\documentclass[reqno]{amsart} \pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 62, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/62\hfil Modified wave operators] {Modified wave operators for nonlinear Schr\"odinger equations in one and two dimensions} \author[N. Hayashi, P. I. Naumkin, A. Shimomura, \& S. Tonegawa \hfil EJDE-2004/62\hfilneg] {Nakao Hayashi, Pavel I. Naumkin, \\ Akihiro Shimomura, \& Satoshi Tonegawa} % in alphabetical order \address{Nakao Hayashi \hfill\break Department of Mathematics\\ Graduate School of Science\\ Osaka University, Osaka\\ Toyonaka, 560-0043, Japan} \email{nhayashi@math.wani.osaka-u.ac.jp} \address{Pavel I. Naumkin\hfill\break Instituto de Matem\'{a}ticas\\ UNAM Campus Morelia, AP 61-3 (Xangari)\\ Morelia CP 58089, Michoac\'{a}n, Mexico} \email{pavelni@matmor.unam.mx} \address{Akihiro Shimomura \hfill\break Department of Mathematics\\ Gakushuin University\\ 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan} \email{simomura@math.gakushuin.ac.jp} \address{Satoshi Tonegawa \hfill\break College of Science and Technology\\ Nihon University\\ 1-8-14 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan} \email{tonegawa@math.cst.nihon-u.ac.jp} \dedicatory{Dedicated to Professor ShigeToshi Kuroda on his 70th birthday\\ and to Professor Masaru Yamaguchi on his 60th birthday} \date{} \thanks{Submitted March 10, 2004. Published April 21, 2004.} \subjclass[2000]{35Q55, 35B40, 35B38} \keywords{Modified wave operators, nonlinear Schr\"{o}dinger equations} \begin{abstract} We study the asymptotic behavior of solutions, in particular the scattering theory, for the nonlinear Schr\"{o}dinger equations with cubic and quadratic nonlinearities in one or two space dimensions. The nonlinearities are summation of gauge invariant term and non-gauge invariant terms. The scattering problem of these equations belongs to the long range case. We prove the existence of the modified wave operators to those equations for small final data. Our result is an improvement of the previous work \cite{Shim-Tone}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper, we study the existence global solutions and scattering theory for the nonlinear Schr\"{o}dinger equations \begin{equation} \mathcal{L}u=\mathcal{N}_n(u) +\mathcal{G}_n(u) ,\quad (t,x) \in \mathbb{R}\times \mathbb{R}^{n}, \label{1.1} \end{equation} in one or two space dimensions $n=1$ and $2$, where $\mathcal{L}=i\partial _{t}+\frac{1}{2}\Delta $ and \begin{gather*} \mathcal{N}_1(u) =\lambda _1u^{3}+\lambda _{2}\overline{u} ^2u+\lambda _3\overline{u}^{3}, \\ \mathcal{N}_{2}(u) =\lambda _1u^2+\lambda _{2}\overline{u}^2, \\ \mathcal{G}_n(u) =\lambda _0| u| ^{\frac{2}{n}}u \end{gather*} with $\lambda _0\in \mathbb{R}$ and $\lambda _{j}\in \mathbb{C}$, $j=1,2,3$. We construct a modified wave operator in $L^2$ to equation (\ref{1.1}) for small final data $\phi \in H^{0,2}\cap \dot{H}^{-\delta }$ with $\frac{n}{2}<\delta <2$, where the weighted Sobolev space is defined by \begin{equation*} H^{m,s}=\left\{ u\in \mathcal{S}';\| u\| _{ H^{m,s}}=\| \left\langle i\nabla \right\rangle ^{m}\left\langle x\right\rangle ^{s}u\| _{L^2}<\infty \right\} , \end{equation*} where $\langle x\rangle =\sqrt{1+|x| ^2}$ and the homogeneous Sobolev space is \begin{equation*} \dot{H}^{m}=\left\{ u\in \mathcal{S}';\| u\| _{ \dot{H}^{m}}=\| (-\Delta) ^{\frac{m}{2}}u\|_{L^2}<\infty \right\} . \end{equation*} We intend to weaken the assumption $\phi \in \dot{H}^{-4}$ from the previous work \cite{Shim-Tone}. Many works have been devoted to the global existence and asymptotic behavior of solutions for the nonlinear Schr\"{o}dinger equations. We remind the definition of the wave operators in the scattering theory for the linear Schr\"{o}dinger equation. Assume that for a solution $u_{f}(t,x) $ of the free Schr\"{o}dinger equation $\mathcal{L} u_{f}=0$ with given initial data $u_{f}(0,x) =\phi (x) $, there exists a unique global in time solution $u(t,x) $ of the perturbed Schr\"{o}dinger equation such that $u(t,x) $ behaves like free solution $u_{f}(t,x) $ as $ t\rightarrow \infty $ (this case is called by the short range case, otherwise it is called by the long range case). Then we define a wave operator $\mathcal{W}_{+}$ by the mapping from $\phi $ to $u|_{t=0}$. In the long range case, ordinary wave operators do not exist and we have to construct modified wave operators including a suitable phase correction in their definition. Analogously we can define the wave operators and introduce the modified wave operators for the nonlinear Schr\"{o}dinger equation. We first recall several known results concerning the scattering problem for the nonlinear Schr\"{o}dinger equation \begin{equation} \mathcal{L}u=\lambda |u|^{p-1}u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^{n} \label{eq:NLS} \end{equation} with $\lambda \in \mathbb{R}$ and $p>1$. We consider the existence of wave operators $W_{\pm }$ for equation (\ref{eq:NLS}). The wave operator $W_{+}$ is defined for equation (\ref{eq:NLS}) as follows. Let $\Sigma $ be $L^2$ or a dense subset of it. Let $ \phi \in \Sigma $, and define the free solution \begin{equation*} u_{f}(t)=\mathcal{U}(t)\phi , \end{equation*} where \begin{equation*} \mathcal{U}(t)\equiv