Scattering for the $L^2$ supercritical point NLS

TL;DR

This paper proves that solutions to the 1D focusing point nonlinear Schrödinger equation in the $L^2$ supercritical case scatter as time approaches infinity, using a modified Kenig-Merle approach.

Contribution

It demonstrates scattering for the $L^2$ supercritical focusing point NLS, adapting the Kenig-Merle method with a specialized function space.

Findings

Global solutions scatter as t approaches infinity

The method adapts Kenig-Merle to non-Strichartz spaces

Provides insight into asymptotic behavior of supercritical NLS

Abstract

We consider the 1D nonlinear Schr\"odinger equation with focusing point nonlinearity. "Point" means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of solution, blow-up occurrence and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e, we show that the global solution scatters as t tends to minus/plus infinity in the $L^2$ supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

Equations

$$\left\{\begin{array}[]{ll}i\partial_{t}\psi+\partial_{x}^{2}\psi+K(x)|\psi|^{p-1}\psi=0,&t\in\mathbb{R},~{}x\in\mathbb{R}\\ \psi(x,0)=\psi_{0}(x)&\end{array}\right.$$

$$i{\partial }_t\psi +{\partial }_x^2\psi =0,t\in \mathbb{R},x\ne 0$$

$$\psi (0,t):=\psi (0-,t)=\psi (0+,t)$$

$${\partial }_x\psi (0+,t)-{\partial }_x\psi (0-,t)=-|\psi (0,t){|}^{p-1}\psi (0,t).$$

$$M(\psi )=\int |\psi {|}^2$$

$$E(\psi )=\frac{1}{2}\int |{\partial }_x\psi {|}^2-\frac{1}{p+1}|\psi (0){|}^{p+1}.$$

$$H^s=\{v\in {\mathcal{S}}^{\prime }(\mathbb{R}),\Vert v{\Vert }_{H^s}:=\Vert (1+|\xi {|}^2{)}^{\frac{s}{2}}\hat{v}(\xi ){\Vert }_{L_{\mathbb{R}}^2}<+\infty \},$$

$${\dot{H}}^s=\{v\in {\mathcal{S}}^{\prime }(\mathbb{R}),\Vert v{\Vert }_{{\dot{H}}^s}:=\Vert |\xi {|}^s\hat{v}(\xi ){\Vert }_{L_{\mathbb{R}}^2}<+\infty \},$$

$${\sigma }_c=\frac{1}{2}-\frac{1}{p-1},$$

$$\frac{1}{4}<\frac{2{\sigma }_c+1}{4}<\frac{1}{2},-\frac{1}{4}<\frac{2{\sigma }_c-1}{4}<0.$$

$$\frac{1}{q}=\frac{1}{2}-\frac{2{\sigma }_c+1}{4},\frac{1}{2}=\frac{1}{\tilde{q}}-\frac{1-2{\sigma }_c}{4},$$

$$q=2(p-1),\tilde{q}=\frac{2(p-1)}{p}.$$

$$\Vert \psi {\Vert }_{L_{\mathbb{R}}^q}\lesssim \Vert \psi {\Vert }_{{\dot{H}}^{\frac{2{\sigma }_c+1}{4}}},\Vert f{\Vert }_{{\dot{H}}^{\frac{2{\sigma }_c-1}{4}}}\lesssim \Vert f{\Vert }_{L_{\mathbb{R}}^{\tilde{q}}}.$$

$$\psi \in C_{[0,T]}^0{H}_x^1\cap C_{\mathbb{R}}^0{H}_{(0,T)}^{\frac{3}{4}},$$

$${\partial }_x\psi \in C_{{\mathbb{R}}_x\setminus \{0\}}^0{H}_{(0,T)}^{\frac{1}{4}}.$$

$${\partial }_x\psi (0^{+},t)-{\partial }_x\psi (0^{-},t)=-|\psi (0,t){|}^{p-1}\psi (0,t)$$

$${\partial }_t^2\int a(x)|\psi {|}^{2}dx=4\int {\partial }_x^{(2)}a|{\partial }_x\psi {|}^2-2{\partial }_x^{(2)}a(0)|\psi (0){|}^{p+1}-\int {\partial }_x^{(4)}a|\psi {|}^2.$$

