \end{equation*} Therefore, if $\lambda \geq 0$, \eqref{203} follows by $\frac{d}{dt}E(t) \leq 0$. If $\lambda < 0$, \eqref{203} follows by the Young inequality with $\varepsilon$. \eqref{204} follows by \eqref{203} and \eqref{202}. \end{proof} With Lemma \ref{lem2.4} in hand, we can obtain the uniform bound on the $\Sigma$-norm of $u(t)$. The proof is analogue to that of Corollary 3.4 in \cite{Anto}, so we omit it. \begin{corollary} \label{coro2.5} Let $u(t)\in \Sigma$ be a solution of \eqref{1.1} defined on the maximal interval $[0,T^*)$. Then \begin{equation*}\label{201} \|u(t)\|_{\Sigma} \leq C(\|u_0\|_{\Sigma}),\quad \forall t\in [0,T^*). \end{equation*} \end{corollary} \section{Proof of main results} \begin{proof}[Proof of Theorem \ref{thm1.1}] Let $I$ be some finite time interval, in the following, we set \[ W(I)=L^{\frac{2(N+2)}{N-2}}(I,L^{\frac{2(N+2)}{N-2}}), \quad V(I)=L^{\frac{2(N+2)}{N}}(I,L^{\frac{2(N+2)}{N}}). \] We divide the proof into two steps: (i) $\frac{2}{N}< \sigma \leq \frac{2}{N-2}$ and (ii) $0< \sigma \leq \frac{2}{N}$. Step 1. We first treat the case (i) $\frac{2}{N}< \sigma \leq \frac{2}{N-2}$. By applying Strichartz's estimates to \eqref{l1} and H\"older's inequality, we can estimate as follows: \begin{equation} \label{303} \begin{aligned} &\|u\|_{L^{q}(I,L^r)} \\ &\leq C|I|^{1/q}\Big(\|u_0\|_{L^2}+\||u|^{2\sigma} u\|_{L^{\frac{2(N+2)}{N+4}}(I,L^{\frac{2(N+2)}{N+4}})}\\ &\quad +\||u|^{\frac{4}{N-2}} u\|_{L^{\frac{2(N+2)}{N+4}}(I,L^{\frac{2(N+2)}{N+4}})}\Big) \\ &\leq C|I|^{1/q}\Big(\|u_0\|_{L^2}+\| u\|^{2\sigma}_{L^{\sigma (N+2)}(I,L^{\sigma (N+2)})}\|u\|_{V(I)}+\|u\|^{\frac{4}{N-2}}_{W(I)}\| u\|_{V(I)}\Big) \\ &\leq C|I|^{1/q}\Big(\|u_0\|_{L^2}+\|u\|^{N\sigma -2}_{W(I)}\| u\|^{3-\sigma (N-2)}_{V(I)}+\|u\|^{\frac{4}{N-2}}_{W(I)}\|u\|_{V(I)}\Big), \end{aligned} \end{equation} where $C$ is independent of $I$. By an analogous argument to that of \eqref{303}, we obtain \begin{equation}\label{305} \begin{aligned} &\|\nabla u\|_{L^{q}(I,L^r)}+\|xu\|_{L^{q}(I,L^r)} \\ &\leq C|I|^{1/q}\Big(\|\nabla u_0\|_{L^2}+\||u|^{2\sigma} \nabla u\|_{L^{\frac{2(N+2)}{N+4}}(I,L^{\frac{2(N+2)}{N+4}})}\\ &\quad +\||u|^{\frac{4}{N-2}}\nabla u\|_{L^{\frac{2(N+2)}{N+4}}(I,L^{\frac{2(N+2)}{N+4}})}\Big) \\ &\quad +C|I|^{1/q}\Big(\|x u_0\|_{L^2}+\||u|^{2\sigma} xu\|_{L^{\frac{2(N+2)}{N+4}}(I,L^{\frac{2(N+2)}{N+4}})}\\ &\quad +\||u|^{\frac{4}{N-2}}x u\|_{L^{\frac{2(N+2)}{N+4}}(I,L^{\frac{2(N+2)}{N+4}})}\Big) \\ &\leq C|I|^{1/q}\Big(\|\nabla u_0\|_{L^2}+\| u\|^{N\sigma -2}_{W(I)}\|u\|^{2-\sigma (N-2)}_{V(I)}\| \nabla u\|_{V(I)}\\ &\quad +\| u\|^{\frac{4}{N-2}}_{W(I)}\|\nabla u\|_{V(I)}\Big) +C|I|^{1/q}\Big(\|x u_0\|_{L^2}\\ &\quad +\|u\|^{N\sigma -2}_{W(I)}\| u\|^{2-\sigma (N-2)}_{V(I)}\|x u\|_{V(I)} +\| u\|^{\frac{4}{N-2}}_{W(I)}\|xu\|_{V(I)}\Big). \end{aligned} \end{equation} Denoting the Strichartz norm in $\Sigma$ by \begin{equation*} \|u\|_{S_\Sigma(I)}:=\| u\|_{S(I)}+\|\nabla u\|_{S(I)}+\|x u\|_{S(I)}, \end{equation*} it follows from \eqref{303} and \eqref{305} that \begin{equation}\label{306} \begin{aligned} &\|u\|_{S_\Sigma(I)}\\ &\leq C\sup_{q} |I|^{1/q}\Big(\|u_0\|_{\Sigma}+\| u\|^{N\sigma -2}_{W(I)} \| u\|_{S_\Sigma(I)}^{3-\sigma (N-2)}+\| u\|^{\frac{4}{N-2}}_{W(I)}\|u\|_{S_\Sigma(I)}\Big). \end{aligned} \end{equation} On the other hand, for every $T\in [0,T^*)$, we deduce from \eqref{204} that there exists $M>0$ such that $\|u\|_{W([0,T])}\leq M$, where $M$ is independent of the length of $I$. Therefore, we can divide $[0,T]$ into subintervals $[0,T]=I_1\cup\ldots \cup I_K$, where $I_k=[t_{k-1},t_k]$ and such that in each $I_k$, we have \begin{equation*} \|u\|_{W(I_k)}\leq \varepsilon,\quad\text{for all }k=1,\ldots,K, \end{equation*} for some $\varepsilon <1$, which only depends on $\|u_0\|_\Sigma$. Considering the first interval, $I_1=[0,t_1]$, from \eqref{306} it follows that \begin{equation*} \|u\|_{S_\Sigma(I_1)}\leq C\sup_{q} |I_1|^{1/q}(\|u_0\|_{\Sigma} +\varepsilon^{N\sigma -2}\| u\|_{S_\Sigma(I_1)}^{3-\sigma (N-2)} +\varepsilon^{\frac{4}{N-2}}\|u\|_{S_\Sigma(I_1)}). \end{equation*} A standard continuity argument yields \begin{equation*} \|u\|_{S_\Sigma (I_1)}\leq C(\|u_0\|_\Sigma,|I_1|). \end{equation*} Similarly, we can show that \begin{equation*} \|u\|_{S_\Sigma (I_k)}\leq C(\|u_{t_{k-1}}\|_\Sigma,|I_k|),\quad k=2,\ldots,K, \end{equation*} which, together with Corollary 2.5 implies \begin{equation*} \|u\|_{S_\Sigma (I_k)}\leq C(\|u_0\|_\Sigma,|I_k|),\quad k=1,\ldots,K. \end{equation*} Summing up all the subintervals $I_k$, it follows that \begin{equation*} \|u\|_{S_\Sigma ([0,T])}\leq C(\|u_0\|_\Sigma,M),\quad \text{for every } T0, \] $r_1=2\sigma +2$, taking $q_1$ such that $(q_1,r_1)$ is an admissible pair. By an analogous argument to that of \eqref{310}, we have \begin{equation} \label{311} \begin{aligned} &\|\nabla u\|_{L^{q}(I,L^r)}+\|xu\|_{L^{q}(I,L^r)} \\ &\leq C|I|^{1/q}\Big(\|\nabla u_0\|_{L^2}+\||u|^{2\sigma} \nabla u\|_{L^{q_1^\prime}(I,L^{r_1^\prime})}+\||u|^{\frac{4}{N-2}}\nabla u\|_{L^{\frac{2(N+2)}{N+4}}(I,L^{\frac{2(N+2)}{N+4}})}\Big) \\ &\quad +C|I|^{1/q}\Big(\|x u_0\|_{L^2}+\||u|^{2\sigma} xu\|_{L^{q_1^\prime}(I,L^{r_1^\prime})}+\||u|^{\frac{4}{N-2}}x u\|_{L^{\frac{2(N+2)}{N+4}}(I,L^{\frac{2(N+2)}{N+4}})}\Big) \\ &\leq C|I|^{1/q}\Big(\|\nabla u_0\|_{L^2}+|I|^{1/\gamma}\| u\|^{1-\theta}_{L^{\infty}(I,L^2)}\| u\|^\theta_{W(I)}\|\nabla u\|_{L^{q_1}(I,L^{r_1})}\\ &\quad +\|u\|^{\frac{4}{N-2}}_{W(I)}\|\nabla u\|_{V(I)}\Big) +C|I|^{1/q}\Big(\|xu_0\|_{L^2}\\ &\quad +|I|^{1/\gamma}\|u\|^{1-\theta}_{L^{\infty}(I,L^2)}\|u\|^\theta_{W(I)} \|x u\|_{L^{q_1}(I,L^{r_1})}+\|u\|^{\frac{4}{N-2}}_{W(I)}\|xu\|_{V(I)}\Big). \end{aligned} \end{equation} It follows from \eqref{310} and \eqref{311} that \begin{equation}\label{312} \|u\|_{S_\Sigma}\leq C\sup_{q} |I|^{1/q} \Big(\|u_0\|_{\Sigma}+|I|^{1/\gamma}\| u\|^\theta_{W(I)} \| u\|^{2-\theta}_{S_\Sigma}+\| u\|^{\frac{4}{N-2}}_{W(I)}\|u\|_{S_\Sigma}\Big). \end{equation} Arguing as Step 1, we can conclude that the Cauchy problem \eqref{1.1} with $0<\sigma \leq 2/N$ is global well-posedness. 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