e^{\frac{it}{2}\Delta }. \end{equation*} Note that $u_{f}$ is the solution to the Cauchy problem of the free Schr\"{o}dinger equation \begin{gather*} \mathcal{L}u=0,\quad (t,x) \in \mathbb{R}\times \mathbb{R}^{n}, \\ u(0,x)=\phi (x),\quad x\in \mathbb{R}^{n}. \end{gather*} If there exists a unique global solution $u$ of equation (\ref{eq:NLS}) such that \begin{equation*} \| u(t)-u_{f}(t)\| _{L^2}\rightarrow 0, \end{equation*} as $t\rightarrow +\infty $, then a mapping \begin{equation*} \mathcal{W}_{+}:\phi \mapsto u(0) \end{equation*} is well-defined on $\Sigma $. We call the mapping $W_{+}$ by the wave operator. The function $\phi $ is called by a final state, final data, a scattered state or scattered data. It is known that, when $p>1+\frac{2}{n}$ and $n\leq 3$, there exist the wave operators $\mathcal{W}_{\pm }$ on a suitable weighted Sobolev space (see \cite{GOV}). In the case of $n\geq 4$, the existence of wave operators is proved if $p>\frac{1}{4}( \sqrt{n^2+4n+36}-n+2) $ in \cite{GOV} and if $p=\frac{1}{4}( \sqrt{n^2+4n+36}-n+2) $ in \cite{NO}. (Note that $1+\frac{2}{n}<\frac{1}{4}( \sqrt{n^2+4n+36}-n+2) $ if $n\geq 4$, so for the case $n\geq 4 $ and $1+\frac{2}{n}1+\frac{2}{n}$. There are several results concerning the long range scattering for equation (\ref{eq:NLS}) in the critical case $p=1+\frac{2}{n}$. In the long range case, as we already mentioned, the usual wave operators do not exist, so we introduce the modified wave operators $\widetilde{\mathcal{W}}_{+}$ as follows. We construct a suitable modified free profile $A_{+}(t) $, and consider a unique solution $u(t) $ of equation (\ref {eq:NLS}) which approaches $A_{+}(t) $ in $L^2$ as $t\rightarrow \infty $: \begin{equation*} \| u(t)-A_{+}(t) \| _{L^2}\rightarrow 0,\quad t\rightarrow \infty . \end{equation*} Then the mapping \begin{equation*} \widetilde{\mathcal{W}}_{+}:A_{+}(0)\mapsto u(0) \end{equation*} is called the modified wave operator. Ozawa \cite{O1} and Ginibre and Ozawa \cite{GO} proved the existence of modified wave operators for small final data in one space dimension and in two and three space dimensions, respectively, by the phase correction method. More precisely, they put a modified free profile of the form $A_{+}(t) =\mathcal{U} (t)e^{-iS(t,-i\nabla )}\phi $, where $\phi $ is a final state, and chose the phase function $S$ such that $\| \mathcal{L}A_{+}(t) -|A(t)|^{ \frac{2}{n}}A(t)\| _{L^2}$ decays faster than $\| |\mathcal{ U}(t)\phi |^{\frac{2}{n}}\mathcal{U}(t)\phi \| _{{L} ^2}=O(t^{-1}) $. Recently, Ginibre and Velo \cite{GV-h3} have partially extended above results removing the size restrictions of the final data in the case of the nonlinearity $a(t)|u|^2u$. where $a(t)$ has a suitable growth rate with respect to $t$. The large time asymptotic behavior of solutions to the initial value problem for equation (\ref{eq:NLS}) with $ 1\leq n\leq 3$ was studied and the asymptotic completeness of the wave operator was partially shown in \cite{HN1}. The phase correction method is applicable only for the gauge invariant nonlinearities, like $\lambda |u|^{p-1}u$, where $\lambda \in \mathbb{R}$, because we can regard $ |u|^{p-1} $ as a time dependent long range potential. We cannot apply the phase correction method to non-gauge invariant nonlinearities of the form $ u^{p}$ or $|u|^{p-1}u+u^{p}$, because we should consider the non-gauge invariant nonlinearity as a time dependent external force. There are some results on the scattering theory for equation (\ref{1.1}) in one or two space dimensions. In \cite{Mori-Tone-Tsu} it was shown the existence of the wave operator for equation (\ref{1.1}) with $\mathcal{G} _n(u) =0$ by using the method by H\"{o}rmander \cite{Hor}, where he studied the life span of solutions of nonlinear Klein-Gordon equations and in \cite{Shim-Tone} it was constructed the modified wave operator for equation (\ref{1.1}) by combining the methods in \cite{Hor} and \cite{O1}. More precisely, the following two propositions were obtained in \cite{Shim-Tone}: \begin{proposition} \label{Prop 1} Let $n=1$, $\phi \in H^{0,3}\cap \dot{H} ^{-4}$ and $\| \phi \| _{H^{0,3}}+\| \phi \|_{\dot{H}^{-4}}$ be sufficiently small. Then there exists a unique global solution $u$ of (\ref{1.1}) such that $u\in C(\mathbb{R}^{+};L^2)$, \begin{equation*} \sup_{t\geq 1}t^{b}\| u(t) -u_{p}(t)\| _{ L^2}+\sup_{t\geq 1}t^{b}\Big( \int_{t}^{\infty }\| u(\tau) -u_{p}(\tau) \| _{L^{\infty }}^4\,d\tau \Big) ^{1/4}<\infty , \end{equation*} where $\frac{1}{2}\frac{n+1}{2}$, provided that $f(0)=0$, \item[(b)] $\| |\cdot |^{-\alpha +1}f\| _{L^2}\leq C\| f\| _{H^{1,0}}$ for $1<\alpha <1+\frac{n}{2}$ with $n=1,2$. \end{itemize} Note that this implies that $\int \phi (x)dx=0$ and $\phi \in H^{0,2}$, then $\phi \in H^{0,2}\cap \dot{H}^{-\alpha }$. \noindent Proof of (a): From the equality \begin{equation*} f(\xi )=f(\xi )-f(0)=\int_0^{1}\frac{d}{dt}f(t\xi )dt=\int_0^{1}\xi \cdot \nabla f(t\xi )dt. \end{equation*} and Schwarz' inequality, it follows that \begin{equation*} |f(\xi )|^2\leq |\xi |^2\int_0^{1}|\nabla f(t\xi )|^2dt. \end{equation*} Therefore, we have \begin{align*} \| |\cdot |^{-\alpha }f\| _{L^2}^2 &=\int \frac{1}{ |\xi |^{2\alpha }}|f(\xi )|^2d\xi \leq \int \frac{1}{|\xi |^{2\alpha -2}} \int_0^{1}|\nabla f(t\xi )|^2dtd\xi \\ &=\int_0^{1}\!