1 Introduction

We study the large time asymptotics of solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}w+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}w=\partial _{x}\left( w^{3}\right) ,\text { }t>0,\, x\in {\mathbb {R}}\textbf{,}\\ w\left( 0,x\right) =w_{0}\left( x\right) ,\,x\in {\mathbb {R}} \textbf{,} \end{array} \right. \end{aligned}$$
(1.1)

where \(\alpha \in \left[ 4,5\right) \) and \(w_{0}\) is a real-valued known function. The case of \(\alpha =3\) corresponds to the classical modified KdV equation. In the case of \(\alpha =2,\) (1.1) is the modified Benjamin–Ono equation. Fractional KdV Eq. (1.1) was proposed in paper [16] as a good toy model to understand the influence of a weak dispersion on the dynamics of a scalar conservation law. Indeed, when \(\alpha =\frac{1}{2},\) Eq. (1.1) is reminiscent for large frequencies, to a modified Whitham equation \(\partial _{t}w+{\mathcal {K}} \partial _{x}w=\partial _{x}\left( w^{3}\right) ,\) where the Fourier multiplier operator \({\mathcal {K}}\) has symbol \(K\left( \xi \right) =\sqrt{\frac{\tanh \xi }{\xi }},\) which corresponds to the full dispersion of gravity waves with finite depth. So that the dispersion in (1.1), when \(\alpha =\frac{1}{2}\) is that of purely gravity waves with infinite depth.

The case of \(\alpha \in \left( 2,4\right) \) was considered in paper [9]. As far as we know the large time asymptotics for the Cauchy problem (1.1) with \(\alpha \in \left[ 4,5\right) \) was not studied previously. In the present paper we fill this gap and develop the factorization technique for (1.1) which was started in papers [6,7,8, 10, 13], to obtain the sharp time decay estimates of solutions. Also we apply the known theorems on the \({\textbf{L}}^{2}\)—boundedness of the pseudodifferential operators.

We have the conservation law \(\int _{{\mathbb {R}}}w\left( t,x\right) dx=\int _{{\mathbb {R}}}w_{0}\left( x\right) dx\) for any \(t>0.\) We assume that the total mass of the initial data vanishes \(\int _{{\mathbb {R}}}w_{0}\left( x\right) dx=0,\) then by the conservation of the total mass we obtain \(\int _{{\mathbb {R}}}w\left( t,x\right) dx=\int _{{\mathbb {R}}}w_{0}\left( x\right) dx=0\) for all \(t>0.\) In this case taking the antiderivative, we introduce the new dependent variable \(u=\partial _{x}^{-1}w=\int _{-\infty } ^{x}w\left( t,x^{\prime }\right) dx^{\prime }\) to obtain the potential form of the fractional modified Korteweg-de Vries equation

$$\begin{aligned} \left\{ \begin{array}{c} {\mathcal {L}}u=\left( u_{x}\right) ^{3},\text { }t>0,\,x\in {\mathbb {R}}\textbf{,}\\ u\left( 0,x\right) =u_{0}\left( x\right) ,\,x\in {\mathbb {R}} \textbf{,} \end{array} \right. \end{aligned}$$
(1.2)

where \({\mathcal {L}}=\partial _{t}+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}\) and \(u_{0}\left( x\right) =\int _{-\infty }^{x}w_{0}\left( x^{\prime }\right) dx^{\prime }.\)

To state our results precisely we introduce Notation and Function Spaces. We denote the Lebesgue space by \({\textbf{L}}^{p}=\left\{ \phi \in {\textbf{S}}^{\prime };\left\| \phi \right\| _{{\textbf{L}}^{p}} <\infty \right\} \), where the norm \(\left\| \phi \right\| _{{\textbf{L}} ^{p}}=\left( \int _{{\mathbb {R}}}\left| \phi \left( x\right) \right| ^{p}dx\right) ^{\frac{1}{p}}\) for \(1\le p<\infty \) and \(\left\| \phi \right\| _{{\textbf{L}}^{\infty }}=ess.\sup _{x\in {\textbf{R}}}\left| \phi \left( x\right) \right| \) for \(p=\infty \). The weighted Sobolev space is

$$\begin{aligned} {\textbf{H}}_{p}^{m,s}=\left\{ \varphi \in {\textbf{S}}^{\prime };\left\| \phi \right\| _{{\textbf{H}}_{p}^{m,s}}=\left\| \left\langle x\right\rangle ^{s}\left\langle i\partial _{x}\right\rangle ^{m}\phi \right\| _{{\textbf{L}} ^{p}}<\infty \right\} , \end{aligned}$$

\(m,s\in {\textbf{R}},1\le p\le \infty ,\) \(\left\langle x\right\rangle =\sqrt{1+x^{2}},\left\langle i\partial _{x}\right\rangle =\sqrt{1-\partial _{x}^{2}}.\) We also use the notations \({\textbf{H}}^{m,s}={\textbf{H}}_{2}^{m,s},\) \({\textbf{H}}^{m}={\textbf{H}}^{m,0}\) shortly, if it does not cause any confusion. Let \({\textbf{C}}({\textbf{I}};{\textbf{B}})\) be the space of continuous functions from an interval \({\textbf{I}}\) to a Banach space \({\textbf{B}}.\) Different positive constants might be denoted by the same letter C.

Define the dilation operator \({\mathcal {D}}_{t}\phi =t^{-\frac{1}{2}}\phi \left( \frac{x}{t}\right) ,\) the scaling operator \(\left( {\mathcal {B}}\phi \right) \left( x\right) =\phi \left( \mu \left( x\right) \right) ,\) \(\mu \left( x\right) =x^{\frac{1}{\alpha -1}},\) the multiplication factor \(M=e^{it\frac{\alpha -1}{\alpha }\left| \eta \right| ^{\alpha }},\) the symbol \(\Lambda \left( \xi \right) =\frac{1}{\alpha }\xi \left| \xi \right| ^{\alpha -1}.\) Let \(\theta \) be the Heaviside function \(\theta \left( \xi \right) =1\) for \(\xi >0\) and \(\theta \left( \xi \right) =0\) for \(\xi \le 0.\)

Theorem 1.1

Assume that the initial data are such that \(u_{0} \in {\textbf{H}}^{3}\cap {\textbf{H}}^{2,1},\) with sufficiently small norm \(\left\| u_{0}\right\| _{{\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}}\). Then there exists a unique global solution \(u\in {\textbf{C}}\left( \left[ 0,\infty \right) ;{\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}\right) \) of the Cauchy problem (1.2). Moreover there exists a unique modified final state \(W_{+},\) such that \(\left| \xi \right| ^{\frac{\alpha -2}{2}} W_{+}\left( \xi \right) \in {\textbf{L}}^{\infty }\) and the asymptotic formula

$$\begin{aligned} \left| \partial _{x}\right| ^{\frac{\alpha -2}{2}}u\left( t\right) =2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}}M\theta \left| \xi \right| ^{\frac{\alpha -2}{2}}W_{+}\left( \xi \right) \exp \left( 3i\xi ^{3}\left| W_{+}\left( \xi \right) \right| ^{2}\log t\right) +O\left( t^{-\frac{1}{2}-\delta }\right) \nonumber \\ \end{aligned}$$
(1.3)

is valid for \(t\rightarrow \infty \) uniformly with respect to \(x\in {\mathbb {R}},\) for some \(\delta >0\).

Remark 1.1

Large time asymptotics (1.3) can be written explicitly in the following form

$$\begin{aligned}&\left| \partial _{x}\right| ^{\frac{\alpha -2}{2}}u\left( t,x\right) =2{\text {Re}}\frac{1}{\sqrt{t}}\theta \left( x\right) \left( \frac{x}{t}\right) ^{\frac{\alpha -2}{2\left( \alpha -1\right) } }W_{+}\left( \left( \frac{x}{t}\right) ^{\frac{1}{\alpha -1}}\right) \\&\quad \times \exp \left( it\frac{\alpha -1}{\alpha }\left( \frac{x}{t}\right) ^{\frac{\alpha }{\alpha -1}}+3i\left( \frac{x}{t}\right) ^{\frac{3}{\alpha -1} }\left| W_{+}\left( \left( \frac{x}{t}\right) ^{\frac{1}{\alpha -1} }\right) \right| ^{2}\log t\right) \\&\qquad +O\left( t^{-\frac{1}{2}-\delta }\right) . \end{aligned}$$

We note that the main term of asymptotics differs from the corresponding linear case by the logarithmic oscillation.

Remark 1.2

The proof of Theorem 1.1 shows that the modified final state \(W_{+}\) can be found as a large-time limit. However this definition is not constructive. Therefore one should develop a method for finding \(W_{+}\) in terms of \(u_{0}\) on the basis of perturbation theory with respect to a parameter characterizing the smallness of the initial data \(u_{0}\) (see the approach developed in [14], also see a discussion of an algorithm for the approximate calculation of \(W_{+}\) in terms of \(u_{0}\) in paper [12]).

As a consequence of Theorem 1.1 using a relation \(w=u_{x}\) we get a result concerning well-posedness of the solution w of the Cauchy problem (1.1).

Corollary 1.1

Assume that the total mass of the initial data vanishes \(\int _{{\mathbb {R}} }w_{0}\left( x\right) dx=0.\) Suppose that the initial data are such that \(w_{0}\in {\textbf{H}}^{2}\cap {\textbf{H}}^{1,1},\) with sufficiently small norm \(\left\| w_{0}\right\| _{{\textbf{H}}^{2}\cap {\textbf{H}}^{1,1}}\). Then there exists a unique global solution \(w\in {\textbf{C}}\left( \left[ 0,\infty \right) ;{\textbf{H}}^{2}\cap {\textbf{H}}^{1,1}\right) \) of the Cauchy problem (1.1). Moreover there exists a unique modified final state \(W_{+},\) such that \(\left| \xi \right| ^{\frac{\alpha -2}{2}}W_{+} \in {\textbf{L}}^{\infty }\) and the asymptotics

$$\begin{aligned} \left| \partial _{x}\right| ^{\frac{\alpha -4}{2}}w\left( t\right) =2{\text {Re}}\theta {\mathcal {D}}_{t}{\mathcal {B}}M\left| \xi \right| ^{\frac{\alpha -4}{2}}W_{+}\exp \left( 3i\xi ^{3}\left| W_{+}\right| ^{2}\log t\right) +O\left( t^{-\frac{1}{2}-\delta }\right) \end{aligned}$$

is valid for \(t\rightarrow \infty \) uniformly with respect to \(x\in {\mathbb {R}},\) for some \(\delta >0\).

Remark 1.3

The above large time asymptotics can be written explicitly as follows

$$\begin{aligned}&\left| \partial _{x}\right| ^{\frac{\alpha -4}{2}}w\left( t,x\right) =2{\text {Re}}\frac{1}{\sqrt{t}}\theta \left( x\right) \left( \frac{x}{t}\right) ^{\frac{\alpha -4}{2\left( \alpha -1\right) } }W_{+}\left( \left( \frac{x}{t}\right) ^{\frac{1}{\alpha -1}}\right) \\&\quad \times \exp \left( it\frac{\alpha -1}{\alpha }\left( \frac{x}{t}\right) ^{\frac{\alpha }{\alpha -1}}+3i\left( \frac{x}{t}\right) ^{\frac{3}{\alpha -1} }\left| W_{+}\left( \left( \frac{x}{t}\right) ^{\frac{1}{\alpha -1} }\right) \right| ^{2}\log t\right) \\&\qquad +O\left( t^{-\frac{1}{2}-\delta }\right) . \end{aligned}$$

We organize the rest of the paper as follows. In Sect. 2 we state the factorization formula for (1.2), \({\textbf{L}}^{2}\)—estimates for pseudodifferential operators and prepare \({\textbf{L}}^{\infty }\) and \({\textbf{L}}^{2}\)—estimates for defect operator \({\mathcal {Q}}\) and its adjoint operator \({\mathcal {Q}}^{*}\). Section 3 is devoted to the proof of a-priori estimates of local solutions u of the Cauchy problem (1.2) in the following norm

$$\begin{aligned} \left\| u\right\| _{{\textbf{X}}_{T}}=\sup _{t\in \left[ 1,T\right] }\left( \left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{\infty }}+t^{-\gamma }\left\| \left\langle \xi \right\rangle ^{3}\widehat{\varphi }\right\| _{{\textbf{L}}^{2}} +t^{-\frac{1}{2\alpha }}\left\| \left\langle \xi \right\rangle ^{2} \partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}^{2}}\right) , \end{aligned}$$

where \(\widehat{\varphi }\left( t\right) =\) \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u\left( t\right) ,\) \(\gamma >0\) is small and depends on the size of the data. Finally, we prove Theorem 1.1 in Sect. 4.

2 Preliminaries

The proofs in this section are omitted since they can be obtained in the same way as in our previous papers [1, 13].

2.1 Factorization techniques

Denote the symbol \(\Lambda \left( \xi \right) =\frac{1}{\alpha }\xi \left| \xi \right| ^{\alpha -1},\) then the free evolution group has the form \({\mathcal {U}}\left( t\right) ={\mathcal {F}}^{-1}e^{-it\Lambda \left( \xi \right) }{\mathcal {F}}.\) We write

$$\begin{aligned} {\mathcal {U}}\left( t\right) {\mathcal {F}}^{-1}\phi&={\mathcal {F}} ^{-1}e^{-it\Lambda \left( \xi \right) }\phi =\frac{1}{\sqrt{2\pi }} \int _{{\mathbb {R}}}e^{it\left( \frac{x}{t}\xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi \\&={\mathcal {D}}_{t}\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{it\left( x\xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi , \end{aligned}$$

where \({\mathcal {D}}_{t}\phi =t^{-\frac{1}{2}}\phi \left( \frac{x}{t}\right) \) is the dilation operator. In the integral \(\int _{{\mathbb {R}}}e^{it\left( x\xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi ,\) the stationary points are defined by the equation \(x=\Lambda ^{\prime }\left( \xi \right) =\left| \xi \right| ^{\alpha -1}\ge 0,\) therefore we find that there are two stationary points \(\xi =\pm \mu \left( x\right) ,\) where \(\mu \left( x\right) =x^{\frac{1}{\alpha -1}},\) \(x>0.\) We note that there are no any stationary point in the region \(x<0\). For convenience we extend \(\mu \left( x\right) \) for all \(x\in {\mathbb {R}}\) by the odd continuation \(\mu \left( x\right) =x\left| x\right| ^{\frac{1}{\alpha -1}-1}.\) Thus we have \(\frac{\mu }{\left| \mu \right| }\Lambda ^{\prime }\left( \mu \right) =x\) for all \(x\in {\mathbb {R}}\). We write

$$\begin{aligned}&{\mathcal {U}}\left( t\right) {\mathcal {F}}^{-1}\phi ={\mathcal {D}}_{t} \sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{it\left( \frac{\mu }{\left| \mu \right| }\Lambda ^{\prime }\left( \mu \right) \xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi \\&\quad ={\mathcal {D}}_{t}{\mathcal {B}}\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}} }e^{it\left( \frac{\eta }{\left| \eta \right| }\Lambda ^{\prime }\left( \eta \right) \xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi , \end{aligned}$$

where the scaling operator \(\left( {\mathcal {B}}\phi \right) \left( x\right) =\phi \left( \mu \left( x\right) \right) \). We assume that \({\mathcal {F}} ^{-1}\phi \) is a real-valued function, then \(u={\mathcal {U}}\left( t\right) {\mathcal {F}}^{-1}\phi \) is a real-valued function and \(\phi \left( -\xi \right) =\overline{\phi \left( \xi \right) }.\) We also have \(\Lambda \left( \xi \right) =-\Lambda \left( -\xi \right) ,\) hence

$$\begin{aligned}&{\mathcal {U}}\left( t\right) {\mathcal {F}}^{-1}\phi ={\mathcal {D}} _{t}{\mathcal {B}}\sqrt{\frac{t}{2\pi }}\int _{0}^{\infty }e^{it\left( \frac{\eta }{\left| \eta \right| }\Lambda ^{\prime }\left( \eta \right) \xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi \\&\qquad +{\mathcal {D}}_{t}{\mathcal {B}}\sqrt{\frac{t}{2\pi }}\int _{-\infty } ^{0}e^{it\left( \frac{\eta }{\left| \eta \right| }\Lambda ^{\prime }\left( \eta \right) \xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi \\&\quad =2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}}\sqrt{\frac{t}{2\pi }}\int _{0}^{\infty }e^{it\left( \frac{\eta }{\left| \eta \right| } \Lambda ^{\prime }\left( \eta \right) \xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi . \end{aligned}$$