$$|\psi (0){|}^2\le \Vert \psi {\Vert }_{L^2}\Vert {\partial }_x\psi {\Vert }_{L^2}.$$

$$M({\psi }_0{)}^{\frac{1-{\sigma }_c}{{\sigma }_c}}E({\psi }_0)<M({\phi }_0{)}^{\frac{1-{\sigma }_c}{{\sigma }_c}}E({\phi }_0).$$

$$\eta (t)=\frac{\Vert \psi {\Vert }_{L^2}^{\frac{1-{\sigma }_c}{{\sigma }_c}}\Vert {\partial }_x\psi (t){\Vert }_{L^2}}{\Vert {\phi }_0{\Vert }_{L^2}^{\frac{1-{\sigma }_c}{{\sigma }_c}}\Vert {\partial }_x{\phi }_0{\Vert }_{L^2}}$$

$$M({\psi }_0{)}^{\frac{1-{\sigma }_c}{{\sigma }_c}}E({\psi }_0)<M({\phi }_0{)}^{\frac{1-{\sigma }_c}{{\sigma }_c}}E({\phi }_0)$$

$$\Vert {\psi }_0{\Vert }_{L^2}^{\frac{1-{\sigma }_c}{{\sigma }_c}}\Vert {\partial }_x{\psi }_0{\Vert }_{L^2}<\Vert {\phi }_0{\Vert }_{L^2}^{\frac{1-{\sigma }_c}{{\sigma }_c}}\Vert {\partial }_x{\phi }_0{\Vert }_{L^2}.$$

$$\underset{t\to \pm \infty }{\lim }\Vert e^{-it{\partial }_x^2}\psi (t)-{\psi }^{\pm }{\Vert }_{H_x^1}=0.$$

$$\psi (x,t)$$

$$\psi (0,t)=[e^{it{\partial }_x^2}{\psi }_0](0)+i{\int }_0^t\frac{1}{\sqrt{4\pi i(t-s)}}|\psi (0,s){|}^{p-1}\psi (0,s)ds.$$

$$[{\mathcal{L}}_{s}f](x,t):={\int }_s^t\frac{e^{\frac{i{x}^2}{4(t-\tau )}}}{\sqrt{4\pi i(t-\tau )}}f(\tau )d\tau .$$

$$[\Lambda f](x,t):={\int }_t^{\infty }\frac{e^{\frac{i{x}^2}{4(t-\tau )}}}{\sqrt{4\pi i(t-\tau )}}f(\tau )d\tau .$$

$$\Vert {\chi }_{[0,t]}(s)f(s){\Vert }_{{\dot{H}}_s^{\mu }}\lesssim \Vert f{\Vert }_{{\dot{H}}_s^{\mu }}$$

$$\Vert {\chi }_{[0,+\infty )}f{\Vert }_{{\dot{H}}_s^{\mu }}\lesssim \Vert f{\Vert }_{{\dot{H}}_s^{\mu }}$$

$$\Vert {\chi }_{[0,t]}f{\Vert }_{{\dot{H}}_s^{\mu }}$$

$$\lesssim \Vert {\chi }_{(-\infty ,t]}f{\Vert }_{{\dot{H}}_s^{\mu }}$$

$$=\Vert {\chi }_{[0,+\infty )}\tilde{f}{\Vert }_{{\dot{H}}_s^{\mu }}$$

Full text

1 DISMA, Politecnico di Torino, Italy;

2 Graduate School of Information Sciences, Tohoku University, Japan;

3 Department of Mathematics, Brown University, USA;

1. Introduction

In this paper, we address a theoretical study on a model, proposed in [16], that describes a wave propagation in a 1D linear medium containing a narrow strip of nonlinear material, where the nonlinear strip is assumed to be much smaller than the typical wavelength. Considering such nonlinear strip may allow to model a wave propagation in nanodevices, in particular the authors in [13] consider some nonlinear quasi periodic super lattices and investigate an interplay between the nonlinearity and the quasi periodicity. Such a strip is described as an impurity, i.e. a delta measure in the nonlinearity of nonlinear Schrödinger equation. For applications in nanodevices, it should be important to study NLS with a quasi periodic location of delta measures, but in this paper, as a first step, we will treat the Schrödinger equation which has only one impurity in the nonlinearity:

[TABLE]

where $p>1$, and $K=\delta$, $\delta$ is the Dirac mass at $x=0.$ This singularity in the nonlinearity is interpreted as the linear Schrödinger equation:

[TABLE]

together with the jump condition at $x=0$

[TABLE]

Remark that this equation (1.1) also appears as a limiting case of nonlinear Schrödinger equation with a concentrated nonlinearity (see [7]).