\!\int \frac{1}{|\xi |^{2\alpha -2}}|\nabla f(t\xi )|^2d\xi dt=\int_0^{1}\!\!\int \frac{t^{2\alpha -2}}{|\eta |^{2\alpha -2} }|\nabla f(\eta )|^2\frac{d\eta }{t^{n}}dt \\ &=\frac{1}{2\alpha -1-n}\| |\cdot |^{-\alpha +1}\nabla f\| _{L^2}^2 \end{align*} for $\alpha >\frac{n+1}{2}$. \noindent Proof of (b) : We split the norm on the left hand side as follows: \begin{equation*} \| |\cdot |^{-\alpha +1}f\| _{L^2}\leq \| |\cdot |^{-\alpha +1}f\| _{L^2(|\cdot |\geq 1)}+\| |\cdot |^{-\alpha +1}f\| _{L^2(|\cdot |<1)}=I_1+I_{2}. \end{equation*} Since $\alpha \geq 1$, it is easily seen that $I_1\leq \| f\| _{L^2}$. By the H\"{o}lder inequality, we have \begin{equation*} I_{2}\leq \| |\cdot |^{-\alpha +1}\| _{L^{p}(|\cdot |<1)}\| f\| _{L^{q}(|\cdot |<1)}, \end{equation*} where $2\leq p,q\leq \infty $ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$. Here, we put $(p,q)=(2,\infty )$ for $n=1$ and $(p,q)=\bigl(\frac{\alpha }{\alpha -1},\frac{2\alpha }{2-\alpha }\bigr)$ for $n=2$ so that we have $\| |\cdot |^{-\alpha +1}\| _{L^{p}(|\cdot |<1)}<\infty $ and $\| f\| _{L^{q}(|\cdot |<1)}\leq \| f\| _{{L}^{q}} \leq C\| f\| _{H^{1,0}}$ by the Sobolev embedding. \end{remark} \begin{remark} \label{rmk3} \rm In the previous paper \cite{H-N}, we considered the Cauchy problem for the cubic nonlinear Schr\"{o}dinger equation \begin{gather*} iu_{t}+\frac{1}{2}u_{xx}=\mathcal{N}(u) ,\quad x\in \mathbb{R},\; t>1 \\ u(1,x) =u_1(x) ,\quad x\in \mathbb{R}, \end{gather*} where $\mathcal{N}(u) =\lambda _1u^{3}+\lambda _{2}\overline{u} ^2u+\lambda _3\overline{u}^{3}$. $\lambda _{j}\in \mathbb{C}$. $j=1,2,3$. It was shown that there exists a global small solution $u\in C([ 1,\infty ) ,L^{\infty })$, if the initial data $u_1$ belong to some analytic function space and are sufficiently small. For the coefficients $\lambda _{j}$ it was assumed that there exists $\theta _0>0$ such that \begin{gather*} \mathop{\rm Re}\big( \frac{\lambda _1}{\sqrt{3}}e^{2ir}-i\lambda _{2}e^{-2ir}+ \frac{\lambda _3}{\sqrt{3}}e^{-4ir}\big) \geq C>0,\\ \mathop{\rm Im}\big( \frac{\lambda _1}{\sqrt{3}}e^{2ir}-i\lambda _{2}e^{-2ir}+ \frac{\lambda _3}{\sqrt{3}}e^{-4ir}\big) r\geq Cr^2, \end{gather*} for all $|r| <\theta _0$. and also it was assumed that the initial data $u_1(x) $ are such that \begin{equation*} \big| \arg e^{-\frac{i}{2}\xi ^2}\widehat{u_1}(\xi) \big| <\theta _0,\text{ }\inf_{|\xi| \leq 1}| \widehat{u_1}(\xi) | \geq C\varepsilon , \end{equation*} where $\varepsilon $ is a small positive constant depending on the size of the initial data in a suitable norm. Moreover it was shown that there exist unique final states $\mathcal{W}_{+},r_{+}\in L^{\infty }$ and $0<\gamma <1/20$ such that the asymptotic statement \begin{equation*} u(t,x) =\frac{(it) ^{-\frac{1}{2}}W_{+}( \frac{x }{t}) e^{\frac{ix^2}{2t}}}{\sqrt{1+\chi \big( \frac{x}{t}\big) | W_{+}\big( \frac{x}{t}\big) | ^2\log \frac{t^2}{t+x^2}} }+O\Big( t^{-\frac{1}{2}}\big( 1+\log \frac{t^2}{t+x^2} \big) ^{-\frac{1}{2}-\gamma }\Big) \end{equation*} is valid for $t\rightarrow \infty $ uniformly with respect to $x\in \mathbf{R }$, where $\gamma >0$ and $\chi (\xi) $ is given by \begin{equation*} \chi (\xi) =\mathop{\rm Re}\Big( \frac{\lambda _1}{\sqrt{3}}\exp ( 2ir_{+}(\xi) ) -i\lambda _{2}\exp ( -2ir_{+}(\xi) ) +\frac{\lambda _3}{\sqrt{3}}\exp ( -4ir_{+}(\xi) ) \Big) . \end{equation*} This asymptotic formula shows that, in the short range region $| x| <\sqrt{t}$. the solution has an additional logarithmic time decay comparing with the corresponding linear case. Thus we can see that the vanishing condition at the origin on the Fourier transform of the final data seems to be essential for our result in the present paper. \end{remark} For the convenience of the reader we now state the strategy of the proof. We consider the linearized version of equation (\ref{1.1}) \begin{equation*} \mathcal{L}u=\mathcal{N}_n(v) +\mathcal{G}_n(v) ,\quad (t,x) \in \mathbb{R}\times \mathbb{R}^{n}. \end{equation*} We take \begin{equation*} u_0(t,x) =\frac{1}{(it) ^{\frac{n}{2}}}e^{\frac{ix^2}{2t}}\widehat{\phi } \big( \frac{x}{t}\big) \exp \big( -i\lambda _0| \widehat{\phi }\big( \frac{x}{t}\big) | ^{\frac{2}{n}}\log t\big) \end{equation*} as the first approximation for solutions to (\ref{1.1}). By a direct calculation we get \begin{equation*} \mathcal{L}u_0=\mathcal{G}_n( u_0) +R_1(t) , \end{equation*} where $R_1(t) $ is a remainder term. Hence \begin{equation*} \mathcal{L}( u-u_0) =\mathcal{N}_n(v) +\mathcal{G} _n(v) -\mathcal{G}_n( u_0) +R_1. \end{equation*} We define the second approximation $u_1$ for solutions of (\ref{1.1}) as \begin{equation*} u_1(t) =-i\int_{\infty }^{t}\mathcal{U}( t-\tau ) \mathcal{N}_n( u_0) \,d\tau \end{equation*} which implies that \begin{equation*} \mathcal{L}u_1=\mathcal{N}_n( u_0) \end{equation*} and \begin{align*} u(t) -u_0(t) &=-i\int_{\infty }^{t}\mathcal{U}( t-\tau ) ( \mathcal{N} _n(v) -\mathcal{N}_n( u_0) +\mathcal{G} _n(v) -\mathcal{G}_n( u_0) ) \,d\tau \\ &\quad -i\int_{\infty }^{t}\mathcal{U}( t-\tau ) R_1(\tau) \,d\tau +u_1(t) . \end{align*} We define the function space \begin{gather*} X =\left\{ f\in C( [ T,\infty) ;\mathbf{ L}^2) ;\| f\| _{X}<\infty \right\} \\ \| f\| _{X} =\sup_{t\in [ T,\infty)}t^{b}\| f(t) -u_0(t) \| _{{L} ^2}+\sup_{t\in [ T,\infty) }t^{b}( \int_{t}^{\infty }\| f(t) -u_0(t) \| _{X _n}^4\,dt) ^{1/4}, \end{gather*} where \begin{equation*} X_1=L^{\infty },\ X_{2}=L^4,\ b> \frac{n}{4}. \end{equation*} In order to get the result we need to prove the following estimate for $u_1(t) $, \[ \| u_1(t) \| +( \int_{t}^{\infty }\| u_1(\tau) \| _{X_n}^4\,d\tau ) ^{1/4} \leq C( \| | \cdot | ^{-\widetilde{\delta }}\widehat{ \phi }\| +\| \phi \| _{H^{0,2}}) ^{1+\frac{2 }{n}}t^{-\widetilde{\delta }/2}, \] for $n/2<\widetilde{\delta }<2$. which is the main estimate of the present paper. Note that the choice of $u_1$ differs from that used in the previous papers. \section{Preliminaries} \begin{lemma} \label{Lemma 2.1} We have for $\omega \neq 1$. $f,g\in L^{1}\cap L^2$ and $h\in C^2$, \begin{align*} &\int_{\infty }^{t}h( i\tau ) \mathcal{U}( t-\tau ) \Delta ( e^{\frac{i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau } ) \log \tau }f( \frac{x}{\tau }) ) \,d\tau \\ &=-\frac{2i\omega }{1-\omega }h(it) e^{\frac{i\omega x^2}{2t} }e^{ig\big( \frac{x}{t}\big) \log t}f\big( \frac{x}{t}\big) \\ &\quad-\frac{2\omega }{( 1-\omega ) ^2}\int_{\infty }^{t}\Big( \sum_{( F,k) }F( i\tau ) e^{\frac{i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau }) \log \tau }k( \frac{x}{\tau } ) \\ &\quad-i\omega \mathcal{U}( t-\tau ) \int_{\infty }^{\tau }\sum_{( F,k) }F'( is) e^{\frac{i\omega x^2 }{2s}}e^{ig( \frac{x}{s}) \log s}k( \frac{x}{s}) \,ds \\ & -i\omega \mathcal{U}( t-\tau ) \int_{\infty }^{\tau }\sum_{( F,k) }F( is) e^{\frac{i\omega x^2}{2s} }e^{ig( \frac{x}{s}) \log s}\frac{1}{s}k( g-\frac{in}{2} ) ( \frac{x}{s}) \,ds\Big) \,d\tau +R(t) , \end{align*} where the summation is taken over $(F,k)=( h',f) ,( h\tau ^{-1},f(g-in/2)) $, \begin{align*} R(t) &=-\frac{i\omega }{( 1-\omega ) ^2} \int_{\infty }^{t}\mathcal{U}( t-\tau ) \int_{\infty }^{\tau }\sum_{( F,k) }F( is) R_{0,k}( s) \,ds\,d\tau \\ &\quad +\frac{1}{1-\omega }\int_{\infty }^{t}h( i\tau ) \mathcal{U} ( t-\tau ) R_{0,f}(\tau) \,d\tau , \end{align*} and \begin{align*} R_{0,k}(t) &=e^{\frac{i\omega x^2}{2t}}k( \frac{x}{t} ) \Delta e^{ig\big( \frac{x}{t}\big) \log t} +2i\frac{1}{t^2}\sum \partial _{j}g\big( \frac{x}{t}\big) \partial _{j}k\big( \frac{x}{t}\big) e^{\frac{i\omega x^2}{2t}}e^{ig( \frac{ x}{t}) \log t}\log t \\ &\quad +\frac{1}{t^2}( \Delta k) \big( \frac{x}{t}\big) e^{\frac{ i\omega x^2}{2t}}e^{ig\big( \frac{x}{t}\big) \log t}. \end{align*} \end{lemma} \begin{proof} By a direct computation we find that \[ ( 2i\omega \partial _{t}+\Delta ) e^{\frac{i\omega x^2}{2t} }e^{ig\big( \frac{x}{t}\big) \log t}f\big( \frac{x}{t}\big) =-2\omega \frac{1}{t}f( g-\frac{id}{2}) ( \frac{x}{t} ) e^{\frac{i\omega x^2}{2t}}e^{ig\big( \frac{x}{t}\big) \log t}+R_{0,f}(t) , \] where \begin{align*} R_{0,f}(t) &=e^{\frac{i\omega x^2}{2t}}f( \frac{x}{t} ) \Delta e^{ig\big( \frac{x}{t}\big) \log t} +2i\frac{1}{t^2}\sum ( \partial _{j}g\cdot \partial _{j}f) \big( \frac{x}{t}\big) e^{\frac{i\omega x^2}{2t}}e^{ig( \frac{x}{t} ) \log t}\log t \\ &\quad +\frac{1}{t^2}( \Delta f) \big( \frac{x}{t}\big) e^{\frac{ i\omega x^2}{2t}}e^{ig\big( \frac{x}{t}\big) \log t}. \end{align*} Therefore, \begin{align*} &\mathcal{U}( -t) \Delta ( e^{\frac{i\omega x^2}{2t} }e^{ig\big( \frac{x}{t}\big) \log t}f\big( \frac{x}{t}\big) ) \\ &=-\partial _{t}( \mathcal{U}( -t) 2i\omega ( e^{\frac{ i\omega x^2}{2t}}e^{ig\big( \frac{x}{t}\big) \log t}f( \frac{x}{t} ) ) ) +\omega \mathcal{U}( -t) \Delta ( e^{\frac{i\omega x^2}{ 2t}}e^{ig\big( \frac{x}{t}\big) \log t}f\big( \frac{x}{t}\big) ) \\ &\quad +\mathcal{U}( -t) ( -2\omega \frac{1}{t}f( g-\frac{in }{2}) \big( \frac{x}{t}\big) e^{\frac{i\omega x^2}{2t} }e^{ig\big( \frac{x}{t}\big) \log t}+R_{0,f}(t) ) \end{align*} from which it follows that \begin{equation} \begin{split} \mathcal{U}& ( -t) \Delta ( e^{\frac{i\omega x^2}{2t} }e^{ig\big( \frac{x}{t}\big) \log t}f\big( \frac{x}{t}\big) ) \\ =& -\frac{2i\omega }{1-\omega }\partial _{t}( \mathcal{U}( -t) e^{\frac{i\omega x^2}{2t}}e^{ig\big( \frac{x}{t}\big) \log t}f\big( \frac{x}{t}\big) ) \\ & -\frac{2\omega }{1-\omega }\mathcal{U}(-t) \Big( \frac{1}{t} f( g-\frac{in}{2}) ( \frac{x}{t}) e^{\frac{i\omega x^2}{2t}}e^{ig\big( \frac{x}{t}\big) \log t}\Big) \\ & +\frac{1}{1-\omega }\mathcal{U}( -t) R_{0,f}(t) . \end{split} \label{2.1} \end{equation} Hence \begin{equation} \begin{split} \int_{\infty }^{t}& h( i\tau ) \mathcal{U}( t-\tau ) \Delta ( e^{\frac{i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau } ) \log \tau }f( \frac{x}{\tau }) ) \,d\tau \\ =& -\frac{2i\omega }{1-\omega }\mathcal{U}(t) \int_{\infty }^{t}h( i\tau ) \partial _{\tau }( \mathcal{U}( -\tau ) e^{\frac{i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau }) \log \tau }f( \frac{x}{\tau }) ) \,d\tau \\ & -\frac{2\omega }{1-\omega }\int_{\infty }^{t}h( i\tau ) \mathcal{U}( t-\tau ) \frac{1}{\tau }f( g-\frac{in}{2} ) ( \frac{x}{\tau }) e^{\frac{i\omega x^2}{2\tau } }e^{ig( \frac{x}{\tau }) \log \tau }\,d\tau \\ & +R_{1,f}(t) \\ =& -\frac{2i\omega }{1-\omega }h(it) e^{\frac{i\omega x^2}{2t} }e^{ig\big( \frac{x}{t}\big) \log t}f\big( \frac{x}{t}\big) \\ & -\frac{2\omega }{1-\omega }\int_{\infty }^{t}h'( i\tau ) \mathcal{U}( t-\tau ) e^{\frac{i\omega x^2}{2\tau } }e^{ig( \frac{x}{\tau }) \log \tau }f( \frac{x}{\tau } ) \,d\tau \\ & -\frac{2\omega }{1-\omega }\int_{\infty }^{t}h( i\tau ) \mathcal{U}( t-\tau ) \frac{1}{\tau }f( g-\frac{in}{2} ) ( \frac{x}{\tau }) e^{\frac{i\omega x^2}{2\tau } }e^{ig( \frac{x}{\tau }) \log \tau }\,d\tau \\ & +R_{1,f}(t) , \end{split} \label{2.2} \end{equation} where \begin{equation*} R_{1,f}(t) =\frac{1}{1-\omega }\int_{\infty }^{t}h( i\tau ) \mathcal{U}( t-\tau ) R_{0,f}(\tau) \,d\tau . \end{equation*} We write \begin{align*} & F(i\tau ) \mathcal{U}( -\tau ) e^{\frac{i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau }) \log \tau }k( \frac{x }{\tau }) \\ &=\partial _{\tau }( \mathcal{U}( -\tau ) \int_{\infty }^{\tau }F( is) e^{\frac{i\omega x^2}{2s}}e^{ig( \frac{x}{s }) \log s}k( \frac{x}{s}) \,ds) \\ &\quad+\frac{i}{2}\mathcal{U}( -\tau ) \int_{\infty }^{\tau }F( is) \Delta ( e^{\frac{i\omega x^2}{2s}}e^{ig( \frac{x}{s} ) \log s}k( \frac{x}{s}) ) \,ds \\ &=\partial _{\tau }( \mathcal{U}( -\tau ) \int_{\infty }^{\tau }F( is) e^{\frac{i\omega x^2}{2s}}e^{ig( \frac{x}{s }) \log s}k( \frac{x}{s}) \,ds) \\ &\quad+\omega F( i\tau ) \mathcal{U}( -\tau ) e^{\frac{ i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau }) \log \tau }k( \frac{x}{\tau }) \\ &\quad -\omega \mathcal{U}( -\tau ) \int_{\infty }^{\tau }iF^{\prime }( is) e^{\frac{i\omega x^2}{2s}}e^{ig( \frac{x}{s}) \log s}k( \frac{x}{s}) \,ds \\ &\quad -i\omega \mathcal{U}( -\tau ) \int_{\infty }^{\tau }F( is) e^{\frac{i\omega x^2}{2s}}e^{ig( \frac{x}{s}) \log s} \frac{1}{s}k( g-\frac{in}{2}) ( \frac{x}{s}) \,ds \\ &\quad +\frac{i}{2}\mathcal{U}( -\tau ) \int_{\infty }^{\tau }F( is) R_{0,k}( s) \,ds \end{align*} hence \begin{equation} \begin{split} & ( 1-\omega ) F( i\tau ) \mathcal{U}( -\tau ) e^{\frac{i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau } ) \log \tau }k( \frac{x}{\tau }) \\ =& \partial _{\tau }( \mathcal{U}( -\tau ) \int_{\infty }^{\tau }F( is) e^{\frac{i\omega x^2}{2s}}e^{ig( \frac{x}{s }) \log s}k( \frac{x}{s}) \,ds) \\ & -\omega \mathcal{U}( -\tau ) \int_{\infty }^{\tau }iF^{\prime }( is) e^{\frac{i\omega x^2}{2s}}e^{ig( \frac{x}{s}) \log s}k( \frac{x}{s}) \,ds \\ & -i\omega \mathcal{U}( -\tau ) \int_{\infty }^{\tau }F( is) e^{\frac{i\omega x^2}{2s}}e^{ig( \frac{x}{s}) \log s} \frac{1}{s}k( g-\frac{in}{2}) ( \frac{x}{s}) \,ds \\ & +\frac{i}{2}\mathcal{U}( -\tau ) \int_{\infty }^{\tau }F( is) R_{0,k}( s) \,ds. \end{split} \label{2.3} \end{equation} We apply (\ref{2.3}) with $( F,k) =( h',f) $ or $( F,k) =( h\tau ^{-1},f(g-in/2)) $ to the right-hand side of (\ref{2.1}) to get the desired result. \end{proof} In the next lemma we state the Strichartz estimate for $\int_{s}^{t}\mathcal{ U}( t-\tau ) f(\tau) \,d\tau $ obtained by Yajima \cite{Yajima}. \begin{lemma} \label{Lemma 2.2} For any pairs $(q,r) $ and $( q',r') $ such that $0\leq \frac{2}{q}=\frac{n}{2}-\frac{n}{r}<1$ and $0\leq \frac{2}{q'}=\frac{n}{2}-\frac{n}{r'}<1$. for any (possibly unbounded) interval $I$ and for any $s\in \overline{I}$ the Strichartz estimate \begin{equation*} ( \int_{I} \Big\| \int_{s}^{t}\mathcal{U}( t-\tau ) f(\tau) \,d\tau \Big\| _{L^{r}}^{q}\,dt) ^{\frac{1}{q} } \leq C( \int_{I}\| f(t) \| _{L^{ \overline{r}'}}^{\overline{q}'}\,dt) ^{\frac{1}{\overline{q}'}}, \end{equation*} is true with a constant $C$ independent of $I$ and $s$, where $\frac{1}{r}+\frac{1}{\overline{r}}=1$ and $\frac{1}{q}+\frac{1}{\overline{q}}=1$. \end{lemma} Denote \begin{gather*} \widetilde{R}_1(t) =\int_{\infty }^{t}\mathcal{U}( t-\tau ) \int_{\infty }^{\tau }F( is) R_{0,k}( s) \,ds\,d\tau \\ \widetilde{R}_{2}(t) =\int_{\infty }^{t}\mathcal{U}( t-\tau ) h( i\tau ) R_{0,k}(\tau) \,d\tau , \end{gather*} where \begin{align*} R_{0,k}(t) &=e^{\frac{i\omega x^2}{2t}}k( \frac{x}{t} ) \Delta e^{ig\big( \frac{x}{t}\big) \log t} +2i\frac{1}{t^2}\sum \partial _{j}g\big( \frac{x}{t}\big) \partial _{j}k\big( \frac{x}{t}\big) e^{\frac{i\omega x^2}{2t}}e^{ig( \frac{ x}{t}) \log t}\log t \\ &\quad +\frac{1}{t^2}( \Delta k) \big( \frac{x}{t}\big) e^{\frac{ i\omega x^2}{2t}}e^{ig\big( \frac{x}{t}\big) \log t}. \end{align*} \begin{lemma} \label{Lemma 2.3} Let \begin{equation*} | F(it) | \leq C| t| ^{-2-\frac{n}{2} },\quad | h(it) | \leq C| t| ^{-1-\frac{n}{2}}. \end{equation*} Then \begin{align*} &\| \widetilde{R}_{j}(t) \| _{L^2}+( \int_{t}^{\infty }\| \widetilde{R}_{j}(t) \| _{X_n}^4\,dt) ^{1/4} \\ &\leq Ct^{-2}( \| \Delta k\| _{L^2}+\| \nabla k\cdot \nabla g\| _{L^2}\log t+\| k\Delta g\| _{L^2}\log t+\| k\nabla g\cdot \nabla g\| _{L^2}( \log t) ^2), \end{align*} where $X_1=L^{\infty },X_{2}={L}^4$. \end{lemma} \begin{proof} We have by the Strichartz estimate (see Lemma \ref{Lemma 2.2}) \begin{align*} &\| \widetilde{R}_{j}(t)\| _{{L}^2}+\Big(\int_{t}^{\infty }\left\| \widetilde{R}_{j}(t) \right\| _{X_n}^4\,dt\Big) ^{1/4} \\ &\leq C\int_{t}^{\infty }\Big( \int_{\tau }^{\infty }\left\vert s\right\vert ^{-2-\frac{2}{n}}\left\| R_{0,k}( s) \right\| _{L^2}\,ds+\left\vert \tau \right\vert ^{-1-\frac{2}{n} }\left\| R_{0,k}(\tau) \right\| _{L^2}\Big) \,d\tau . \end{align*} It is easy to see that \begin{align*} &\left\| R_{0,k}(t) \right\| _{L^2} \\ &\leq Ct^{-2+\frac{2}{n}}( \left\| \Delta