Thus denoting the multiplication factor \(M=e^{-it\left( \frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\left| \eta \right| \Lambda ^{\prime }\left( \eta \right) \right) }=e^{it\frac{\alpha -1}{\alpha }\left| \eta \right| ^{\alpha }},\) the phase function \(S\left( \xi ,\eta \right) =\Lambda \left( \xi \right) -\frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\frac{\eta }{\left| \eta \right| }\Lambda ^{\prime }\left( \eta \right) \left( \xi -\eta \right) \) and the defect operator

$$\begin{aligned} {\mathcal {Q}}\left( t\right) \phi =\sqrt{\frac{t}{2\pi }}\int _{0}^{\infty }e^{-itS\left( \xi ,\eta \right) }\phi \left( \xi \right) d\xi , \end{aligned}$$

we obtain the factorization formula for the evolution group \({\mathcal {U}} \left( t\right) {\mathcal {F}}^{-1}\phi =2{\text {Re}}{\mathcal {D}} _{t}{\mathcal {B}}M{\mathcal {Q}}\phi .\) Also we need the representation for the inverse evolution group

$$\begin{aligned} {{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \phi&=\frac{1}{\sqrt{2\pi }}\int _{{\mathbb {R}} }e^{it\Lambda \left( \xi \right) -ix\xi }\phi \left( x\right) dx\\&=\frac{1}{\sqrt{2\pi }}\int _{{\mathbb {R}}}e^{it\Lambda \left( \xi \right) -ix\xi }\phi \left( \frac{\mu }{\left| \mu \right| }\Lambda ^{\prime }\left( \mu \right) \right) \left| \Lambda ^{\prime \prime }\left( \mu \right) \right| d\mu \\&=\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }{\overline{M}}\left( {\mathcal {B}}^{-1}{\mathcal {D}}_{t}^{-1}\phi \right) \left( \eta \right) \left| \Lambda ^{\prime \prime }\left( \eta \right) \right| d\eta \end{aligned}$$

since \(x=\frac{\mu }{\left| \mu \right| }\Lambda ^{\prime }\left( \mu \right) ,\) \(\frac{dx}{d\mu }=\left| \Lambda ^{\prime \prime }\left( \mu \right) \right| ,\) where the inverse dilation operator \({\mathcal {D}} _{t}^{-1}\phi =t^{\frac{1}{2}}\phi \left( xt\right) ,\) the inverse scaling operator \(\left( {\mathcal {B}}^{-1}\phi \right) \left( \mu \right) =\phi \left( x\right) \). Then denoting the adjoint defect operator

$$\begin{aligned} {\mathcal {Q}}^{*}\left( t\right) \phi =\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \left| \Lambda ^{\prime \prime }\left( \eta \right) \right| d\eta , \end{aligned}$$

we obtain the factorization formula for the inverse evolution group \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \phi ={\mathcal {Q}}^{*}{\overline{M}} {\mathcal {B}}^{-1}{\mathcal {D}}_{t}^{-1}\phi .\)

Define the new dependent variable \(\widehat{\varphi }\left( t\right) =\) \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u\left( t\right) \). Since \({{\mathcal {F}}}{{\mathcal {U}}} \left( -t\right) {\mathcal {L}}=\partial _{t}{{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) ,\) where \({\mathcal {L}}=\partial _{t}+i\Lambda \left( -i\partial _{x}\right) ,\) applying the operator \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \) to equation (1.2) we get \(\partial _{t}\widehat{\varphi }={{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u_{x}^{3}.\) Then since \(u_{x}=2{\text {Re}}{\mathcal {D}} _{t}{\mathcal {B}}Mv,\) \(v={\mathcal {Q}}i\xi \widehat{\varphi },\) we obtain

$$\begin{aligned} \partial _{t}\widehat{\varphi }&={{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u_{x} ^{3}={\mathcal {Q}}^{*}{\overline{M}}{\mathcal {B}}^{-1}{\mathcal {D}}_{t}^{-1}\left( 2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}}Mv\right) ^{3}=t^{-1} {\mathcal {Q}}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\\&=t^{-1}{\mathcal {Q}}^{*}M^{2}v^{3}+3t^{-1}{\mathcal {Q}}^{*}\left| v\right| ^{2}v+3t^{-1}{\mathcal {Q}}^{*}{\overline{M}}^{2}\left| v\right| ^{2}{\overline{v}}+t^{-1}{\mathcal {Q}}^{*}{\overline{M}} ^{4}{\overline{v}}^{3}. \end{aligned}$$

By the definition of the operator \({\mathcal {Q}}^{*}\left( t\right) \) we obtain \({\mathcal {Q}}^{*}\)\(\left( t\right) M^{k}\phi =e^{it\Omega _{k+1} }{\mathcal {D}}_{k+1}{\mathcal {Q}}^{*}\left( \left( k+1\right) t\right) \phi ,\) where \(\Omega _{k+1}=\Lambda \left( \xi \right) -\left( k+1\right) \Lambda \left( \frac{\xi }{k+1}\right) \) for \(k\ne -1.\) We denote below \(\Omega =\Omega _{3}.\) Then we find \({\mathcal {Q}}^{*}\left( t\right) M^{2}v^{3}=e^{it\Omega }{\mathcal {D}}_{3}{\mathcal {Q}}^{*}\left( 3t\right) v^{3},\)

$$\begin{aligned} {\mathcal {Q}}^{*}\left( t\right) {\overline{M}}^{2}\left| v\right| ^{2}v=e^{it\Omega _{-1}}{\mathcal {D}}_{-1}{\mathcal {Q}}^{*}\left( -t\right) \left| v\right| ^{2}{\overline{v}}=\overline{{\mathcal {D}}_{-1} {\mathcal {Q}}^{*}\left( t\right) \left| v\right| ^{2}v} \end{aligned}$$

since \(\Omega _{-1}=\Lambda \left( \xi \right) +\Lambda \left( -\xi \right) =0\) and

$$\begin{aligned} {\mathcal {Q}}^{*}\left( t\right) {\overline{M}}^{4}{\overline{v}} ^{3}=e^{it\Omega _{-3}}{\mathcal {D}}_{-3}{\mathcal {Q}}^{*}\left( -3t\right) {\overline{v}}^{3}=e^{it\Omega }\overline{{\mathcal {D}}_{-3}{\mathcal {Q}}^{*}\left( 3t\right) v^{3}} \end{aligned}$$

since \(\Omega _{-3}=\Lambda \left( \xi \right) +3\Lambda \left( -\frac{\xi }{3}\right) =\Lambda \left( \xi \right) -3\Lambda \left( \frac{\xi }{3}\right) =\) \(\Omega _{3},\) where \({\mathcal {D}}_{-1}\phi =e^{-i\frac{\pi }{2}}\phi \left( -x\right) \) and \({\mathcal {D}}_{-3}\phi =\frac{1}{\sqrt{3}}e^{-i\frac{\pi }{2} }\phi \left( -\frac{x}{3}\right) .\) Thus we get the following equation

$$\begin{aligned} \partial _{t}\widehat{\varphi }&=t^{-1}e^{it\Omega }{\mathcal {D}}_{3} {\mathcal {Q}}^{*}\left( 3t\right) v^{3}+3t^{-1}{\mathcal {Q}}^{*}\left( t\right) \left| v\right| ^{2}v\nonumber \\&\quad +3t^{-1}\overline{{\mathcal {D}}_{-1}{\mathcal {Q}}^{*}\left( t\right) \left| v\right| ^{2}v}+t^{-1}e^{it\Omega }\overline{{\mathcal {D}} _{-3}{\mathcal {Q}}^{*}\left( 3t\right) v^{3}}. \end{aligned}$$
(2.1)

Finally we mention some important identities. We have \({\mathcal {A}} _{1}{\mathcal {Q}}={\mathcal {Q}}i\xi \) and \(i\xi {\mathcal {Q}}^{*}\) \(={\mathcal {Q}} ^{*}{\mathcal {A}}_{1}\), where \({\mathcal {A}}_{1}={\overline{M}}{\mathcal {A}}_{0}M\), and \({\mathcal {A}}_{0}=\frac{1}{t\Lambda ^{\prime \prime }\left( \eta \right) }\partial _{\eta }.\) The operator \({\mathcal {J}}={\mathcal {U}}\left( t\right) x{\mathcal {U}}\left( -t\right) =x-t\Lambda ^{\prime }\left( -i\partial _{x}\right) ,\) plays a crucial role in the large time asymptotic estimates. Note that \({\mathcal {J}}\) commutes with \({\mathcal {L}}=\partial _{t}+i\Lambda \left( -i\partial _{x}\right) ,\) namely, \(\left[ {\mathcal {J}},{\mathcal {L}} \right] =0\). The symbol \(\Lambda \left( \xi \right) \) satisfies the identity \(\xi \partial _{\xi }\Lambda =\xi \left| \xi \right| ^{\alpha -1} =\alpha \Lambda .\) Hence we have the commutator \(\left[ \widehat{{\mathcal {P}} },e^{-it\Lambda \left( \xi \right) }\right] =0\) with \(\widehat{{\mathcal {P}} }=\alpha t\partial _{t}-\xi \partial _{\xi }.\) Also we define the operator \({\mathcal {P}}=\alpha t\partial _{t}+\partial _{x}x\). The commutator relation \(\left[ {\mathcal {L}},{\mathcal {P}}\right] =\alpha {\mathcal {L}}\) holds. Using \(u\left( t\right) ={\mathcal {U}}\left( t\right) {\mathcal {F}}^{-1} \widehat{\varphi }={\mathcal {F}}^{-1}e^{-it\Lambda \left( \xi \right) } \widehat{\varphi },\) we get

$$\begin{aligned} {\mathcal {P}}u={{\mathcal {P}}}{{\mathcal {F}}}^{-1}e^{-it\Lambda \left( \xi \right) } \widehat{\varphi }={\mathcal {F}}^{-1}e^{-it\Lambda \left( \xi \right) } \widehat{{\mathcal {P}}}\widehat{\varphi }={\mathcal {U}}\left( t\right) {\mathcal {F}}^{-1}\widehat{{\mathcal {P}}}\widehat{\varphi }. \end{aligned}$$

Also we have the identity \({\mathcal {P}}=\alpha t{\mathcal {L}}+\partial _{x}{\mathcal {J}}\) since by a direct calculation \(\alpha t{\mathcal {L}} +\partial _{x}{\mathcal {J}}=\alpha t\partial _{t}+\partial _{x}x.\)

2.2 Estimates for the operator \({\mathcal {Q}}\) in the uniform norm

Define the cut off functions \(\chi _{j}\left( x\right) \in {\textbf{C}} ^{4}\left( {\mathbb {R}}\right) ,\) such that \(\chi _{1}\left( x\right) =1\) for \(\frac{2}{3}\le x\le 2\), \(\chi _{1}\left( x\right) =0\) for \(x\le \frac{1}{3}\) or \(x\ge 3,\) also \(\chi _{2}\left( x\right) =1-\chi _{1}\left( x\right) .\) Denote the operators

$$\begin{aligned} {\mathcal {Q}}_{k}\phi =\sqrt{\frac{t}{2\pi }}\int _{0}^{\infty }e^{-itS\left( \xi ,\eta \right) }\chi _{k}\left( \xi \eta ^{-1}\right) \phi \left( \xi \right) d\xi \end{aligned}$$

for \(\eta >0,\) \(k=1,2.\) Define the kernel

$$\begin{aligned} A_{j}\left( t,\eta \right) =\theta \left( \eta \right) \sqrt{\frac{t}{2\pi } }\int _{0}^{\infty }e^{-itS\left( \xi ,\eta \right) }\chi _{1}\left( \xi \eta ^{-1}\right) \xi ^{j}d\xi \end{aligned}$$

for \(\eta >0.\) We change \(\xi =\eta y\), then we get

$$\begin{aligned} A_{j}\left( t,\eta \right)&=\theta \left( \eta \right) \left| \eta \right| \eta ^{j}\sqrt{\frac{t}{2\pi }}\int _{\frac{1}{3}}^{3} e^{-itS\left( \eta y,\eta \right) }\chi _{1}\left( y\right) y^{j}dy\\&=\theta \left( \eta \right) \left| \eta \right| \eta ^{j}\sqrt{\frac{t}{2\pi }}\int _{\frac{1}{3}}^{3}e^{-it\left| \eta \right| ^{\alpha }S\left( y,1\right) }\chi _{1}\left( y\right) y^{j}dy. \end{aligned}$$

We get \(\left| A_{j}\left( t,\eta \right) \right| \le Ct^{\frac{1}{2}}\theta \left( \eta \right) \left| \eta \right| ^{1+j},\) if \(\left| \eta \right| \le t^{-\frac{1}{\alpha }}.\) To compute the asymptotics of the kernel \(A_{j}\left( t,\eta \right) \) for large \(t\left| \eta \right| ^{\alpha }\) we apply the stationary phase method (see [5], p. 110)

$$\begin{aligned} \int _{{\mathbb {R}}}e^{itg\left( y\right) }f\left( y\right) dy=e^{itg\left( y_{0}\right) }f\left( y_{0}\right) \sqrt{\frac{2\pi }{t\left| g^{\prime \prime }\left( y_{0}\right) \right| }}e^{i\frac{\pi }{4}\text {sgn}g^{\prime \prime }\left( y_{0}\right) }+O\left( t^{-\frac{3}{2} }\right) \end{aligned}$$
(2.2)

for \(t\rightarrow +\infty ,\) where the stationary point \(y_{0}\) is defined by the equation \(g^{\prime }\left( y_{0}\right) =0.\) By virtue of formula (2.2) with \(g\left( y\right) =-S\left( y,1\right) ,\) \(f\left( y\right) =\chi _{1}\left( y\right) y^{j},\) \(y_{0}=1,\) we get

$$\begin{aligned} A_{j}\left( t,\eta \right) =\frac{\eta ^{j}\theta \left( \eta \right) }{\sqrt{i\Lambda ^{\prime \prime }\left( \eta \right) }}+O\left( t^{\frac{1}{2} }\eta ^{1+j}\left\langle t\left| \eta \right| ^{\alpha }\right\rangle ^{-\frac{3}{2}}\right) \end{aligned}$$

for \(t\left| \eta \right| ^{\alpha }\rightarrow +\infty .\) Hence \(\left| A_{j}\left( t,\eta \right) \right| \le Ct^{\frac{1}{2} }\theta \left( \eta \right) \left| \eta \right| ^{1+j}\left\langle t\left| \eta \right| ^{\alpha }\right\rangle ^{-\frac{1}{2}}\) for all \(t\ge 1,\) \(\eta \in {\mathbb {R}}\).

In the next lemma we find the estimate for the defect operator \({\mathcal {Q}}\) in the uniform metrics.

Lemma 2.1

Let \(4\le \alpha <5\). Then the estimate

$$\begin{aligned} \eta ^{\delta }\left| {\mathcal {Q}}_{1}\xi ^{j}\phi -A_{j}\phi \right| \le Ct^{\frac{1}{2}-\min \left( \frac{3}{4},\frac{1}{\alpha }\left( \delta +j+\frac{3}{2}\right) \right) }\left\langle \eta \right\rangle ^{\delta +j-\frac{1}{2}-\frac{3}{4}\alpha }\left\| \left\langle \xi \right\rangle ^{2}\partial _{\xi }\phi \right\| _{{\textbf{L}}^{2}} \end{aligned}$$

is true for all \(t\ge 1,\) \(\eta >0,\) where \(j=1,2,3,\) \(\delta \ge 0.\)

Consider the estimate of \({\mathcal {Q}}_{2}\left( t\right) .\)

Lemma 2.2

Let \(4\le \alpha <5\). Then the estimate

$$\begin{aligned} \eta ^{\delta }\left| {\mathcal {Q}}_{2}\xi ^{j}\phi \right|&\le Ct^{\frac{1}{2}-\min \left( 1,\frac{1}{\alpha }\left( \delta +j+\frac{3}{2}\right) \right) }\log \left\langle t\right\rangle \left\langle \eta \right\rangle ^{\delta +j+1-\alpha }\left\| \left\langle \xi \right\rangle ^{2}\partial _{\xi }\phi \right\| _{{\textbf{L}}^{2}}\\&\quad +Ct^{\frac{1}{2}-\min \left( 1,\frac{1}{\alpha }\left( \delta +j+1\right) \right) }\left\langle \eta \right\rangle ^{\delta +j+1-\alpha }\left\| \left\langle \xi \right\rangle ^{2}\phi \right\| _{{\textbf{L}}^{\infty }} \end{aligned}$$

is true for all \(t\ge 1,\) \(\eta >0,\) where \(j=1,2,3,\) \(\delta \ge 0.\)

Next we find the estimate of \({\mathcal {Q}}\left( t\right) \) for \(\eta \le 0.\)

Lemma 2.3

Let \(4\le \alpha <5\). Then the estimate

$$\begin{aligned} \left| \eta \right| ^{\delta }\left| {\mathcal {Q}}\xi ^{j} \phi \right|&\le Ct^{\frac{1}{2}-\min \left( 1,\frac{1}{\alpha }\left( \delta +j+\frac{3}{2}\right) \right) }\log \left\langle t\right\rangle \left\langle \eta \right\rangle ^{\delta +j+1-\alpha }\left\| \left\langle \xi \right\rangle ^{2}\partial _{\xi }\phi \right\| _{{\textbf{L}}^{2}}\\&\quad +Ct^{\frac{1}{2}-\min \left( 1,\frac{1}{\alpha }\left( \delta +j+1\right) \right) }\left\langle \eta \right\rangle ^{\delta +j+1-\alpha }\left\| \left\langle \xi \right\rangle ^{2}\phi \right\| _{{\textbf{L}}^{\infty }} \end{aligned}$$

is true for all \(t\ge 1,\) \(\eta \le 0,\) where \(j=1,2,3,\) \(\delta \ge 0.\)

As a consequence of Lemma 2.1, Lemma 2.2 and Lemma 2.3 we obtain the following result. Define the norm \(\left\| \phi \right\| _{{\textbf{Y}}}=\left\| \left\langle \xi \right\rangle ^{2}\phi \right\| _{{\textbf{L}}^{\infty }}+t^{-\frac{1}{2\alpha }}\left\| \left\langle \xi \right\rangle ^{2}\partial _{\xi }\phi \right\| _{{\textbf{L}} ^{2}}.\) Denote \(\widetilde{\eta }=\eta t^{\frac{1}{\alpha }}.\)

Corollary 2.1

The estimates

$$\begin{aligned} \left| {\mathcal {Q}}\xi \phi \right|\le & {} Ct^{\frac{\alpha -4}{2\alpha } }\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-2}\left\| \phi \right\| _{{\textbf{Y}}},\\ \left| {\mathcal {Q}}\xi ^{2}\phi \right|\le & {} Ct^{-\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{\frac{\alpha -4}{2} }\left\| \phi \right\| _{{\textbf{Y}}},\\ \left| {\mathcal {Q}}\xi ^{3}\phi \right|\le & {} Ct^{-\frac{1}{2\alpha } }\left\langle \widetilde{\eta }\right\rangle ^{\frac{1}{2}}\left\| \phi \right\| _{{\textbf{Y}}},\\ \left| {\mathcal {Q}}\xi \phi \right| \left| {\mathcal {Q}}\xi ^{2} \phi \right|\le & {} C\left( \left| \eta \right| ^{5-\alpha } +t^{-\frac{5-\alpha }{\alpha }}\right) \left\| \phi \right\| _{{\textbf{Y}} }^{2} \end{aligned}$$

and

$$\begin{aligned} \left\langle \widetilde{\eta }\right\rangle ^{-1}\left| {\mathcal {Q}}\xi \phi \right| \left| {\mathcal {Q}}\xi ^{3}\phi \right| \le C\left\| \phi \right\| _{{\textbf{Y}}}^{2} \end{aligned}$$

are true.

2.3 \({\textbf{L}}^{\infty }\) - estimates for the adjoint defect operator \({\mathcal {Q}}^{*}\)

Denote the operators

$$\begin{aligned} {\mathcal {Q}}_{j}^{*}\left( t\right) \phi =\sqrt{\frac{t}{2\pi }} \int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \chi _{j}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta \end{aligned}$$

for \(\xi \ne 0\). Denote the adjoint kernel

$$\begin{aligned} A^{*}\left( t,\xi \right) =\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}} }e^{itS\left( \xi ,\eta \right) }\chi _{1}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta \end{aligned}$$

for \(\xi \ne 0\). We change \(\eta =\xi y\), then we get

$$\begin{aligned} A^{*}\left( t,\xi \right)&=\left| \xi \right| \sqrt{\frac{t}{2\pi }}\int _{\frac{1}{3}}^{3}e^{itS\left( \xi ,\xi y\right) }\chi _{1}\left( y\right) \Lambda ^{\prime \prime }\left( \xi y\right) dy\\&=\left| \xi \right| ^{\alpha -1}\sqrt{\frac{t}{2\pi }}\int _{\frac{1}{3}}^{3}e^{it\left| \xi \right| ^{\alpha }S\left( 1,y\right) } \chi _{1}\left( y\right) y^{\alpha -2}dy. \end{aligned}$$

We have \(\left| A^{*}\left( t,\xi \right) \right| \le Ct^{\frac{1}{2}}\left| \xi \right| ^{\alpha -1},\) if \(\left| \xi \right| \le t^{-\frac{1}{\alpha }}.\) By virtue of formula (2.2) with \(g\left( y\right) =S\left( 1,y\right) ,\) \(f\left( y\right) =\chi _{1}\left( y\right) y^{\alpha -2},\) \(y_{0}=1,\) we find

$$\begin{aligned} A^{*}\left( t,\xi \right) =\left( \sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }+O\left( t^{\frac{1}{2}}\left| \xi \right| ^{\alpha -1}\left\langle t\left| \xi \right| ^{\alpha }\right\rangle ^{-\frac{3}{2}}\right) \right) \end{aligned}$$

for \(t\left| \xi \right| ^{\alpha }\rightarrow +\infty .\) Hence

$$\begin{aligned} \left| A^{*}\left( t,\xi \right) \right| \le Ct^{\frac{1}{2} }\left| \xi \right| ^{\alpha -1}\left\langle t\left| \xi \right| ^{\alpha }\right\rangle ^{-\frac{1}{2}} \end{aligned}$$

for \(t\ge 1,\) \(\xi \in {\mathbb {R}}\).

In the next lemma we estimate the adjoint defect operator \({\mathcal {Q}} _{1}^{*}\) in the uniform metrics.

Lemma 2.4

Let \(4\le \alpha <5\). Then the estimate

$$\begin{aligned} \left| {\mathcal {Q}}_{1}^{*}\phi -A^{*}\phi \right| \le Ct^{-\min \left( \frac{1}{4},\frac{\alpha -1-2\delta }{2\alpha }\right) }\left\langle \xi \right\rangle ^{\frac{\alpha -2}{4}-\delta }\left\| \left| \eta \right| ^{\delta }\partial _{\eta }\phi \right\| _{{\textbf{L}}^{2}} \end{aligned}$$

is true for all \(t\ge 1,\) \(\xi \in {\mathbb {R}}\), where \(\delta \ge 0.\)

Next we estimate \({\mathcal {Q}}_{2}^{*}\phi .\)

Lemma 2.5

Let \(4\le \alpha <5\). Then the estimate

$$\begin{aligned} \left| {\mathcal {Q}}_{2}^{*}\phi \right| \le Ct^{-\frac{1}{2} +\frac{1+2\delta }{2\alpha }}\left( \left\| \left| \eta \right| ^{\delta }\partial _{\eta }\phi \right\| _{{\textbf{L}}^{2}}+\left\| \left| \eta \right| ^{\delta -1}\phi \right\| _{{\textbf{L}}^{2} }\right) \end{aligned}$$

is true for all \(t\ge 1,\) \(\xi \in {\mathbb {R}}\), where \(0\le \delta <\alpha -\frac{1}{2}.\)

2.4 Estimates for pseudodifferential operators

There are many papers devoted to the \({\textbf{L}}^{2}\) - estimates of pseudodifferential operators (see, e.g. [2,3,4, 11]). Consider the following weighted defect operator

$$\begin{aligned} {\mathcal {V}}_{h}\phi =\frac{t^{\frac{1}{2}}}{\sqrt{2\pi }}\int _{0}^{\infty }e^{-itS\left( \xi ,\eta \right) }h\left( t,\xi ,\eta \right) \phi \left( \xi \right) d\xi . \end{aligned}$$

Lemma 2.6

Let \(h\left( t,\xi ,\eta \right) \) satisfy the estimate

\(\sup _{\xi ,\eta \in {\mathbb {R}},t\ge 1} \left| \left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left( \eta \partial _{\eta }\right) ^{k}h\left( t,\xi ,\eta \right) \right| \le C\) for \(k=0,1,2,\) with some \(\nu \in \left( 0,1\right) .\) Then there exists a positive constant C such that the inequality \(\left\| \sqrt{\Lambda ^{\prime \prime }}{\mathcal {V}}_{h}\phi \right\| _{{\textbf{L}}^{2}}\le C\left\| \phi \right\| _{{\textbf{L}}^{2}}\) holds.

Similarly we estimate the \({\textbf{L}}^{2}\) - norm of the adjoint weighted defect operator

$$\begin{aligned} {\mathcal {V}}_{h}^{*}\phi =\frac{t^{\frac{1}{2}}}{\sqrt{2\pi }}\int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }h\left( t,\xi ,\eta \right) \phi \left( \eta \right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta . \end{aligned}$$

Lemma 2.7

Let \(h\left( t,\xi ,\eta \right) \) satisfy the estimate

\(\sup _{\xi ,\eta \in {\mathbb {R}},t\ge 1}\left| \left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left( \eta \partial _{\eta }\right) ^{k}h\left( t,\xi ,\eta \right) \right| \le C\) for \(k=0,1,2\) with some \(\nu \in \left( 0,1\right) .\) Then there exists a positive constant C such that the inequality \(\left\| {\mathcal {V}} _{h}^{*}\phi \right\| _{{\textbf{L}}^{2}}\le C\left\| \sqrt{\Lambda ^{\prime \prime }}\phi \right\| _{{\textbf{L}}^{2}}\) holds.

2.5 Estimate for derivative of the defect operator

Lemma 2.8

The estimate

$$\begin{aligned} \left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{\nu }\left\langle \widetilde{\eta }\right\rangle ^{-\nu }\eta ^{-1} \partial _{\eta }{\mathcal {Q}}\xi \phi \right\| _{{\textbf{L}}^{2}}\le C\left\| \left\langle \xi \right\rangle \partial _{\xi }\phi \right\| _{{\textbf{L}}^{2} }+C\left\| \phi \right\| _{{\textbf{L}}^{2}}+Ct^{\frac{1}{2\alpha } }\left| \phi \left( 0\right) \right| \end{aligned}$$

is true for all \(t\ge 1,\) where \(\nu >0\) is small.

2.6 Estimates for the derivative of \({\mathcal {Q}}^{*}\)

Define the cut off functions \(\chi _{3}\left( x\right) ,\chi _{4}\left( x\right) \in {\textbf{C}}^{4}\left( {\mathbb {R}}\right) ,\) such that \(\chi _{3}\left( x\right) =1\) for \(x\ge 3\), \(\chi _{3}\left( x\right) =0\) for \(x\le 2,\) also \(\chi _{4}\left( x\right) =1-\chi _{3}\left( x\right) .\) Denote

$$\begin{aligned} {\mathcal {Q}}_{5}^{*}\phi&=\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}} }e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \left( \chi _{3}\left( \frac{\eta }{\xi }\right) +\chi _{4}\left( \widetilde{\xi }\right) \chi _{4}\left( \frac{\eta }{\xi }\right) \right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta ,\\ {\mathcal {Q}}_{6}^{*}\phi&=\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}} }e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \chi _{3}\left( \widetilde{\xi }\right) \chi _{4}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta , \end{aligned}$$

where \(\widetilde{\xi }=\xi t^{\frac{1}{\alpha }}.\) In the next lemma we obtain the estimates of the derivative of the adjoint defect operator \(\partial _{\xi }{\mathcal {Q}}_{5}^{*}\phi .\)

Lemma 2.9

The estimate

$$\begin{aligned}&\left\| \partial _{\xi }{\mathcal {Q}}_{5}^{*}\phi \right\| _{{\textbf{L}} _{\xi }^{2}}\\&\le C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\partial _{\eta }\phi \right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }+C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\frac{1}{\eta }\phi \right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad +Ct^{\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha }}\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }\phi \right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) } \end{aligned}$$

is true for all \(t\ge 1\), where \(0<\nu <1.\)

We now estimate the adjoint defect operators \(\xi {\mathcal {Q}}_{5}^{*} \eta ^{-1}\phi \) and \(\frac{1}{\xi }{\mathcal {Q}}_{6}^{*}\eta \phi .\)

Lemma 2.10

The estimate

$$\begin{aligned}&\left\| \xi ^{j}{\mathcal {Q}}_{5}^{*}\eta ^{-j}\phi \right\| _{{\textbf{L}}_{\xi }^{2}}+\left\| \frac{1}{\widetilde{\xi }^{j}} {\mathcal {Q}}_{6}^{*}\left\langle \widetilde{\eta }\right\rangle ^{j} \phi \right\| _{{\textbf{L}}_{\xi }^{2}}\\&\quad \le C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu } \phi \right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }+C\left\| \sqrt{\Lambda ^{\prime \prime }} \phi \right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) } \end{aligned}$$

is true for all \(t\ge 1\), where \(0<\nu <1,\) \(j=0,1.\)

We also have the estimate for \({\mathcal {Q}}_{5}^{*}\phi .\)

Lemma 2.11

The estimate

$$\begin{aligned} \left\| {\mathcal {Q}}_{5}^{*}\phi \right\| _{{\textbf{L}}_{\xi }^{2}}&\le Ct^{-1}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left| \eta \right| ^{1-\alpha }\partial _{\eta }\phi \right\| _{{\textbf{L}} ^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad +Ct^{-1}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left| \eta \right| ^{-\alpha }\phi \left( \eta \right) \right\| _{{\textbf{L}} ^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad +Ct^{-\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha }}\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }\phi \right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) } \end{aligned}$$

is true for all \(t\ge 1\), where \(0<\nu <1.\)

3 A-priori estimates

3.1 Estimates for the nonlinearity

In the next lemma we calculate the asymptotic representation for the nonlinearity \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \partial _{x}\left( \left| u\right| ^{2}u\right) .\) Define the norm

$$\begin{aligned} \left\| u\right\| _{{\textbf{X}}_{T}}=\sup _{t\in \left[ 1,T\right] }\left( \left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{\infty }}+t^{-\gamma }\left\| \left\langle \xi \right\rangle ^{3}\widehat{\varphi }\right\| _{{\textbf{L}}^{2}} +t^{-\frac{1}{2\alpha }}\left\| \left\langle \xi \right\rangle ^{2} \partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}^{2}}\right) , \end{aligned}$$

where \(\widehat{\varphi }\left( t\right) =\) \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u\left( t\right) ,\) \(\gamma >0\) is small.