k\right\| _{{L} ^2}+\left\| \nabla k\cdot \nabla g\right\| _{L^2}\log t+\left\| k\Delta g\right\| _{L^2}\log t+\left\| k\nabla g\cdot \nabla g\right\| _{L^2}( \log t) ^2). \end{align*} Therefore, we have the result of the lemma. \end{proof} \begin{lemma} \label{Lemma 2.4} Assume that $| G(it) | +|t| | G'(it) | \leq C| t| ^{-q-\frac{n}{2}}$, then \begin{equation*} \begin{split} & \Big\| \int_{\infty }^{t}G( i\tau ) e^{\frac{i\omega x^2}{ 2\tau }}e^{ig( \frac{x}{\tau }) \log s}k( \frac{x}{\tau } ) \,d\tau \Big\| _{L^{p}} \\ & \leq \begin{cases} Ct^{-\frac{\delta }{2}-q+1-\frac{n}{2}( 1-\frac{2}{p}) }\| | \cdot | ^{-\delta }k\| _{L^{p}} \\ +Ct^{-\frac{\widetilde{\delta }}{2}-q+1-\frac{n}{2}( 1-\frac{2}{p} ) }( \| | \cdot | ^{1-\widetilde{\delta }}\nabla k\| _{L^{p}}+\| | \cdot | ^{1-\widetilde{ \delta }}k\nabla g\| _{L^{p}}\log t) , \\ \qquad \text{for } 0<\delta ,\widetilde{\delta }<2,\ 1\leq p<\infty , \\[5pt] Ct^{-\frac{\delta }{2}-q+1-\frac{n}{2}( 1-\frac{1}{p}) }\| | \cdot | ^{-\delta }k\| _{L^{\infty }} \\ +Ct^{-\frac{\widetilde{\delta }}{2}-q+1-\frac{n}{2}( 1-\frac{1}{p} ) }( \| | \cdot | ^{1-\widetilde{\delta }}\nabla k\| _{L^{\infty }}+\| | \cdot | ^{1- \widetilde{\delta }}k\nabla g\| _{L^{\infty }}\log t) ,\\ \qquad \text{for } 0<\delta ,\widetilde{\delta }<2-\frac{n}{p},\ 1\leq p<\infty . \end{cases} \end{split} \end{equation*} \end{lemma} \begin{proof} Using the identity \begin{equation*} \frac{1}{1-\frac{i\omega x^2}{2\tau }}\partial _{t}\tau e^{\frac{i\omega x^2}{2\tau }}=e^{\frac{i\omega x^2}{2\tau }} \end{equation*} we have \begin{align*} &\int_{\infty }^{t}G( i\tau ) e^{\frac{i\omega x^2}{2\tau } }e^{ig( \frac{x}{\tau }) \log \tau }k( \frac{x}{\tau }) \,d\tau \\ &=\int_{\infty }^{t}G( i\tau ) e^{ig( \frac{x}{\tau } ) \log \tau }k( \frac{x}{\tau }) \big( \frac{1}{1-\frac{ i\omega x^2}{2\tau }}\partial _{\tau }\tau e^{\frac{i\omega x^2}{2\tau } }\Big) \,d\tau \\ &=G(it) k\big( \frac{x}{t}\big) e^{ig( \frac{x}{t} ) \log t}\Big( \frac{1}{1-\frac{i\omega x^2}{2t}}te^{\frac{i\omega x^2}{2t}}\Big) \\ &-\int_{\infty }^{t}\tau e^{\frac{i\omega x^2}{2\tau }}\partial _{\tau }\Big( G( i\tau ) k( \frac{x}{\tau }) \frac{1}{1- \frac{i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau }) \log \tau }\Big) \,d\tau . \end{align*} We also obtain \begin{align*} &\big\| G(it) k\big( \frac{x}{t}\big) e^{ig( \frac{x }{t}) \log t}\big( \frac{1}{1-\frac{i\omega x^2}{2t}}te^{\frac{ i\omega x^2}{2t}}) \Big\| _{L^{p}} \\ &\leq Ct^{-\frac{\delta }{2}-q+1-\frac{n}{2}}\Big( \int \big( \frac{ \left\vert \frac{x}{t^{1/2}}\right\vert ^{\delta }}{1+\left\vert \frac{x}{ t^{1/2}}\right\vert ^2}\left\vert \frac{x}{t}\right\vert ^{-\delta }k\big( \frac{x}{t}\big) \big) ^{p}dx\Big) ^{1/p} \\ &\leq \begin{cases} Ct^{-\frac{\delta }{2}-q+1-\frac{n}{2}( 1-\frac{2}{p}) }\| \vert \cdot \vert ^{-\delta }k\| _{{L} ^{p}},& 0<\delta <2,1\leq p<\infty \\ Ct^{-\frac{\delta }{2}-q+1-\frac{n}{2}( 1-\frac{1}{p}) }\| \vert \cdot \vert ^{-\delta }k\| _{{L} ^{\infty }},& 0<\delta <2-\frac{n}{p},1\leq p<\infty \end{cases} \end{align*} and in the same way we get \begin{equation*} \begin{split} & \Big\| te^{\frac{i\omega x^2}{2t}}\partial _{t} \Big( G(it) k\big( \frac{x}{t}\big) \frac{1}{1-\frac{i\omega x^2}{2t}} e^{ig\big( \frac{x}{t}\big) \log t}\Big) \Big\| _{L^{p}} \\ & \leq \begin{cases} Ct^{-\frac{\delta }{2}-q-\frac{n}{2}( 1-\frac{2}{p}) }\left\| \left\vert \cdot \right\vert ^{-\delta }k\right\| _{{L} ^{p}} \\ +Ct^{-\frac{\widetilde{\delta }}{2}-q-\frac{n}{2}( 1-\frac{2}{p} ) }( \left\| \left\vert \cdot \right\vert ^{1-\widetilde{ \delta }}\nabla k\right\| _{L^{p}}+\left\| \left\vert \cdot \right\vert ^{1-\widetilde{\delta }}k\nabla g\right\| _{{L} ^{p}}\log t) , \\ \qquad \text{for } 0<\delta ,\widetilde{\delta }<2,\ 1\leq p<\infty , \\[5pt] Ct^{-\frac{\delta }{2}-q-\frac{n}{2}( 1-\frac{1}{p}) }\left\| \left\vert \cdot \right\vert ^{-\delta }k\right\| _{{L} ^{\infty }} \\ +Ct^{-\frac{\widetilde{\delta }}{2}-q-\frac{n}{2}( 1-\frac{1}{p} ) }( \left\| \left\vert \cdot \right\vert ^{1-\widetilde{ \delta }}\nabla k\right\| _{L^{\infty }}+\left\| \left\vert \cdot \right\vert ^{1-\widetilde{\delta }}k\nabla g\right\| _{{L} ^{\infty }}\log t) , \\ \qquad \text{for}\ 0<\delta ,\widetilde{\delta }<2-\frac{n}{2},\ 1\leq p<\infty . \end{cases} \end{split} \end{equation*} Hence we have the result of the lemma. \end{proof} \section{Proof of Theorem~\ref{Th 1}} We consider the linearized version of equation (\ref{1.1}) \begin{equation} \mathcal{L}u=\mathcal{N}_n(v) +\mathcal{G}_n(v) ,\quad (t,x) \in \mathbb{R}\times \mathbb{R}^{n}. \label{3.0} \end{equation} We take \begin{equation*} u_0(t,x) =\frac{1}{(it) ^{\frac{n}{2}}}e^{\frac{ ix^2}{2t}}\widehat{\phi }\big( \frac{x}{t}\big) \exp \Big( -i\lambda _0| \widehat{\phi }\big( \frac{x}{t}\big) | ^{\frac{2}{n} }\log t\Big) \end{equation*} as the first approximation for solutions of (\ref{3.0}). By a direct calculation we get \begin{equation*} \mathcal{L}u_0=\mathcal{G}_n( u_0) +R_1, \end{equation*} where \begin{align*} R_1(t) &=\frac{1}{(it) ^{\frac{n}{2}}}e^{\frac{ ix^2}{2t}}\widehat{\phi }\big( \frac{x}{t}\big) \frac{1}{2}\Delta \exp ( -i\lambda _0| \widehat{\phi }\big( \frac{x}{t}\big) | ^{\frac{2}{n}}\log t) \\ &\quad -\frac{2}{n}\lambda _0\frac{1}{t^2}\frac{1}{(it) ^{\frac{n }{2}}}e^{\frac{ix^2}{2t}}\nabla \widehat{\phi }\big( \frac{x}{t}\big) \exp ( -i\lambda _0| \widehat{\phi }\big( \frac{x}{t}\big) | ^{\frac{2}{n}}\log t) \\ &\quad \times 2\mathop{\rm Re}\nabla \widehat{\phi }\big( \frac{x}{t}\big) \overline{ \widehat{\phi }\big( \frac{x}{t}\big) }| \widehat{\phi }( \frac{ x}{t}) | ^{\frac{2}{n}-2}\log t \\ &\quad +\frac{1}{2}\frac{1}{(it) ^{\frac{n}{2}}}e^{\frac{ix^2}{2t} }t^{-2}\Delta \widehat{\phi }\big( \frac{x}{t}\big) \exp ( -i\lambda _0| \widehat{\phi }\big( \frac{x}{t}\big) | ^{\frac{2}{n} }\log t) . \end{align*} Hence \begin{equation*} \mathcal{L}( u-u_0) =\mathcal{N}_n(v) +\mathcal{G} _n(v) -\mathcal{G}_n( u_0) +R_1. \end{equation*} By Lemma \ref{Lemma 2.2} we obtain \begin{equation} \begin{split} & \Big\| \int_{t}^{\infty }\mathcal{U}( t-\tau ) R_1(\tau) \,d\tau \Big\| _{L^2}+\Big( \int_{t}^{\infty} \Big\| \int_{t}^{\infty }\mathcal{U}( t-\tau ) R_1(\tau) \,d\tau \Big\|_{X_n}^4\,dt\Big) ^{1/4} \\ &\leq C\int_{t}^{\infty }\| R_1(\tau) \| _{L^2}\,d\tau \leq Ct^{-1}( \log t) ^2\| \phi \| _{H^{0,2}}^{1+\frac{2}{n}} \end{split} \label{3.1} \end{equation} since by the H\"{o}lder inequality we have \begin{align*} &\| R_1(t) \| _{L^2}\\ &\leq Ct^{-2}\|\Delta \widehat{\phi }\| _{L^2}+Ct^{-2}( \log t) ^2\| \widehat{\phi }\| _{L^{\infty }}^{\frac{2}{n} -1}\| \nabla \widehat{\phi }\| _{L^4}^2 +Ct^{-2}( \log t) \| \widehat{\phi }\| _{{L} ^{\infty }}^{\frac{2}{n}}\| \Delta \widehat{\phi }\| _{{L}^2}\\ &\leq Ct^{-2}( \log t) ^2\| \phi \| _{H^{0,2}}^{1+\frac{2}{n}}. \end{align*} We now define $u_1$ as \begin{equation*} u_1(t) =-i\int_{\infty }^{t}\mathcal{U}( t-\tau ) \mathcal{N}_n( u_0) \,d\tau \end{equation*} which implies $\mathcal{L}u_1=\mathcal{N}_n( u_0)$ and \begin{equation} \label{3.a} \begin{aligned} u(t) -u_0(t) &=-i\int_{\infty }^{t}\mathcal{U}( t-\tau ) ( \mathcal{N} _n(v) -\mathcal{N}_n( u_0) +\mathcal{G} _n(v) -\mathcal{G}_n( u_0) ) \,d\tau \\ &\quad -i\int_{\infty }^{t}\mathcal{U}( t-\tau ) R_1(\tau) \,d\tau +u_1(t) . \end{aligned} \end{equation} Note that \begin{equation} i\partial _{t}u_1(t) =\mathcal{N}_n( u_0) + \frac{i}{2}\int_{\infty }^{t}\mathcal{U}( t-\tau ) \Delta \mathcal{N}_n( u_0) \,d\tau . \label{3.s} \end{equation} Now, we define the function space \begin{gather*} X =\left\{ f\in C( [ T,\infty) ;{L} ^2) ;\| f\| _{X}<\infty \right\} ,\text{ where}\\ \| f\| _{X} =\sup_{t\in [ T,\infty)}t^{b}\| f(t) -u_0(t) \| _{{L} ^2}+\sup_{t\in [ T,\infty) }t^{b}\Big( \int_{t}^{\infty }\| f(t) -u_0(t) \| _{X_n}^4\,dt\Big) ^{1/4}, \end{gather*} and \begin{equation*} X_1=L^{\infty },\quad X_{2}=L^4,\quad b>\frac{n}{4}. \end{equation*} Let $X_{\rho }$ be a closed ball in $X$ with a radius $ \rho $ and a center $u_0.$ Let $v\in X_{\rho }$. From (\ref{3.s}) and Lemma \ref{Lemma 2.1} it follows that \begin{align*} i\partial _{t}u_1(t) &=\mathcal{N}_n( u_0) + \frac{i}{2}\sum_{( \omega ,h,g,f) }\Big( -\frac{2i\omega }{ 1-\omega }h(it) e^{\frac{i\omega x^2}{2t}}e^{ig( \frac{x}{ t}) \log t}f(\frac{x}{t}) \\ &\quad -\frac{2\omega }{( 1-\omega ) ^2}\int_{\infty }^{t}\Big( \sum_{( F,k) }F( i\tau ) e^{\frac{i\omega x^2}{2\tau }}e^{ig( \frac{x}{\tau }) \log \tau }k( \frac{x}{\tau } ) \\ &\quad -i\omega \mathcal{U}( t-\tau ) \int_{\infty }^{\tau }\sum_{( F,k) }F'( is) e^{\frac{i\omega x^2 }{2s}}e^{( \frac{x}{s}) \log s}k( \frac{x}{s}) \,ds \\ &\quad -i\omega \mathcal{U}( t-\tau ) \int_{\infty }^{\tau }\sum_{( F,k) }F( is) e^{\frac{i\omega x^2}{2s} }e^{ig( \frac{x}{s}) \log s}\frac{1}{s}k( g-\frac{in}{2} ) ( \frac{x}{s}) \,ds\Big) \,d\tau +R(t) , \end{align*} where the summation with respect to $( \omega ,h,g,f) $ is taken over \begin{align*} ( \omega ,h,g,f) &=\Big( 3,(it)^{-3/2},\lambda _0| \hat{ \phi}\big( \frac{x}{t}\big) | ^2,\lambda _1\hat{\phi}\big( \frac{x}{t}\big)^{3}\Big) , \\ &\quad\Big( -1,(-i)^{-1/2}t^{-3/2},\lambda _0| \hat{\phi}( \frac{x}{ t}) | ^2,\lambda _{2}\hat{\phi}\big( \frac{x}{t}\big) \overline{\hat{\phi}\big( \frac{x}{t}\big) }^2\Big) , \\ &\quad \Big( -3,(-it)^{-3/2},\lambda _0| \hat{\phi}( \frac{x}{t} ) | ^2,\lambda _3\overline{\hat{\phi}( \frac{x}{t} ) }^{3}\Big) , \end{align*} when $n=1$, and \begin{gather*} (\omega ,h,g,f) =\Big( 2,(it)^{-1},\lambda _0| \hat{\phi}\big( \frac{x}{t}\big) | ,\lambda _1\hat{\phi}\big( \frac{x}{t}\big) ^2\Big) , \Big( -2,(-it)^{-1},\lambda _0| \hat{\phi}( \frac{x}{t} ) | ,\lambda _{2}\overline{\hat{\phi}\big( \frac{x}{t}\big) } ^2\Big) , \end{gather*} when $n=2$, and the summation with respect to $( F,k) $ is taken over $(F,k)=( h',f) ,( h\tau ^{-1},f(g-in/2)) $. We have \begin{align*} &\mathcal{G}_n(v) -\mathcal{G}_n( u_0) \\ &=\lambda _0| v| ^{\frac{2}{n}}v-\lambda _0| u_0| ^{\frac{2}{n}}u_0 \\ &=\lambda _0( | v| ^{\frac{2}{n}}-| u_0| ^{ \frac{2}{n}}) ( v-u_0) +\lambda _0( | v| ^{\frac{2}{n}}-| u_0| ^{\frac{2}{n}}) u_0+\lambda _0| u_0| ^{\frac{2}{n}}( v-u_0)\,. \end{align*} Therefore, by the Strichartz estimate we get \begin{equation} \label{3.5} \begin{aligned} {}&\Big\| \int_{t}^{\infty }\mathcal{U}( t-\tau ) ( \mathcal{G}_n(v) -\mathcal{G}_n( u_0) ) \,d\tau \Big\| _{L^2} \\ &+\Big( \int_{t}^{\infty } \Big\| \int_{t}^{\infty }\mathcal{U}( t-\tau ) ( \mathcal{G}_n(v) -\mathcal{G}_n(u_0) ) \,d\tau \Big\| _{L^4}^4\,dt\Big) ^{1/4} \\ &\leq C\Big( \int_{t}^{\infty }\| v(\tau) -u_0(\tau) \| _{L^2}^2\,d\tau \Big) ^{\frac{1}{2}}\Big( \int_{t}^{\infty }\| v(\tau) -u_0(\tau) \| _{L^4}^4\,d\tau \Big) ^{1/4}\\ &\quad +C\int_{t}^{\infty }\| v(\tau) -u_0(\tau)\| _{L^2}\| u_0(\tau) \| _{L^{\infty }}\,d\tau \\ &\leq C\rho ^{2}t^{-2b+\frac{1}{2}}+Ct^{-b}\rho \| \phi \| _{L^{1}}, \end{aligned} \end{equation} for $n=2$. Also \begin{equation} \label{3.6} \begin{aligned} {}& \Big\| \int_{t}^{\infty }\mathcal{U}( t-\tau ) ( \mathcal{ G}_n(v) -\mathcal{G}_n( u_0) ) \,d\tau \Big\| _{L^2} \\ &\quad+\Big( \int_{t}^{\infty } \Big\| \int_{t}^{\infty }\mathcal{U}( t-\tau ) ( \mathcal{G}_n(v) -\mathcal{G}_n( u_0) ) \,d\tau \Big\| _{X_1}^4\,dt\Big) ^{1/4} \\ &\leq C\Big( \int_{t}^{\infty }\| | v(\tau) -u_0(\tau) | ^{3}\| _{L^{1}}^{\frac{4}{3}}\,d\tau \Big) ^{3/4} \\ &\quad +C\int_{t}^{\infty }\| | v(\tau) -u_0(\tau) | | u_0(\tau) | ^2\| _{ L^2}\,d\tau \\ &\leq C\Big( \int_{t}^{\infty }\| v(\tau) -u_0(\tau) \| _{L^{\infty }}^{\frac{4}{3}}\,\| v(\tau) -u_0(\tau) \| _{L^2}^{\frac{8}{3 }}d\tau \Big) ^{3/4} \\ &\quad +C\int_{t}^{\infty }\| v(\tau) -u_0(\tau) \| _{L^2}\| u_0(\tau) \| _{ L^{\infty }}^2\,d\tau \\ &\leq C\Big( \int_{t}^{\infty }\| v(\tau) -u_0(\tau) \| _{L^{\infty }}^4\,d\tau \Big) ^{\frac{1}{4}} \Big( \int_{t}^{\infty }\| v(\tau) -u_0(\tau) \| _{L^2}^4d\tau \Big) ^{1/2} \\ &\quad +C\int_{t}^{\infty }\| v(\tau) -u_0(\tau) \| _{L^2}\| u_0(\tau) \| _{L^{\infty }}^2\,d\tau \\ &\leq C\rho t^{-b} \Big( \int_{t}^{\infty }\rho ^4\tau ^{-4b}d\tau \Big) ^{1/2} +C\rho \| \phi \| _{L^{1}}^2\int_{t}^{\infty }\tau ^{-b-1}d\tau \\ &\leq C\rho ^{3}t^{-3b+\frac{1}{2}}+Ct^{-b}\rho \| \phi \| _{ L^{1}}^2, \end{aligned} \end{equation} for $n=1$, where we have used the facts that $b>n/4$ and \begin{equation*} | \mathcal{G}_n(v) -\mathcal{G}_n( u_0) | \leq C( | v-u_0| ^{\frac{2}{n}}+| u_0| ^{\frac{2}{n}}) | v-u_0| . \end{equation*} Similarly, we see that the above estimate holds valid with $\mathcal{G}_n$ replaced by $\mathcal{N}_n$. Thus by (\ref{3.1}), (\ref{3.a}), (\ref{3.5}) and (\ref{3.6}) \begin{equation} \begin{split} &\| u (t) -u_0(t) \| _{{L} ^2}+\Big( \int_{t}^{\infty }\| u(\tau) -u_0(\tau) \| _{X_n}^4\,d\tau \Big) ^{1/4} \\ &\leq C\rho ^{1+\frac{2}{n}}t^{-( 1+\frac{2}{n}) b+\frac{1}{2} }+Ct^{-b}\rho \| \phi \| _{L^{1}}^{\frac{2}{n} }+Ct^{-1}( \log t) ^2\| \phi \| _{H^{0,2}}^{1+\frac{2}{n}} \\ & \quad +\| u_1(t) \| _{L^2}+\Big(\int_{t}^{\infty }\| u_1(\tau) \| _{X _n}^4\,d\tau \Big) ^{1/4}. \end{split} \label{3.2} \end{equation} To get the result we now estimate $u_1(t) $. By Lemma \ref{Lemma 2.1}, Lemma \ref{Lemma 2.3} and Lemma \ref{Lemma 2.4} we get \begin{equation} \| u_1(t) \| _{L^2}+\Big(\int_{t}^{\infty }\| u_1(\tau) \| _{X_n}^4d\tau \Big) ^{1/4} \leq C( \| | \cdot | ^{-\widetilde{\delta }}\widehat{ \phi }\| _{L^2}+\| \phi \| _{H^{0,2}}) ^{1 +\frac{2}{n}}t^{-\frac{\widetilde{\delta }}{2}}, \label{3.3} \end{equation} for $\frac{n}{2}<\widetilde{\delta }<2$. where we have used the fact that \begin{align*} &\Big\| \int_{t}^{\infty }\int_{s}^{\infty }\mathcal{U}(s-\tau) f(\tau) \,d\tau \,ds\Big\| _{X_n} \\ &\leq C\int_{t}^{\infty }s^{-\alpha }s^{\alpha } \Big\| \int_{s}^{\infty } \mathcal{U}(s-\tau) f(\tau) \,d\tau \Big\| _{X_n}\,ds \\ &\leq C\Big( \int_{t}^{\infty }s^{-\frac{4}{3}\alpha }\,ds\Big) ^{3/4} \Big( \int_{t}^{\infty }s^{4\alpha } \Big\| \int_{s}^{\infty }\mathcal{ U}(s-\tau) f(\tau) \,d\tau \Big\| _{X_n}^4\,ds\Big) ^{1/4} \\ &\leq Ct^{-\alpha +\frac{3}{4}}\Big( \int_{t}^{\infty }s^{4\alpha } \Big\| \int_{s}^{\infty }\mathcal{U}(s-\tau) f(\tau)\,d\tau \Big\| _{X_n}^4\,ds\Big) ^{1/4}\end{align*} with $\alpha \geq 1$. from which it follows that \begin{align*} &\Big( \int_{\widetilde{t}}^{\infty } \Big\| \int_{t}^{\infty }\int_{s}^{\infty }\mathcal{U}(s-\tau) f(\tau) \,d\tau \,ds \Big\| _{X_n}^4\,dt\Big) ^{1/4} \\ &\leq C\Big( \int_{\widetilde{t}}^{\infty }t^{-4\alpha +3}\Big( \int_{t}^{\infty } \Big\| \int_{s}^{\infty }\mathcal{U}(s-\tau) \tau ^{\alpha }f(\tau) \,d\tau \Big\| _{X_n}^4\,ds\Big) \,dt\Big) ^{1/4} \\ &\leq C\Big( \int_{\widetilde{t}}^{\infty }t^{-4\alpha +3}\Big( \int_{t}^{\infty }\| \tau ^{\alpha }f(\tau) \| _{ L^2}\,d\tau \Big) ^4\,dt\Big) ^{1/4} \\ &\leq Ct^{-\alpha +1-\beta }\sup_{t}t^{\beta }\int_{t}^{\infty }\| \tau ^{\alpha }f(\tau) \| _{L^2}\,d\tau \\ &\leq Ct^{-\beta }\sup_{t}t^{\beta }\int_{t}^{\infty }\| \tau ^{\alpha }f(\tau) \| _{L^2}\,d\tau . \end{align*} By virtue of (\ref{3.2}) and (\ref{3.3}), taking $\frac{n}{2}<\widetilde{ \delta }<2,b=\frac{\widetilde{\delta }}{2}$. we get \begin{equation} \label{3.4} \begin{aligned} {}&\| u(t) -u_0(t) \| _{L^2} +\Big( \int_{t}^{\infty }\| u(\tau) -u_0(\tau) \| _{X_n}^4d\tau \Big) ^{1/4}\\ &\leq C( \| | \cdot | ^{-\widetilde{\delta }}\widehat{ \phi }\| +\| \phi \| _{H^{0,2}}) ^{1+\frac{2}{n}}t^{-b}. \end{aligned} \end{equation} Since the norm of the final state $\| \phi \| _{H ^{0,2}}+\| \phi \| _{\dot{H}^{-\delta }}$ is sufficiently small, estimate (\ref{3.4}) implies that there exists a sufficiently small radius $\rho >0$ such that the mapping $\mathcal{M}v=u$. defined by equation (\ref{3.0}), transforms the set $X_{\rho }$ into itself. 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