Lemma 3.1

Suppose that \(\left\| u\right\| _{{\textbf{X}}_{T}}\le C\varepsilon .\) Then the asymptotic representation

$$\begin{aligned} \left\langle \xi \right\rangle ^{-\frac{1}{2}}\partial _{t}\widehat{\varphi }&=\frac{3i\xi ^{3}\theta \left( \xi \right) }{t\Lambda ^{\prime \prime }\left( \xi \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left| \widehat{\varphi }\left( \xi \right) \right| ^{2}\widehat{\varphi }\left( \xi \right) \\&\quad -e^{it\Omega }{\mathcal {D}}_{3}\frac{\xi ^{3}\theta \left( \xi \right) }{t\Lambda ^{\prime \prime }\left( \xi \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\widehat{\varphi }^{3}\left( \xi \right) +O\left( \varepsilon ^{3}t^{-1-\delta _{1}}\right) \end{aligned}$$

is true for all \(t\ge 1\), \(\xi \in {\mathbb {R}},\) with some \(\delta _{1}>0.\)

Proof

By equation (2.1) we have

$$\begin{aligned} \partial _{t}\widehat{\varphi }&=3t^{-1}{\mathcal {Q}}^{*}\left( t\right) \left| v\right| ^{2}v+t^{-1}e^{it\Omega }{\mathcal {D}}_{3}{\mathcal {Q}} ^{*}\left( 3t\right) v^{3}\\&\quad +3t^{-1}\overline{{\mathcal {D}}_{-1}{\mathcal {Q}}^{*}\left( t\right) \left| v\right| ^{2}v}+t^{-1}e^{it\Omega }\overline{{\mathcal {D}} _{-3}{\mathcal {Q}}^{*}\left( 3t\right) v^{3}}. \end{aligned}$$

We define the cut off function \(\chi _{5}\left( x\right) \in {\textbf{C}} ^{4}\left( {\mathbb {R}}\right) ,\) such that \(\chi _{5}\left( x\right) =1\) for \(\left| x\right| \le 1\) and \(\chi _{5}\left( x\right) =0\) for \(\left| x\right| \ge 2,\) \(\chi _{6}\left( x\right) =1-\chi _{5}\left( x\right) .\) We represent

$$\begin{aligned} \left\langle \xi \right\rangle ^{-\frac{1}{2}}{\mathcal {Q}}^{*}\left( \left| v\right| ^{2}v\right) =\left\langle \xi \right\rangle ^{-\frac{1}{2}}{\mathcal {Q}}^{*}\chi _{5}\left( \widetilde{\eta }\right) \left| v\right| ^{2}v+\left\langle \xi \right\rangle ^{-\frac{1}{2} }{\mathcal {Q}}^{*}\chi _{6}\left( \widetilde{\eta }\right) \left| v\right| ^{2}v. \end{aligned}$$

By Corollary 2.1 we have \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\le C\varepsilon \left| \eta \right| ^{2-\frac{\alpha }{2}}\) in the domain \(\left| \eta \right| \le 2t^{-\frac{1}{\alpha }}.\) Then we estimate the first term as follows

$$\begin{aligned}&\left| \left\langle \xi \right\rangle ^{-\frac{1}{2}}{\mathcal {Q}}^{*}\chi _{5}\left( \widetilde{\eta }\right) \left| v\right| ^{2}v\right| \le Ct^{\frac{1}{2}}\int _{\left| \eta \right| \le 2t^{-\frac{1}{\alpha }}}\left| v\right| ^{3}\Lambda ^{\prime \prime }\left( \eta \right) d\eta \\&\quad \le C\varepsilon ^{3}t^{\frac{1}{2}}\int _{\left| \eta \right| \le 2t^{-\frac{1}{\alpha }}}\left( \left| \eta \right| ^{2-\frac{\alpha }{2}}\right) ^{3}\left| \eta \right| ^{\alpha -2}d\eta \le C\varepsilon ^{3}t^{-\frac{5-\alpha }{\alpha }}. \end{aligned}$$

Next we find the asymptotics of the second term. By virtue of Lemma 2.4 and Lemma 2.5 with \(\delta =3\left( \frac{\alpha }{2}-2\right) -\nu ,\) we get

$$\begin{aligned}&\left\langle \xi \right\rangle ^{-\frac{1}{2}}{\mathcal {Q}}^{*}\chi _{6}\left( \widetilde{\eta }\right) \left| v\right| ^{2} v=\left\langle \xi \right\rangle ^{-\frac{1}{2}}A^{*}\chi _{6}\left( \widetilde{\eta }\right) \left| v\right| ^{2}v\\&\qquad +O\left( t^{-\frac{1}{2}+\frac{1+2\delta }{2\alpha }}\left\| \left| \eta \right| ^{\delta -1}v^{3}\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\right) \\&\qquad +O\left( t^{-\min \left( \frac{1}{4},\frac{\alpha -1-2\delta }{2\alpha }\right) }\left\| \left| \eta \right| ^{\delta }v^{2}\partial _{\eta }v\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\right) . \end{aligned}$$

We use the estimate \(\left| v\right| \le C\varepsilon t^{\frac{1}{2}-\frac{2}{\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{2-\frac{\alpha }{2}}\) for \(\left| \eta \right| \ge t^{-\frac{1}{\alpha }}\). Then, we find

$$\begin{aligned}&t^{-\frac{1}{2}+\frac{1+2\delta }{2\alpha }}\left\| \left| \eta \right| ^{\delta -1}v^{3}\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\varepsilon ^{3}t^{1-\frac{\nu }{\alpha }-\frac{11}{2\alpha }}\left\| \left| \eta \right| ^{-\nu -1}\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}t^{1-\frac{2\nu }{\alpha }-\frac{6}{\alpha }}. \end{aligned}$$

By Lemma 2.8 we have

$$\begin{aligned} \left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \eta \right\} ^{\nu }\left\langle \eta \right\rangle ^{-\nu }\eta ^{-1}\partial _{\eta }v\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}. \end{aligned}$$

Hence using \(\left| v\right| \le C\varepsilon t^{\frac{1}{2}-\frac{2}{\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{2-\frac{\alpha }{2}} \) for \(\left| \eta \right| \ge t^{-\frac{1}{\alpha }},\) we get

$$\begin{aligned}&t^{-\min \left( \frac{1}{4},\frac{\alpha -1-2\delta }{2\alpha }\right) }\left\| \left| \eta \right| ^{\delta }\left| v\right| ^{2}\partial _{\eta }v\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\varepsilon ^{3}t^{\frac{1}{2\alpha }+1-\frac{4}{\alpha }-\min \left( \frac{1}{4},\frac{\alpha -1-2\delta }{2\alpha }\right) }\left\| \left\{ \eta \right\} ^{\alpha -4-2\nu }\left\langle \eta \right\rangle ^{\alpha -4}\left\langle \widetilde{\eta }\right\rangle ^{4-\alpha }\right\| _{{\textbf{L}}^{\infty }\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\varepsilon ^{3}t^{-\min \left( \frac{\alpha -2-8\nu }{4\alpha } ,\frac{5-\alpha -\nu }{\alpha }\right) }. \end{aligned}$$

Thus we find

$$\begin{aligned} \left\langle \xi \right\rangle ^{-\frac{1}{2}}{\mathcal {Q}}^{*}\chi _{6}\left( \widetilde{\eta }\right) \left| v\right| ^{2}v=\left\langle \xi \right\rangle ^{-\frac{1}{2}}A^{*}\chi _{6}\left( \widetilde{\xi }\right) \left| v\right| ^{2}v+O\left( \varepsilon ^{3}t^{-\delta _{1}}\right) \end{aligned}$$

with some \(\delta _{1}>0.\) We use the asymptotics of the kernel \(A^{*}\), so that

$$\begin{aligned}&\left\langle \xi \right\rangle ^{-\frac{1}{2}}A^{*}\chi _{6}\left( \widetilde{\xi }\right) \left| v\right| ^{2}v=\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left| v\right| ^{2}v\\&\qquad +O\left( t^{\frac{1}{2}}\left| \xi \right| ^{\alpha -1}\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left\langle t\left| \xi \right| ^{\alpha }\right\rangle ^{-\frac{3}{2}}\chi _{6}\left( \widetilde{\xi }\right) \left| v\right| ^{2}v\right) +O\left( \left| \xi \right| ^{\frac{\alpha -2}{2}}\left\langle \xi \right\rangle ^{-\frac{1}{2}}\chi _{5}\left( \widetilde{\xi }\right) \left| v\right| ^{2}v\right) \\&\quad =\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left| v\right| ^{2}v+O\left( \varepsilon ^{3}t^{-\delta _{1}}\right) . \end{aligned}$$

Also by Lemma 2.1 and Lemma 2.2 with \(j=1,\) \(\delta =\frac{\alpha -2}{6},\) and by the asymptotics of the kernel \(A_{1}\) we find

$$\begin{aligned} \left| \xi \right| ^{\delta }v&=\frac{i\left| \xi \right| ^{\delta }\xi }{\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }} \widehat{\varphi }+O\left( \varepsilon t^{\frac{1}{2}}\xi ^{2}\left| \xi \right| ^{\delta }\left\langle \widetilde{\xi }\right\rangle ^{-\frac{3}{2}\alpha }\right) +O\left( \varepsilon t^{\frac{1}{2}+\frac{1}{2\alpha }-\min \left( \frac{3}{4},\frac{1}{\alpha }\left( \delta +\frac{5}{2}\right) \right) }\right) \\&=\frac{i\left| \xi \right| ^{\delta }\xi }{\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }}\widehat{\varphi }+O\left( \varepsilon t^{-\min \left( \frac{\alpha -2}{4\alpha },\frac{5-\alpha }{3\alpha }\right) }\right) . \end{aligned}$$

Hence we have

$$\begin{aligned} \left\langle \xi \right\rangle ^{-\frac{1}{2}}{\mathcal {Q}}^{*}\chi _{6}\left( \widetilde{\eta }\right) \left| v\right| ^{2}v&=\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left| \xi \right| ^{-3\delta }\left| \left| \xi \right| ^{\delta }v\right| ^{2}\left| \xi \right| ^{\delta }v+O\left( \varepsilon ^{3}t^{-\delta _{1}}\right) \\&=\frac{i\xi ^{3}}{\Lambda ^{\prime \prime }\left( \xi \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left| \widehat{\varphi }\right| ^{2}\widehat{\varphi }+O\left( \varepsilon ^{3}t^{-\delta _{1}}\right) . \end{aligned}$$

Similarly we get

$$\begin{aligned} \left\langle \xi \right\rangle ^{-\frac{1}{2}}{\mathcal {D}}_{3}{\mathcal {Q}}^{*}\left( 3t\right) v^{3}{=}\left\langle \xi \right\rangle ^{{-}\frac{1}{2} }{\mathcal {D}}_{3}\sqrt{i\left| \Lambda ^{\prime \prime }\left( \xi \right) \right| }v^{3}{=}{-}\left\langle \xi \right\rangle ^{-\frac{1}{2}} {\mathcal {D}}_{3}\frac{\xi ^{3}\theta \left( \xi \right) }{\Lambda ^{\prime \prime }\left( \xi \right) }\widehat{\varphi }^{3}+O\left( \varepsilon ^{3} t^{-\delta _{1}}\right) . \end{aligned}$$

Also \(\left\langle \xi \right\rangle ^{-\frac{1}{2}}\overline{{\mathcal {D}} _{-1}{\mathcal {Q}}^{*}\left( t\right) \left| v\right| ^{2} v}=O\left( \varepsilon ^{3}t^{-\delta _{1}}\right) \) and \(\left\langle \xi \right\rangle ^{-\frac{1}{2}}\overline{{\mathcal {D}}_{-3}{\mathcal {Q}}^{*}\left( 3t\right) v^{3}}=O\left( \varepsilon ^{3}t^{-\delta _{1}}\right) \) since \({\mathcal {D}}_{-1}\theta \left( \xi \right) =0\) in the domain \(\xi >0,\) so that we have

$$\begin{aligned} \left| \xi \right| ^{\delta }{\mathcal {D}}_{-1}v=\left| \xi \right| ^{\delta }{\mathcal {D}}_{-1}\frac{\theta \left( \xi \right) \xi }{\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }}\widehat{\varphi }\left( t,\xi \right) +O\left( \varepsilon t^{-\delta _{1}}\right) =O\left( \varepsilon t^{-\delta _{1}}\right) \end{aligned}$$

and similarly \(\left| \xi \right| ^{\delta }{\mathcal {D}}_{-3}v=O\left( \varepsilon t^{-\delta _{1}}\right) .\) Hence we get the result of the lemma. Lemma 3.1 is proved. \(\square \)

3.2 Estimate for the derivative \(\partial _{\xi }\widehat{\varphi }\)

We have \({\mathcal {Q}}^{*}={\mathcal {Q}}_{5}^{*}+{\mathcal {Q}}_{6}^{*}.\) Define \(\widehat{\varphi }_{1}\) and \(\widehat{\varphi }_{2}\) such that

$$\begin{aligned} \partial _{t}\widehat{\varphi }_{1}=t^{-1}{\mathcal {Q}}_{5}^{*}\overline{M}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}=t^{-1}e^{it\Omega }\Phi _{1}+3t^{-1}\Phi _{2} \end{aligned}$$
(3.1)

and

$$\begin{aligned} \partial _{t}\widehat{\varphi }_{2}=t^{-1}{\mathcal {Q}}_{6}^{*}\overline{M}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}=t^{-1}e^{it\Omega }\Phi _{3}+3t^{-1}\Phi _{4}, \end{aligned}$$
(3.2)

where \(\Phi _{1}={\mathcal {D}}_{3}{\mathcal {Q}}_{5}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}}_{-3}{\mathcal {Q}}_{5}^{*}\left( 3t\right) v^{3}},\) \(\Phi _{2}={\mathcal {Q}}_{5}^{*}\left( t\right) \left| v\right| ^{2}v+\overline{{\mathcal {D}}_{-1}{\mathcal {Q}}_{5}^{*}\left( t\right) \left| v\right| ^{2}v},\) \(\Phi _{3}={\mathcal {D}} _{3}{\mathcal {Q}}_{6}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}} _{-3}{\mathcal {Q}}_{6}^{*}\left( 3t\right) v^{3}},\) \(\Phi _{4} ={\mathcal {Q}}_{6}^{*}\left( t\right) \left| v\right| ^{2}v+\overline{{\mathcal {D}}_{-1}{\mathcal {Q}}_{6}^{*}\left( t\right) \left| v\right| ^{2}v}.\) Then we have \(\widehat{\varphi }=\) \(\widehat{\varphi }_{1}+\widehat{\varphi }_{2},\) since \(v={\mathcal {Q}}\left( i\xi \left( \widehat{\varphi }_{1}+\widehat{\varphi }_{2}\right) \right) .\)

In the next lemma we find a priori estimate of \(\partial _{\xi }\widehat{\varphi }_{1}\left( t,\xi \right) \).

Lemma 3.2

Suppose that \(\left\| u\right\| _{{\textbf{X}}_{T}}\le C\varepsilon .\) Then the estimate \(\left\| \partial _{\xi }\widehat{\varphi }_{1}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}\) is true for all \(t\in \left[ 1,T\right] .\)

Proof

Let us prove the following estimates \(\left\| \partial _{\xi }\widehat{\varphi }_{1}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}\) and \(\left\| \frac{1}{\xi }\widehat{{\mathcal {P}}} \widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }},\) where \(\widehat{{\mathcal {P}}}=\alpha t\partial _{t} -\xi \partial _{\xi }\). Then it follows that

$$\begin{aligned} \left\| \partial _{\xi }\widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}&\le \left\| \frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi } _{2}\right\| _{{\textbf{L}}^{2}}+\alpha \left\| \frac{1}{\xi }t\partial _{t}\widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}\\&\le C\varepsilon t^{\frac{1}{2\alpha }}+C\left\| \frac{1}{\xi }{\mathcal {Q}}_{6}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}, \end{aligned}$$

since by Lemma 2.10 and by the estimate of Corollary 2.1\(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}}\) we have

$$\begin{aligned}&\left\| \frac{1}{\xi }{\mathcal {Q}}_{6}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{3}{2}-\frac{5}{\alpha }}\left\| \left| \eta \right| ^{\frac{\alpha -2}{2}-\nu }\left\langle \widetilde{\eta }\right\rangle ^{-3\frac{\alpha -4}{2}-1}\left\langle \eta \right\rangle ^{\nu -\frac{3}{2}}\right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\qquad +C\varepsilon ^{3}t^{\frac{3}{2}-\frac{5}{\alpha }}\left\| \left| \eta \right| ^{\frac{\alpha -2}{2}}\right\| _{{\textbf{L}}_{\eta } ^{2}\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }}. \end{aligned}$$

Differentiating Eq. (3.1) we get

$$\begin{aligned} \partial _{t}\partial _{\xi }\widehat{\varphi }_{1}=R_{1}+i\Omega ^{\prime }e^{it\Omega }\Phi _{1}, \end{aligned}$$
(3.3)

where \(R_{1}=t^{-1}e^{it\Omega }\partial _{\xi }\Phi _{1}+3t^{-1}\partial _{\xi }\Phi _{2}.\)

The derivative \(\partial _{\xi }{\mathcal {Q}}_{5}^{*}\) was estimated in Lemma 2.9

$$\begin{aligned}&\left\| \partial _{\xi }{\mathcal {Q}}_{5}^{*}v^{3}\right\| _{{\textbf{L}}_{\xi }^{2}}\le Ct^{\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha } }\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }v^{3}\right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\\&\quad +C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \eta \right\} ^{-\nu }\left\langle \eta \right\rangle ^{\nu }\frac{1}{\eta }v^{3}\right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad +C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \eta \right\} ^{-\nu }\left\langle \eta \right\rangle ^{\nu }v^{2}\partial _{\eta }v\right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }. \end{aligned}$$

By Corollary 2.1 we have \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\le C\varepsilon \left| \eta \right| ^{2-\frac{\alpha }{2}}\) in the domain \(\left| \eta \right| \le 2t^{-\frac{1}{\alpha }}.\) Then

$$\begin{aligned} t^{\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha }}\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }v^{3}\right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}t^{\frac{1}{2\alpha } -\frac{5-\alpha }{\alpha }}. \end{aligned}$$

Also by Corollary 2.1 we find \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}}\) in the domain \(\left| \eta \right| \ge t^{-\frac{1}{\alpha }}.\) Hence

$$\begin{aligned}&\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\frac{1}{\eta } v^{3}\right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\varepsilon ^{3}t^{3\frac{\alpha -4}{2\alpha }}\left\| \left| \eta \right| ^{\frac{\alpha -4}{2}-\nu }\left\langle \widetilde{\eta }\right\rangle ^{-3\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{2\nu -\frac{3}{2}}\right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }}, \end{aligned}$$

since \(4\le \alpha <5\). Next we use Lemma 2.8

$$\begin{aligned} \left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{\nu }\left\langle \widetilde{\eta }\right\rangle ^{-\nu }\eta ^{-1} \partial _{\eta }v\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon t^{\frac{1}{2\alpha }} \end{aligned}$$

and Corollary 2.1\(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}},\ \)to estimate

$$\begin{aligned}&\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }v^{2}\partial _{\eta }v\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\left\| \eta \left\{ \widetilde{\eta }\right\} ^{-2\nu }\left\langle \widetilde{\eta }\right\rangle ^{2\nu }v^{2}\right\| _{{\textbf{L}}^{\infty }\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{\nu }\left\langle \widetilde{\eta }\right\rangle ^{-\nu }\eta ^{-1}\partial _{\eta }v\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\varepsilon ^{3}t^{\frac{1}{2\alpha }+\frac{\alpha -4}{\alpha } }\left\| \left\{ \eta \right\} ^{1-2\nu }\left\langle \widetilde{\eta }\right\rangle ^{4-\alpha }\right\| _{{\textbf{L}}^{\infty }\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }}, \end{aligned}$$

since \(4\le \alpha <5\) and \(\nu >0\) is small. Therefore we obtain\(\left\| R_{1}\right\| _{{\textbf{L}}_{\xi }^{2}}\le Ct^{-1}\left\| \partial _{\xi }\Phi _{1}\right\| _{{\textbf{L}}_{\xi }^{2}}+Ct^{-1}\left\| \partial _{\xi }\Phi _{2}\right\| _{{\textbf{L}}_{\xi }^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) We need to transform the last term \(i\Omega ^{\prime }e^{it\Omega }\Phi _{1}\) in Eq. (3.3). Denote

$$\begin{aligned} {\mathcal {Q}}_{7}^{*}\phi =\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \chi _{3}\left( \widetilde{\eta }\right) \chi _{3}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta \end{aligned}$$

and

$$\begin{aligned} {\mathcal {Q}}_{8}^{*}\phi =\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \left( \chi _{4}\left( \widetilde{\eta }\right) \chi _{3}\left( \frac{\eta }{\xi }\right) +\chi _{4}\left( \widetilde{\xi }\right) \chi _{4}\left( \frac{\eta }{\xi }\right) \right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta . \end{aligned}$$

So that \({\mathcal {Q}}_{5}^{*}={\mathcal {Q}}_{7}^{*}+{\mathcal {Q}}_{8}^{*}\) and \(\Phi _{1}=\Phi _{5}+\Phi _{6},\) where \(\Phi _{5}={\mathcal {D}}_{3} {\mathcal {Q}}_{7}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}} _{-3}{\mathcal {Q}}_{7}^{*}\left( 3t\right) v^{3}},\) \(\Phi _{6} ={\mathcal {D}}_{3}{\mathcal {Q}}_{8}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}}_{-3}{\mathcal {Q}}_{8}^{*}\left( 3t\right) v^{3}}.\) The term \(R_{2}=i\Omega ^{\prime }e^{it\Omega }\Phi _{6}\) can be estimated easily as follows

$$\begin{aligned} \left\| R_{2}\right\| _{{\textbf{L}}_{\xi }^{2}}&\le Ct^{\frac{1}{\alpha }-\frac{1}{2}}\left\| \int _{\left| \eta \right| \le 9t^{-\frac{1}{\alpha }}}\left| v\left( \eta \right) \right| ^{3}\left| \eta \right| ^{\alpha -2}d\eta \right\| _{{\textbf{L}}_{\xi }^{2}\left( \left| \xi \right| \le 9t^{-\frac{1}{\alpha }}\right) }\\&\le Ct^{\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha }-1}\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }v^{3}\right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}. \end{aligned}$$

Next we represent

$$\begin{aligned} i\Omega ^{\prime }e^{it\Omega }\Phi _{5}&=i\Omega ^{\prime }e^{it\Omega }\left( {\mathcal {D}}_{3}{\mathcal {Q}}_{7}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}}_{-3}{\mathcal {Q}}_{7}^{*}\left( 3t\right) v^{3}}\right) \\&=i\Omega ^{\prime }{\mathcal {Q}}_{7}^{*}\left( t\right) M^{2}v^{3} +i\Omega ^{\prime }{\mathcal {Q}}_{7}^{*}\left( t\right) {\overline{M}} ^{4}{\overline{v}}^{3}. \end{aligned}$$

We have

$$\begin{aligned} i\Omega ^{\prime }{\mathcal {Q}}_{7}^{*}\left( t\right) M^{2}v^{3} =\partial _{t}\left( {\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta } v^{3}\right) -R_{3} \end{aligned}$$

and

$$\begin{aligned} i\Omega ^{\prime }{\mathcal {Q}}_{7}^{*}\left( t\right) {\overline{M}} ^{4}{\overline{v}}^{3}=\partial _{t}\left( {\mathcal {V}}_{h_{6}}^{*} {\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}\right) -R_{4}, \end{aligned}$$

where \(h_{5}\left( t,\xi ,\eta \right) =\frac{i\eta \Omega ^{\prime }\left( \xi \right) }{G_{1}\left( \xi ,\eta \right) }\chi _{3}\left( \widetilde{\eta }\right) \chi _{3}\left( \frac{\eta }{\xi }\right) \) and \(h_{6}\left( t,\xi ,\eta \right) =\frac{i\eta \Omega ^{\prime }\left( \xi \right) }{G_{2}\left( \xi ,\eta \right) }\chi _{3}\left( \widetilde{\eta }\right) \chi _{3}\left( \frac{\eta }{\xi }\right) ,\) with \(G_{1}\left( \xi ,\eta \right) =S\left( \xi ,\eta \right) -2i\left( \frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\left| \eta \right| \Lambda ^{\prime }\left( \eta \right) \right) \) and

\(G_{2}\left( \xi ,\eta \right) =S\left( \xi ,\eta \right) +4i\left( \frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\left| \eta \right| \Lambda ^{\prime }\left( \eta \right) \right) ,\) also we denote

$$\begin{aligned} R_{3}&=\frac{1}{2t}{\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta } v^{3}+3{\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta }v^{2}\partial _{t}v,\\ R_{4}&=\frac{1}{2t}{\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}+3{\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{2}\partial _{t}{\overline{v}}. \end{aligned}$$

Note that

$$\begin{aligned}&G_{1}\left( \xi ,\eta \right) =S\left( \xi ,\eta \right) -2\left( \frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\left| \eta \right| \Lambda ^{\prime }\left( \eta \right) \right) \\&\quad =\Lambda \left( \xi \right) -\frac{\eta }{\left| \eta \right| }\Lambda ^{\prime }\left( \eta \right) \xi -3\frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) +3\left| \eta \right| \Lambda ^{\prime }\left( \eta \right) \\&\quad =\left| \eta \right| ^{\alpha }\left( 3-\frac{3}{\alpha }-\frac{\xi }{\eta }\left( 1-\frac{1}{\alpha }\left| \frac{\xi }{\eta }\right| ^{\alpha -1}\right) \right) \ge 2\left| \eta \right| ^{\alpha } \end{aligned}$$

in the domain \(\left| \eta \right| \ge 2\left| \xi \right| .\) Similarly \(G_{1}\left( \xi ,\eta \right) \ge 2\left| \eta \right| ^{\alpha }\) in the domain \(\left| \eta \right| \ge 2\left| \xi \right| .\)Therefore \(h_{5}\left( t,\xi ,\eta \right) =O\left( 1\right) \) and \(h_{6}\left( t,\xi ,\eta \right) =O\left( 1\right) .\) Then as in Lemma 2.10 we get

$$\begin{aligned}&\left\| {\mathcal {V}}_{h_{k}}^{*}\phi \right\| _{{\textbf{L}}_{\xi } ^{2}}\le C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu } \phi \right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }+C\left\| \sqrt{\Lambda ^{\prime \prime }} \phi \right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu } \phi \right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }+Ct^{\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha } }\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }\phi \right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) } \end{aligned}$$

for \(k=5,6.\) Thus

$$\begin{aligned}&\left\| {\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta }v^{3}\right\| _{{\textbf{L}}_{\xi }^{2}}+\left\| {\mathcal {V}}_{h_{6}}^{*}{\overline{M}} ^{4}\frac{1}{\eta }{\overline{v}}^{3}\right\| _{{\textbf{L}}_{\xi }^{2}}\\&\quad \le C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\frac{1}{\eta }v^{3}\right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }+Ct^{\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha }-1}\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }v^{3}\right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}, \end{aligned}$$

since by Corollary 2.1 we have \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\) in the domain \(\left| \eta \right| \le 2t^{-\frac{1}{\alpha }}.\) We have

$$\begin{aligned} \left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }v^{3}\right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}. \end{aligned}$$

Also by Corollary 2.1 we find \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}}\) in the domain \(\left| \eta \right| \ge t^{-\frac{1}{\alpha }}.\) Hence

$$\begin{aligned}&\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \eta \right\} ^{-\nu }\left\langle \eta \right\rangle ^{\nu }\frac{1}{\eta }v^{3}\right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) } \\&\quad \le C\varepsilon ^{3}t^{3\frac{\alpha -4}{2\alpha } }\left\| \left| \eta \right| ^{\frac{\alpha -4}{2}-\nu }\left\langle \widetilde{\eta }\right\rangle ^{-3\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{2\nu -\frac{3}{2}}\right\| _{{\textbf{L}}_{\eta } ^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\quad \le C\varepsilon ^{3}t^{\frac{1}{2\alpha }}. \end{aligned}$$

Therefore we get

$$\begin{aligned} \partial _{t}\partial _{\xi }\widehat{\varphi }_{1}&=R_{1}+R_{2} +i\Omega ^{\prime }e^{it\Omega }\Phi _{5}\\&=R_{1}+R_{2}+\partial _{t}\left( {\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta }v^{3}+{\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}\right) -R_{3}-R_{4}. \end{aligned}$$

Denoting \(y_{1}=\) \(\partial _{\xi }\widehat{\varphi }_{1}-{\mathcal {V}}_{h_{5} }^{*}M^{2}\frac{1}{\eta }v^{3}-{\mathcal {V}}_{h_{6}}^{*}{\overline{M}} ^{4}\frac{1}{\eta }{\overline{v}}^{3},\) we find

$$\begin{aligned} \partial _{t}y_{1}=R_{1}+R_{2}-R_{3}-R_{4}. \end{aligned}$$

To estimate \(\partial _{t}v\) we write \(\alpha t\partial _{t}v=\widehat{{\mathcal {P}}}v+\eta \partial _{\eta }v.\) Then we obtain

$$\begin{aligned} \widehat{{\mathcal {P}}}v&=\widehat{{\mathcal {P}}}{\mathcal {Q}}\left( i\xi \widehat{\varphi }\right) ={\mathcal {Q}}\left( \widehat{{\mathcal {P}}} i\xi \widehat{\varphi }\right) +\frac{\alpha -2}{2}v\\&=i{\mathcal {A}}_{1}^{2}{\mathcal {Q}}\partial _{\xi }\widehat{\varphi } +\frac{\alpha -4}{2}v+\alpha {\mathcal {A}}_{1}{\mathcal {Q}}\left( {\mathcal {Q}} ^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right) \\&=i{\mathcal {A}}_{1}^{2}{\mathcal {Q}}\partial _{\xi }\widehat{\varphi } +\frac{\alpha -4}{2}v+\alpha {\mathcal {A}}_{1}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}. \end{aligned}$$

Hence \(\alpha t\partial _{t}v=\eta \partial _{\eta }v+\frac{\alpha -4}{2}v+i{\mathcal {A}}_{1}^{2}{\mathcal {Q}}\partial _{\xi }\widehat{\varphi } +\alpha {\mathcal {A}}_{1}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\) and also \({\mathcal {A}}_{1}{\overline{M}}\left( Mv+{\overline{M}}\overline{v}\right) ^{3}=3\left( Mv+{\overline{M}}{\overline{v}}\right) ^{2}\left( {\mathcal {A}}_{1}v+{\overline{M}}^{2}\overline{{\mathcal {A}}_{1}v}\right) .\) Then

$$\begin{aligned}&3{\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta }v^{2}\partial _{t}v=\frac{3}{\alpha t}{\mathcal {V}}_{h_{5}}^{*}M^{2}v^{2}\partial _{\eta }v+\frac{\alpha -4}{2}\frac{3}{\alpha t}{\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta }v^{3}\\&\qquad +\frac{3}{t}{\mathcal {V}}_{h_{5}}^{*}\frac{1}{\eta }\left( Mv\right) ^{2}{\mathcal {A}}_{1}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}+\frac{3i}{\alpha t}{\mathcal {V}}_{h_{5}}^{*}\frac{1}{\eta }\left( Mv\right) ^{2}{\mathcal {A}}_{1}^{2}{\mathcal {Q}}\partial _{\xi }\widehat{\varphi }. \end{aligned}$$

The last term can be represented as

$$\begin{aligned}&{\mathcal {V}}_{h_{5}}^{*}\frac{1}{\eta }\left( Mv\right) ^{2} {\mathcal {A}}_{1}^{2}{\mathcal {Q}}\partial _{\xi }\widehat{\varphi }\\&\quad =\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }\left( i\xi \right) ^{2}h_{5}\left( t,\xi ,\eta \right) \frac{1}{\eta } \chi _{3}\left( \frac{\eta }{\xi }\right) \left( Mv\right) ^{2}\left( {\mathcal {Q}}\partial _{\xi }\widehat{\varphi }\right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta \\&\qquad +\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }2\left( i\xi \right) {\mathcal {A}}_{0}\left( h_{5}\left( t,\xi ,\eta \right) \frac{1}{\eta }\chi _{3}\left( \frac{\eta }{\xi }\right) \left( Mv\right) ^{2}\right) \left( {\mathcal {Q}}\partial _{\xi }\widehat{\varphi }\right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta \\&\qquad +\sqrt{\frac{t}{2\pi }}\int _{{\mathbb {R}}}e^{itS\left( \xi ,\eta \right) }{\mathcal {A}}_{0}^{2}\left( h_{5}\left( t,\xi ,\eta \right) \frac{1}{\eta } \chi _{3}\left( \frac{\eta }{\xi }\right) \left( Mv\right) ^{2}\right) \left( {\mathcal {Q}}\partial _{\xi }\widehat{\varphi }\right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta . \end{aligned}$$

Therefore we get

$$\begin{aligned}&\left\| {\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta }v^{2}\partial _{t}v\right\| _{{\textbf{L}}_{\xi }^{2}}+\left\| {\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{2}\partial _{t}\overline{v}\right\| _{{\textbf{L}}_{\xi }^{2}}\\&\quad \le Ct^{-1}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\eta v^{2}\frac{1}{\eta }\partial _{\eta }v\right\| _{{\textbf{L}}_{\eta }^{2} }+Ct^{-1}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\eta v^{2}{\mathcal {Q}}\partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}_{\eta }^{2}}\\&\qquad +Ct^{-1}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\frac{1}{\eta }v^{3}\right\| _{{\textbf{L}}_{\eta }^{2}}+Ct^{-1}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\frac{1}{\eta } v^{4}{\mathcal {A}}_{1}v\right\| _{{\textbf{L}}_{\eta }^{2}}\\&\quad \le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}. \end{aligned}$$

Hence it follows that \(\left\| \partial _{t}y_{1}\right\| _{{\textbf{L}} _{\xi }^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) Integrating we find \(\left\| y_{1}\right\| _{{\textbf{L}}_{\xi }^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }}.\) Thus

$$\begin{aligned} \left\| \partial _{\xi }\widehat{\varphi }_{1}\right\| _{{\textbf{L}}_{\xi }^{2}}\le \left\| {\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta } v^{3}\right\| _{{\textbf{L}}_{\xi }^{2}}+\left\| {\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}\right\| _{{\textbf{L}}_{\xi }^{2}}+C\varepsilon ^{3}t^{\frac{1}{2\alpha }}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }}. \end{aligned}$$

Lemma 3.2 is proved. \(\square \)

Next we estimate \(\partial _{\xi }\widehat{\varphi }_{2}\left( t,\xi \right) \).

Lemma 3.3

Suppose that \(\left\| u\right\| _{{\textbf{X}}_{T}}\le C\varepsilon .\) Then the estimate \(\left\| \partial _{\xi }\widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}\) is true for all \(t\in \left[ 1,T\right] .\)

Proof

Let us prove the following estimate \(\left\| \frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }},\) where \(\widehat{{\mathcal {P}}}=\alpha t\partial _{t}-\xi \partial _{\xi }\). Then it follows that

$$\begin{aligned} \left\| \partial _{\xi }\widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}&\le \left\| \frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi } _{2}\right\| _{{\textbf{L}}^{2}}+\alpha \left\| \frac{1}{\xi }t\partial _{t}\widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}\\&\le C\varepsilon t^{\frac{1}{2\alpha }}+C\left\| \frac{1}{\xi }{\mathcal {Q}}_{6}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}, \end{aligned}$$

since by Lemma 2.10 and by the estimate of Corollary 2.1\(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}}\) we have

$$\begin{aligned}&\left\| \frac{1}{\xi }{\mathcal {Q}}_{6}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{3}{2}-\frac{5}{\alpha }}\left\| \left| \eta \right| ^{\frac{\alpha -2}{2}-\nu }\left\langle \widetilde{\eta }\right\rangle ^{-3\frac{\alpha -4}{2}-1}\left\langle \eta \right\rangle ^{\nu -\frac{3}{2}}\right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\qquad +C\varepsilon ^{3}t^{\frac{3}{2}-\frac{5}{\alpha }}\left\| \left| \eta \right| ^{\frac{\alpha -2}{2}}\right\| _{{\textbf{L}}_{\eta } ^{2}\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }}. \end{aligned}$$

Thus we apply the operator \(\frac{1}{\xi }\widehat{{\mathcal {P}}}\), with \(\widehat{{\mathcal {P}}}=\alpha t\partial _{t}-\xi \partial _{\xi },\) to Eq. (3.2). Then we get

$$\begin{aligned} \partial _{t}\frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi }_{2}=\frac{1}{\xi }\widehat{{\mathcal {P}}}t^{-1}{\mathcal {Q}}_{6}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}=\frac{1}{\xi t}\left( \widehat{{\mathcal {P}}}-\alpha \right) {\mathcal {Q}}_{6}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}. \end{aligned}$$

Note that \(\left( \alpha t\partial _{t}-\xi \partial _{\xi }-\eta \partial _{\eta }\right) e^{itS\left( \xi ,\eta \right) }=0,\) therefore we have the commutator\(\left[ \widehat{{\mathcal {P}}},{\mathcal {Q}}_{6}^{*}\right] =\frac{2-\alpha }{2}{\mathcal {Q}}_{6}^{*}.\) Similarly, we find \(\left[ \widehat{{\mathcal {P}}},{\mathcal {Q}}\right] =\frac{\alpha -2}{2}{\mathcal {Q}}.\) Then we obtain

$$\begin{aligned} \partial _{t}\frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi }_{2}&=\frac{1}{\xi t}{\mathcal {Q}}_{6}^{*}\widehat{{\mathcal {P}}}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}+R_{5}\\&=\frac{3}{\xi t}{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{2}\left( M\widehat{{\mathcal {P}} }v+{\overline{M}}\overline{\widehat{{\mathcal {P}}}v}\right) \right) +R_{5}, \end{aligned}$$

where \(R_{5}=\frac{2-3\alpha }{2}\frac{1}{\xi t}{\mathcal {Q}}_{6}^{*} {\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\) has the estimate similarly to the above

$$\begin{aligned} \left\| R_{5}\right\| _{{\textbf{L}}^{2}}=Ct^{-1}\left\| \frac{1}{\xi }{\mathcal {Q}}_{6}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha } -1}. \end{aligned}$$

Denote \(y=\frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi }_{2}-y_{1},\) \(y_{1}=\) \(\partial _{\xi }\widehat{\varphi }_{1}-{\mathcal {V}}_{h_{5}}^{*} M^{2}\frac{1}{\eta }v^{3}-{\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}\), then since \(\widehat{{\mathcal {P}}}=\alpha t\partial _{t}-\xi \partial _{\xi }\), we get

$$\begin{aligned} \widehat{{\mathcal {P}}}v&=\widehat{{\mathcal {P}}}{\mathcal {Q}}\left( i\xi \widehat{\varphi }\right) ={\mathcal {Q}}\left( \widehat{{\mathcal {P}}} i\xi \widehat{\varphi }\right) +\frac{\alpha -2}{2}v\\&=-i{\mathcal {A}}_{1}^{2}{\mathcal {Q}}y+\frac{\alpha -4}{2}v+\alpha {\mathcal {A}}_{1}{\mathcal {Q}}\left( {\mathcal {Q}}_{5}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right) \\&\quad -{\mathcal {A}}_{1}{\mathcal {Q}}\left( \xi {\mathcal {V}}_{h_{5}}^{*}M^{2} \frac{1}{\eta }v^{3}+\xi {\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}\right) =-i{\mathcal {A}}_{1}^{2}{\mathcal {Q}}y+Y, \end{aligned}$$

where \(Y=\frac{\alpha -4}{2}v+\alpha {\mathcal {A}}_{1}{\mathcal {Q}}\left( {\mathcal {Q}}_{5}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right) \) \(-{\mathcal {A}}_{1}{\mathcal {Q}}\left( \xi {\mathcal {V}}_{h_{5} }^{*}M^{2}\frac{1}{\eta }v^{3}+\xi {\mathcal {V}}_{h_{6}}^{*}{\overline{M}} ^{4}\frac{1}{\eta }{\overline{v}}^{3}\right) .\) Then we obtain

$$\begin{aligned} \partial _{t}\frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi }_{2} =-\frac{12i}{\xi t}{\mathcal {Q}}_{6}^{*}\left( \left( {\text {Re}} Mv\right) ^{2}{\mathcal {A}}_{1}^{2}{\mathcal {Q}}y\right) +\frac{12i}{\xi t}{\mathcal {Q}}_{6}^{*}\left( \left( {\text {Re}}Mv\right) ^{2}{\overline{M}}^{2}\overline{{\mathcal {A}}_{1}^{2}{\mathcal {Q}}y}\right) +R_{5}+R_{6}, \end{aligned}$$

where \(R_{6}=\frac{24}{\xi t}{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}}Mv\right) ^{2}\left( {\text {Re}}MY\right) \right) .\) We represent

$$\begin{aligned} R_{6}=\left( \alpha -4\right) \frac{12}{\xi t}{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}}Mv\right) ^{3}\right) +\frac{24}{\xi t}{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}} Mv\right) ^{2}\left( {\text {Re}}M{\mathcal {A}}_{1}Z\right) \right) , \end{aligned}$$

where we denote

$$\begin{aligned} Z=\alpha {\mathcal {Q}}\left( {\mathcal {Q}}_{5}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right) -{\mathcal {Q}}\left( \xi {\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta }v^{3}+\xi {\mathcal {V}}_{h_{6} }^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}\right) . \end{aligned}$$

As above we find \(\left\| \frac{1}{\xi t}{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}}Mv\right) ^{3}\right) \right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) By Lemma 2.10 we have the estimate

$$\begin{aligned}&\left\| \xi ^{j}{\mathcal {Q}}_{5}^{*}\eta ^{-j}\phi \right\| _{{\textbf{L}}_{\xi }^{2}}+\left\| \frac{1}{\widetilde{\xi }^{j}} {\mathcal {Q}}_{6}^{*}\left\langle \widetilde{\eta }\right\rangle ^{j} \phi \right\| _{{\textbf{L}}_{\xi }^{2}}\\&\quad \le C\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \eta \right\} ^{-\nu }\left\langle \eta \right\rangle ^{\nu }\phi \right\| _{{\textbf{L}} _{\eta }^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha } }\right) }+C\left\| \sqrt{\Lambda ^{\prime \prime }}\phi \right\| _{{\textbf{L}}_{\eta }^{2}\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) } \end{aligned}$$

and by Corollary 2.1\(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}},\) then we get

$$\begin{aligned}&\left\| \frac{24}{\xi t}{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}}Mv\right) ^{2}\left( {\text {Re}} M{\mathcal {A}}_{1}Z\right) \right) \right\| _{{\textbf{L}}^{2}}\le Ct^{-1}\left\| {\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}}Mv\right) ^{2}\left( {\text {Re}}MZ\right) \right) \right\| _{{\textbf{L}}^{2}}\\&\qquad +Ct^{-1}\left\| \frac{1}{\xi }{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}}Mv\right) \left( {\text {Re}} M{\mathcal {A}}_{1}v\right) \left( {\text {Re}}MZ\right) \right) \right\| _{{\textbf{L}}^{2}}\le Ct^{\frac{\nu +1}{\alpha }-1}\left\| \sqrt{\Lambda ^{\prime \prime }}Z\right\| _{{\textbf{L}}_{\eta }^{2}}\\&\quad \le Ct^{\frac{\nu +1}{\alpha }-1}\left\| {\mathcal {Q}}_{5}^{*} {\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right\| _{{\textbf{L}}^{2}}+Ct^{\frac{\nu +1}{\alpha }-1}\left\| \xi {\mathcal {V}}_{h_{5} }^{*}M^{2}\frac{1}{\eta }v^{3}+\xi {\mathcal {V}}_{h_{6}}^{*}{\overline{M}} ^{4}\frac{1}{\eta }{\overline{v}}^{3}\right\| _{{\textbf{L}}_{\eta }^{2}}. \end{aligned}$$

Also using Lemma 2.11 we find

$$\begin{aligned}&\left\| {\mathcal {Q}}_{5}^{*}\phi \right\| _{{\textbf{L}}_{\xi }^{2} }+\left\| \xi {\mathcal {V}}_{h_{5}}^{*}M^{2}\frac{1}{\eta }\phi \right\| _{{\textbf{L}}_{\xi }^{2}}+\left\| \xi {\mathcal {V}}_{h_{6}}^{*}\overline{M}^{4}\frac{1}{\eta }\phi \right\| _{{\textbf{L}}_{\xi }^{2}}\\&\quad \le Ct^{-1}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left| \eta \right| ^{1-\alpha }\partial _{\eta }\phi \right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha } }\right) }\\&\qquad +Ct^{-1}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left| \eta \right| ^{-\alpha }\phi \right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\qquad +Ct^{-\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha }}\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }\phi \right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }. \end{aligned}$$

Then we obtain

$$\begin{aligned}&Ct^{\frac{\nu +1}{\alpha }-1}\left\| {\mathcal {Q}}_{5}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right\| _{{\textbf{L}} ^{2}}+Ct^{\frac{\nu +1}{\alpha }-1}\left\| \xi {\mathcal {V}}_{h_{5}}^{*} M^{2}\frac{1}{\eta }v^{3}+\xi {\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}\right\| _{{\textbf{L}}_{\eta }^{2}}\\&\quad \le C\varepsilon ^{2}t^{\frac{\nu +1}{\alpha }-2}\left\| \left\langle \widetilde{\eta }\right\rangle ^{2\nu }\left| \eta \right| ^{2-\alpha }v^{2}\right\| _{{\textbf{L}}^{\infty }\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\left\| \sqrt{\Lambda ^{\prime \prime } }\left\{ \widetilde{\eta }\right\} ^{\nu }\left\langle \widetilde{\eta }\right\rangle ^{-\nu }\eta ^{-1}\partial _{\eta }v\right\| _{{\textbf{L}} ^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\qquad +C\varepsilon ^{3}t^{\frac{\nu +1}{\alpha }-2}\left\| \sqrt{\Lambda ^{\prime \prime }}\left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left| \eta \right| ^{-\alpha } v^{3}\right\| _{{\textbf{L}}^{2}\left( \left| \eta \right| \ge t^{-\frac{1}{\alpha }}\right) }\\&\qquad +C\varepsilon ^{3}t^{\frac{\nu +1}{\alpha }-1-\frac{1}{2\alpha }-\frac{5-\alpha }{\alpha }}\left\| \left| \eta \right| ^{3\left( \frac{\alpha }{2}-2\right) }v^{3}\right\| _{{\textbf{L}}_{\eta }^{\infty }\left( \left| \eta \right| \le t^{-\frac{1}{\alpha }}\right) }\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}. \end{aligned}$$

Hence \(\left\| R_{6}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) Finally, we transform

$$\begin{aligned}&\qquad -\frac{12i}{\xi t}{\mathcal {Q}}_{6}^{*}\left( \left( {\text {Re}} Mv\right) ^{2}{\mathcal {A}}_{1}^{2}{\mathcal {Q}}y\right) +\frac{12i}{\xi t}{\mathcal {Q}}_{6}^{*}\left( \left( {\text {Re}}Mv\right) ^{2}{\overline{M}}^{2}\overline{{\mathcal {A}}_{1}^{2}{\mathcal {Q}}y}\right) \\&\qquad \quad =-\frac{12i}{\xi t}{\mathcal {Q}}^{*}\left( \left( {\text {Re}} Mv\right) ^{2}\left( {\mathcal {Q}}y-{\overline{M}}^{2}\overline{{\mathcal {Q}} y}\right) \right) +\sum _{j=7}^{10}R_{j}, \end{aligned}$$

where

$$\begin{aligned} R_{7}&=-\frac{12i}{\xi t}{\mathcal {Q}}_{5}^{*}\eta \left( \eta \left( {\text {Re}}Mv\right) ^{2}\left( {\mathcal {Q}}y-{\overline{M}}^{2} \overline{{\mathcal {Q}}y}\right) \right) ,\\ R_{8}&=-\frac{12}{\xi t}{\mathcal {Q}}^{*}\left[ {\mathcal {A}}_{0}^{2} \chi _{4}\left( \frac{\eta }{\xi }\right) \right] \left( \left( {\text {Re}}Mv\right) ^{2}\left( {\mathcal {Q}}y-{\overline{M}}^{2} \overline{{\mathcal {Q}}y}\right) \right) \\&\quad \ -\frac{24i}{t}{\mathcal {Q}}^{*}\left[ {\mathcal {A}}_{0}\chi _{4}\left( \frac{\eta }{\xi }\right) \right] \left( \left( {\text {Re}}Mv\right) ^{2}\left( {\mathcal {Q}}y-{\overline{M}}^{2}\overline{{\mathcal {Q}}y}\right) \right) ,\\ R_{9}&=-\frac{24}{t}{\mathcal {Q}}_{6}^{*}\left( \left( {\mathcal {A}} _{0}\left( {\text {Re}}Mv\right) ^{2}\right) \left( {\mathcal {Q}} y-{\overline{M}}^{2}\overline{{\mathcal {Q}}y}\right) \right) ,\\ R_{10}&=-24\frac{t^{\frac{1}{\alpha }-1}}{\widetilde{\xi }}{\mathcal {Q}} _{6}^{*}\left\langle \widetilde{\eta }\right\rangle \left( \left\langle \widetilde{\eta }\right\rangle ^{-1}\left( {\text {Re}}\left( Mv\right) \left( M{\mathcal {A}}_{1}^{2}v\right) +{\text {Re}}\left( M{\mathcal {A}} _{1}v\right) ^{2}\right) \right. \\&\left. \left( {\mathcal {Q}}y-{\overline{M}}^{2} \overline{{\mathcal {Q}}y}\right) \right) . \end{aligned}$$

Using estimate of Corollary 2.1\(\left\langle \widetilde{\eta }\right\rangle ^{-1}\left| v\right| \left| {\mathcal {A}}_{1} ^{2}v\right| \le C\varepsilon ^{2},\) as above we have the estimates \(\sum _{j=7}^{10}\left\| R_{j}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) Thus we get the equation

$$\begin{aligned} \partial _{t}y=-12t^{-1}i\xi {\mathcal {Q}}^{*}\left( \left( {\text {Re}} Mv\right) ^{2}{\mathcal {Q}}y\right) +12t^{-1}i\xi {\mathcal {Q}}^{*}\left( \left( {\text {Re}}Mv\right) ^{2}{\overline{M}}^{2}\overline{{\mathcal {Q}}y}\right) +R, \end{aligned}$$

where the remainder term has the estimate \(\left\| R\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) Multiplying the above equation by \({\overline{y}}\), taking the real part and integrating over \({\mathbb {R}}\) we get

$$\begin{aligned}&\frac{d}{dt}\left\| y\right\| _{{\textbf{L}}^{2}}^{2}=12t^{-1} {\text {Re}}i\int _{{\mathbb {R}}}\xi {\overline{y}}{\mathcal {Q}}^{*}\left( \left( {\text {Re}}Mv\right) ^{2}{\overline{M}}^{2}\overline{{\mathcal {Q}}y}\right) d\xi \\&\qquad -12t^{-1}{\text {Re}}i\int _{{\mathbb {R}}}\xi {\overline{y}}{\mathcal {Q}} ^{*}\left( \left( {\text {Re}}Mv\right) ^{2}{\mathcal {Q}}y\right) d\xi +{\text {Re}}\int _{{\mathbb {R}}}{\overline{y}}Rd\xi . \end{aligned}$$

Note that

$$\begin{aligned}&{\text {Re}}i\int _{{\mathbb {R}}}\xi {\overline{y}}{\mathcal {Q}}^{*}\left( \left( {\text {Re}}Mv\right) ^{2}{\overline{M}}^{2}\overline{{\mathcal {Q}}y}\right) d\xi \\&\quad ={\text {Re}}i\int _{{\mathbb {R}}}\left( \overline{M{\mathcal {Q}}y}\right) \left( \left( {\text {Re}}Mv\right) ^{2}{\mathcal {A}}_{0}\overline{M{\mathcal {Q}}y}\right) \Lambda ^{\prime \prime }d\eta \\&\quad =-\frac{1}{2}{\text {Re}}i\int _{{\mathbb {R}}}\left( \overline{M{\mathcal {Q}}y}\right) ^{2}{\mathcal {A}}_{0}\left( {\text {Re}}Mv\right) ^{2}\Lambda ^{\prime \prime }d\eta \le C\left\| v{\mathcal {A}}_{1}v\right\| _{{\textbf{L}}^{\infty }}\left\| \sqrt{\Lambda ^{\prime \prime }}{\mathcal {Q}} y\right\| _{{\textbf{L}}^{2}}^{2}\\&\quad \le C\varepsilon ^{2}\left\| y\right\| _{{\textbf{L}}^{2}}^{2} \end{aligned}$$

and

$$\begin{aligned}&{\text {Re}}i\int _{{\mathbb {R}}}\xi {\overline{y}}{\mathcal {Q}}^{*}\left( \left( {\text {Re}}Mv\right) ^{2}{\mathcal {Q}}y\right) d\xi =-{\text {Re}}\int _{{\mathbb {R}}}\left( {\text {Re}}Mv\right) ^{2}\left( \overline{{\mathcal {Q}}i\xi y}\right) \left( {\mathcal {Q}}y\right) \Lambda ^{\prime \prime }d\eta \\&\quad {=}-\frac{1}{2}\int _{{\mathbb {R}}}\left| {\mathcal {Q}}y\right| ^{2}{\mathcal {A}}_{0}\left( {\text {Re}}Mv\right) ^{2}\Lambda ^{\prime \prime }d\eta {\le } C\left\| \sqrt{\Lambda ^{\prime \prime }} {\mathcal {Q}}y\right\| _{{\textbf{L}}^{2}}^{2}\left\| v{\mathcal {A}} _{1}v\right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}\left\| y\right\| _{{\textbf{L}}^{2}}^{2}, \end{aligned}$$

since by Corollary 2.1 we have \(\left\| v{\mathcal {A}} _{1}v\right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}.\) Thus we find

$$\begin{aligned} \frac{d}{dt}\left\| y\right\| _{{\textbf{L}}^{2}}^{2}\le C\varepsilon ^{2}t^{-1}\left\| y\right\| _{{\textbf{L}}^{2}}^{2}+C\varepsilon ^{4}t^{\frac{1}{2\alpha }-1}. \end{aligned}$$

Integrating in time we obtain the estimate of the lemma. Lemma 3.3 is proved. \(\square \)

3.3 A-priori estimates of local solutions

We first state the local existence of solutions to the Cauchy problem (1.2) which can be obtained by the classical energy method (see [14, 15]).

Theorem 3.1

Assume that the initial data \(u_{0}\in {\textbf{H}}^{3} \cap {\textbf{H}}^{2,1},\) and the norm \(\varepsilon =\left\| u_{0}\right\| _{{\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}}\) is sufficiently small. Then there exists a time \(T>1\) such that the Cauchy problem (1.2) has a unique solution \(u\in {\textbf{C}}\left( \left[ 0,T\right] ;{\textbf{H}}^{3} \cap {\textbf{H}}^{2,1}\right) \) such that \(\left\| u\right\| _{{\textbf{X}}_{T}}\le C\varepsilon .\)

To prove the global result, we need a priori estimate of the norm \(\left\| u\right\| _{{\textbf{X}}_{T}}\) uniformly with respect to \(T\ge 1.\)

Lemma 3.4

Let the initial data \(u_{0}\in {\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}\) have a small norm \(\left\| u_{0}\right\| _{{\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}}\). Then the estimate \(\left\| u\right\| _{{\textbf{X}}_{T}}<C\varepsilon \) is true for all \(T\ge 1.\)

Proof

Arguing by the contradiction, we can find a time interval \(T\ge 1\) such that \(\left\| u\right\| _{{\textbf{X}}_{T}}=C\varepsilon .\) We need the estimate the norm \(\left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{\infty }}.\) In the domain \(\left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\), by the Sobolev embedding inequality we get

$$\begin{aligned}&\left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{\infty }\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }\le C\left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2} }\left\| \partial _{\xi }\left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2}}\\&\quad \le C\left\langle t\right\rangle ^{-\frac{\nu }{2}}\left\| \left\langle \xi \right\rangle ^{3}\widehat{\varphi }\right\| _{{\textbf{L}}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2}}\left\| \partial _{\xi }\left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2} }<C\varepsilon \left\langle t\right\rangle ^{-\frac{\nu }{2}+\gamma }<C\varepsilon \end{aligned}$$

if \(\frac{\nu }{2}>\gamma .\) Therefore we need to estimate \(\left\langle \xi \right\rangle ^{2}\widehat{\varphi }\left( t,\xi \right) \) in the domain \(\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }.\) Applying the operator \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \) to Eq. (1.2), we get (2.1). Then by virtue of Lemma 3.1 since \(\left\langle \xi \right\rangle ^{2}\le C\left\langle t\right\rangle ^{2\nu }\), we obtain

$$\begin{aligned} \left\langle \xi \right\rangle ^{2}\partial _{t}\widehat{\varphi }=-e^{it\Omega }\left\langle \xi \right\rangle ^{2}{\mathcal {D}}_{3}\frac{\xi ^{3}}{t\Lambda ^{\prime \prime }\left( \xi \right) }\widehat{\varphi }^{3} +\frac{3i\left\langle \xi \right\rangle ^{2}\xi ^{3}}{t\left| \Lambda ^{\prime \prime }\left( \xi \right) \right| }\left| \widehat{\varphi }\right| ^{2}\widehat{\varphi }+O\left( \varepsilon ^{3}t^{-\delta _{1} -1}\right) . \end{aligned}$$
(3.4)

Multiplying the above equation by \(\overline{\left\langle \xi \right\rangle ^{2}\widehat{\varphi }}\), and taking the real part of the result, we find

$$\begin{aligned} \frac{d}{dt}\left| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\left( t\right) \right| ^{2}=-2{\text {Re}}e^{it\Omega }\left\langle \xi \right\rangle ^{4}\overline{\widehat{\varphi }}{\mathcal {D}} _{3}\frac{\xi ^{3}}{t\Lambda ^{\prime \prime }\left( \xi \right) }\widehat{\varphi }^{3}+{\text {Re}}O\left( \varepsilon ^{3}t^{-\delta _{1} -1}\right) \left\langle \xi \right\rangle ^{2}\overline{\widehat{\varphi }} \end{aligned}$$

in the domain \(\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }.\) Define \(t_{1}\) such that \(\left\langle t_{1}\right\rangle ^{\nu }=\left| \xi \right| ,\) then integrating in time from \(t_{1}\) to t we obtain

$$\begin{aligned} \left| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\left( t\right) \right| ^{2}&\le \left| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\left( t_{1}\right) \right| ^{2}+2{\text {Re}} \int _{t_{1}}^{t}e^{i\tau \Omega }\left\langle \xi \right\rangle ^{4} \overline{\widehat{\varphi }}{\mathcal {D}}_{3}\frac{\xi ^{3}}{\Lambda ^{\prime \prime }\left( \xi \right) }\widehat{\varphi }^{3}\frac{d\tau }{\tau }\\&\qquad +{\text {Re}}O\left( \varepsilon ^{3}\int _{t_{1}}^{t}\tau ^{-\delta _{1}-1}\left\langle \xi \right\rangle ^{2}\overline{\widehat{\varphi }} d\tau \right) . \end{aligned}$$

Integration by parts via the identity \(e^{i\tau \Omega }=\left( 1+i\tau \Omega \right) ^{-1}\partial _{\tau }\left( \tau e^{i\tau \Omega }\right) \) yields the estimate \(\left| \left\langle \xi \right\rangle ^{2} \widehat{\varphi }\left( t\right) \right| <C\varepsilon \). Since the solution u is real, we have \(\overline{\widehat{\varphi }\left( t,\xi \right) }=\widehat{\varphi }\left( t,-\xi \right) .\) Therefore \(\left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{\infty }}<C\varepsilon .\) By Corollary 2.1 we get

$$\begin{aligned} \left\| u_{x}u_{xx}\right\| _{{\textbf{L}}^{\infty }}&=\left\| \left( {\mathcal {D}}_{t}{\mathcal {B}}M{\mathcal {Q}}\xi \widehat{\varphi }\right) \left( {\mathcal {D}}_{t}{\mathcal {B}}M{\mathcal {Q}}\xi ^{2}\widehat{\varphi }\right) \right\| _{{\textbf{L}}^{\infty }}\\&\le Ct^{-1}\left\| \left| {\mathcal {Q}}\xi \widehat{\varphi }\right| \left| {\mathcal {Q}}\xi ^{2}\widehat{\varphi }\right| \right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}t^{-1}. \end{aligned}$$

Then applying the energy method to Eq. (1.2), we have

$$\begin{aligned} \frac{d}{dt}\left\| u\left( t\right) \right\| _{{\textbf{H}}^{3}}\le C\left\| u_{x}u_{xx}\right\| _{{\textbf{L}}^{\infty }}\left\| u\right\| _{{\textbf{H}}^{3}}\le C\varepsilon t^{-1}\left\| u\left( t\right) \right\| _{{\textbf{H}}^{3}}. \end{aligned}$$

Using the Grönwall inequality we get \(\left\| u\left( t\right) \right\| _{{\textbf{H}}^{3}}<C\varepsilon \left\langle t\right\rangle ^{C\varepsilon }.\) To estimate the norm \(\left\| \xi ^{2}\partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}_{\xi }^{2}}\) we apply the operator \(\partial _{x}{\mathcal {P}}\) to Eq. (1.2). In view of the commutators \(\left[ {\mathcal {L}},{\mathcal {P}}\right] =\alpha {\mathcal {L}},\) \(\left[ {\mathcal {P}},\partial _{x}\right] =-\partial _{x},\) we obtain \({\mathcal {L}} \partial _{x}{\mathcal {P}}u=\partial _{x}\left( {\mathcal {P}}+\alpha \right) {\mathcal {L}}u=\partial _{x}\left( {\mathcal {P}}+\alpha \right) \left| u_{x}\right| ^{2}u_{x}.\) Application of the energy method yields

$$\begin{aligned} \frac{d}{dt}\left\| \partial _{x}{\mathcal {P}}u\right\| _{{\textbf{L}}^{2} }^{2}\le C\left\| u_{x}u_{xx}\right\| _{{\textbf{L}}^{\infty }}\left\| \partial _{x}{\mathcal {P}}u\right\| _{{\textbf{L}}^{2}}^{2}+C\varepsilon ^{2}t^{\gamma -1}\left\| \partial _{x}{\mathcal {P}}u\right\| _{{\textbf{L}} ^{2}}. \end{aligned}$$

Hence integrating in time we obtain \(\left\| \partial _{x}{\mathcal {P}} u\right\| _{{\textbf{L}}^{2}}<C\varepsilon t^{\frac{1}{2\alpha }}\). Therefore, using the identity \({\mathcal {J}}=\partial _{x}^{-1}{\mathcal {P}}+i\alpha t\partial _{x}^{-1}{\mathcal {L}}\), we find

$$\begin{aligned} \left\| \partial _{x}^{2}{\mathcal {J}}u\right\| _{{\textbf{L}}^{2}}\le C\left\| \partial _{x}{\mathcal {P}}u\right\| _{{\textbf{L}}^{2}}+Ct\left\| \partial _{x}\left( \left| u_{x}\right| ^{2}u_{x}\right) \right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}. \end{aligned}$$

Also using Lemma 3.2 and Lemma 3.3 we obtain \(\left\| \partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}_{\xi }^{2} }<C\varepsilon t^{\frac{1}{2\alpha }}.\) Hence we get the estimate \(\left\| \left\langle \xi \right\rangle ^{2}\partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}_{\xi }^{2}}<C\varepsilon t^{\frac{1}{2\alpha }}.\) Thus we find \(\left\| u\right\| _{{\textbf{X}}_{T_{1}}}<C\varepsilon ,\) which yields a desired contradiction. Lemma 3.4 is proved. \(\square \)

Next lemma states the asymptotics of the solution.

Lemma 3.5

Let \(\left\| u\right\| _{{\textbf{X}}_{\infty }}\le C\varepsilon .\) Then the asymptotics

$$\begin{aligned} \left| \partial _{x}\right| ^{\frac{\alpha -2}{2}}u\left( t\right) =2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}}M\theta \left| \xi \right| ^{\frac{\alpha -2}{2}}W_{+}\left( \xi \right) \exp \left( 3i\xi ^{3}\left| W_{+}\left( \xi \right) \right| ^{2}\log t\right) +O\left( \varepsilon t^{-\frac{1}{2}-\delta }\right) \end{aligned}$$

is true for \(t\rightarrow \infty \) uniformly with respect to \(x\in {\mathbb {R}},\) where \(\delta >0.\)

Proof

In the domain \(\left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\), by the Sobolev embedding inequality we get

$$\begin{aligned}&\left\| \widehat{\varphi }\right\| _{{\textbf{L}}^{\infty }\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }\le C\left\| \widehat{\varphi }\right\| _{{\textbf{L}}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2} }\left\| \partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2}}\\&\quad \le C\left\langle t\right\rangle ^{-\nu }\left\| \left\langle \xi \right\rangle \widehat{\varphi }\right\| _{{\textbf{L}}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2}}\left\| \partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2}}<C\varepsilon \left\langle t\right\rangle ^{-\nu +\gamma } \end{aligned}$$

with \(\nu >\gamma .\) Therefore we need to compute the asymptotics of the function \(\widehat{\varphi }\) in the domain \(\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }.\) Applying the operator \({{\mathcal {F}}}{{\mathcal {U}}} \left( -t\right) \) to Eq. (1.2), we get Eq. (2.1). Then by virtue of Lemma 3.1, we obtain

$$\begin{aligned} \partial _{t}\widehat{\varphi }&=\frac{3i\xi ^{3}\theta \left( \xi \right) }{t\Lambda ^{\prime \prime }\left( \xi \right) }\left| \widehat{\varphi }\left( \xi \right) \right| ^{2}\widehat{\varphi }\left( t,\xi \right) \\&\quad -e^{it\Omega }{\mathcal {D}}_{3}\frac{\xi ^{3}\theta \left( \xi \right) }{t\Lambda ^{\prime \prime }\left( \xi \right) }\widehat{\varphi }^{3}\left( t,\xi \right) +O\left( \varepsilon ^{3}t^{\frac{\nu }{2}-1-\delta _{1}}\right) . \end{aligned}$$

Then we change the dependent variable \(\widehat{\varphi }\left( t,\xi \right) =y\left( t,\xi \right) \Psi \left( t,\xi \right) \) with

$$\begin{aligned} \Psi \left( t,\xi \right) =\exp \left( \frac{3i\xi ^{3}\theta \left( \xi \right) }{\Lambda ^{\prime \prime }\left( \xi \right) }\int _{1} ^{t}\left| \widehat{\varphi }\left( \tau ,\xi \right) \right| ^{2} \frac{d\tau }{\tau }\right) \end{aligned}$$

to get

$$\begin{aligned} \partial _{t}y=-e^{it\Omega }\overline{\Psi }{\mathcal {D}}_{3}\frac{\xi ^{3} \theta \left( \xi \right) }{t\Lambda ^{\prime \prime }\left( \xi \right) }\widehat{\varphi }^{3}\left( t,\xi \right) +O\left( \varepsilon ^{3} t^{\frac{\nu }{2}-1-\delta _{1}}\right) . \end{aligned}$$

Integration by parts via the identity \(e^{i\tau \Omega }=\left( 1+i\tau \Omega \right) ^{-1}\partial _{\tau }\left( \tau e^{i\tau \Omega }\right) \) yields the estimate \(\left\| y\left( t\right) -y\left( s\right) \right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon s^{-\delta _{2}}\) for all \(t>s>0,\) with some \(\delta _{2}>0.\) Therefore there exists a unique final state \(y_{+}\in {\textbf{L}}^{\infty }\) such that \(\left\| y\left( t\right) -y_{+}\right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon t^{-\delta _{2}}\) for all \(t>0.\) Denote the remainder term \(\Phi \left( t\right) =\int _{1} ^{t}\left| y\left( \tau \right) \right| ^{2}\frac{d\tau }{\tau }-\left| y_{+}\right| ^{2}\log t,\) then we have

$$\begin{aligned} \Phi \left( t\right) -\Phi \left( s\right) =\int _{s}^{t}\left( \left| y\left( \tau \right) \right| ^{2}-\left| y\left( t\right) \right| ^{2}\right) \frac{d\tau }{\tau }+\left( \left| y\left( t\right) \right| ^{2}-\left| y_{+}\right| ^{2}\right) \log \frac{t}{s} \end{aligned}$$

and \(\left\| \Phi \left( t\right) -\Phi \left( s\right) \right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}s^{-\delta _{2}}\) for all \(t>s>0.\) Hence there exists a unique real-valued function \(\Phi _{+}\), such that \(\Phi _{+}\in {\textbf{L}}^{\infty }\) and \(\left\| \Phi \left( t\right) -\Phi _{+}\right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}t^{-\delta _{2}}.\) Therefore we obtain

$$\begin{aligned} \int _{1}^{t}\left| \widehat{\varphi }\left( \tau ,\xi \right) \right| ^{2}\frac{d\tau }{\tau }&=\int _{1}^{t}\left| y\left( \tau ,\xi \right) \right| ^{2}\frac{d\tau }{\tau }\\&=\Phi _{+}+\left| y_{+}\right| ^{2}\log t+O\left( \varepsilon ^{2}t^{-\delta _{2}}\right) \end{aligned}$$

for all \(t>0.\) Then we have

$$\begin{aligned} \left\| \Psi \left( t,\xi \right) -\exp \left( \frac{3i\xi ^{3}\theta \left( \xi \right) }{\Lambda ^{\prime \prime }\left( \xi \right) }\left| y_{+}\right| ^{2}\log t+\frac{3i\xi ^{3}\theta \left( \xi \right) }{\Lambda ^{\prime \prime }\left( \xi \right) }\Phi _{+}\right) \right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}t^{-\delta _{2}} \end{aligned}$$

for all \(t>0.\) Thus we get the large time asymptotics

$$\begin{aligned} \widehat{\varphi }=y_{+}\Psi +O\left( \varepsilon t^{-\delta _{2}}\right) =\widetilde{W_{+}}\exp \left( \frac{3i\xi ^{3}\theta \left( \xi \right) }{\Lambda ^{\prime \prime }\left( \xi \right) }\left| \widetilde{W_{+} }\right| ^{2}\log t\right) +O\left( \varepsilon t^{-\delta _{2}}\right) , \end{aligned}$$

where \(\widetilde{W_{+}}\left( \xi \right) =y_{+}\left( \xi \right) \exp \left( \frac{3i\xi ^{3}\theta \left( \xi \right) }{\Lambda ^{\prime \prime }\left( \xi \right) }\Phi _{+}\left( \xi \right) \right) .\) Note that \(\widetilde{W_{+}}\in {\textbf{L}}^{\infty }.\) Finally, by the factorization formula we get \(\left| \partial _{x}\right| ^{\frac{\alpha -2}{2} }u\left( t\right) =2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}} M{\mathcal {Q}}\left( \left| \xi \right| ^{\frac{\alpha -2}{2}} \widehat{\varphi }\right) \). Hence applying Lemma 2.1, Lemma 2.2 and Lemma 2.3, we find

$$\begin{aligned} \left| \partial _{x}\right| ^{\frac{\alpha -2}{2}}u\left( t\right)&=2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}}M\frac{\theta \left( \xi \right) \left| \xi \right| ^{\frac{\alpha -2}{2}}}{\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }}\widehat{\varphi }+O\left( \varepsilon t^{-\frac{1}{2}-\delta }\right) \\&=2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}}M\frac{\theta \left( \xi \right) \left| \xi \right| ^{\frac{\alpha -2}{2}}}{\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }}\widetilde{W_{+}}\left( \xi \right) \exp \left( \frac{3i\xi ^{3}}{\Lambda ^{\prime \prime }\left( \xi \right) }\left| \widetilde{W_{+}}\left( \xi \right) \right| ^{2}\log t\right) \\&\quad +O\left( \varepsilon t^{-\frac{1}{2}-\delta }\right) \\&=2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}}M\theta \left( \xi \right) \left| \xi \right| ^{\frac{\alpha -2}{2}}W_{+}\left( \xi \right) \exp \left( 3i\xi ^{3}\left| W_{+}\left( \xi \right) \right| ^{2}\log t\right) \\&\quad +O\left( \varepsilon t^{-\frac{1}{2}-\delta }\right) , \end{aligned}$$

where \(W_{+}\left( \xi \right) =\frac{1}{\sqrt{i\Lambda ^{\prime \prime }\left( v\right) }}\widetilde{W_{+}}\left( \xi \right) .\) Note that \(\left| \xi \right| ^{\frac{\alpha -2}{2}}W_{+}\left( \xi \right) \in {\textbf{L}}^{\infty }.\) This completes the proof of the asymptotics. If we substitute the definitions of the operators \({\mathcal {D}}_{t}\) and \({\mathcal {B}}\), then the asymptotics has the form

$$\begin{aligned}&\left| \partial _{x}\right| ^{\frac{\alpha -2}{2}}u\left( t,x\right) =2{\text {Re}}\frac{1}{\sqrt{t}}\theta \left( x\right) \left( \frac{x}{t}\right) ^{\frac{\alpha -2}{2\left( \alpha -1\right) } }W_{+}\left( \left( \frac{x}{t}\right) ^{\frac{1}{\alpha -1}}\right) \\&\quad \times \exp \left( it\frac{\alpha -1}{\alpha }\left( \frac{x}{t}\right) ^{\frac{\alpha }{\alpha -1}}+3i\left( \frac{x}{t}\right) ^{\frac{3}{\alpha -1} }\left| W_{+}\left( \left( \frac{x}{t}\right) ^{\frac{1}{\alpha -1} }\right) \right| ^{2}\log t\right) \\&\qquad +O\left( t^{-\frac{1}{2}-\delta }\right) . \end{aligned}$$

Lemma 3.5 is proved. \(\square \)

4 Proof of Theorem 1.1

By Lemma 3.4 we see that a priori estimate \(\left\| u\right\| _{{\textbf{X}}_{\infty }}\le C\varepsilon \) is true. Therefore global existence of solutions of the Cauchy problem (1.2) satisfying estimate \(\left\| u\right\| _{{\textbf{X}}_{\infty }}\le C\varepsilon ,\) follows by a standard continuation argument via the local existence Theorem 3.1. Then the asymptotic formula (1.3) for the solutions u of the Cauchy problem (1.2) follows from Lemma 3.5. Theorem 1.1 is proved.