In [3, 11], it was proved that the equation (1.1) is locally well-posed for any $\psi_{0}\in H^{1}(\mathbb{R})$ for $p>1$, and Equation (1.1) has two conservative quantities: the mass

[TABLE]

and the energy

[TABLE]

The mass condition for the global existence/blow-up, further an analysis of the blow-up profile were established in [11, 12]. Furthermore, the problem of asymptotic stability of the standing waves of equation (1.1) has been treated in [5] and [14].

As far as we know, the asymptotic behavior, in particular, the scattering of the solution is not known for (1.1). For the standard NLS, i.e. $K\equiv 1$, in one dimensional case, such a result in $H^{1}$ was firstly established in [17]. This topic has been very active these decades thanks to a breakthrough result by Kenig-Merle [15]. Our proof therefore essentially will be based on Kenig-Merle [15], and some results after [15], for example [10]. However, it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation to (1.1).

Higher-dimensional models with a generalization of the delta potential have been introduced in [2] and in [6] for the three and two-dimensional setting, respectively. While, at a qualitative level, the model in dimension three behaves like that in dimension one, the two-dimensional setting displays some uncommon features still to be understood (for the analysis of the blow-up, see [1]).

We remark that the model of a NLS with a standard power nonlinearity and a linear point interaction has been studied in [4].

Notation. If $I$ is an interval of $\mathbb{R}$, and $1\leq r\leq\infty$, then $L^{r}_{I}$ is the space of strongly Lebesgue measurable, complex-valued functions $v$ from $I$ into $\mathbb{C}$ satisfying $\|v\|_{L^{r}_{I}}:=\int_{I}|v(t)|^{r}dt<+\infty$ if $r<+\infty$, when $r=+\infty$, $\|v\|_{L^{\infty}_{I}}:=\sup_{t\in I}|v(t)|<+\infty$. The space $C_{I}^{0}E$ denotes the space of continuous functions on $I$ with values in a Banach space $E$.

For $s\in\mathbb{R},$ we define the Sobolev space

[TABLE]

and the homogeneous Sobolev space

[TABLE]

where $\widehat{f}$ is the Fourier transform of the function $f$. Thus, $H^{0}=\dot{H}^{0}=L^{2}_{\mathbb{R}},$ and this will be simply denoted as $L^{2}$. Sometimes we put an index $t$ or $x$ like $\dot{H}^{s}_{t}$ or $\dot{H}^{s}_{x}$ to enlighten which variable concerns. For $\alpha\in\mathbb{R}$, $|\nabla|^{\alpha}$ denotes the Fourier multiplier with symbol $|\xi|^{\alpha}.$ For $s\geq 0$, define $v\in H^{s}_{I}$ if, when $v(x)$ is extended to $\tilde{v}(x)$ on $\mathbb{R}$ by setting $\tilde{v}(x)=0$ for $x\notin I$, then $\tilde{v}\in H^{s}$; in this case we set $\|v\|_{H^{s}_{I}}=\|\tilde{v}\|_{H^{s}}.$ Finally, $\chi_{I}$ denotes the characteristic function for the interval $I\subset\mathbb{R}$.

The equation (1.1) has a scaling invariance: if $\psi(x,t)$ is a solution to (1.1) then $\lambda^{\frac{1}{p-1}}\psi(\lambda x,\lambda^{2}t)$, $\lambda>0$ is also. The scale-invariant Sobolev space for (1.1) is $\dot{H}^{\sigma_{c}}$ with

[TABLE]

thus, for (1.1), $p=3$ is the $L^{2}$ critical setting. If $p>3$, then $0<\sigma_{c}<\frac{1}{2}$ and

[TABLE]

We take $q$ and $\tilde{q}$ to be given by

[TABLE]

and from the definition of $\sigma_{c}$, we find that

[TABLE]

In the remainder of the paper, once $p>3$ is selected, we will take $\sigma_{c}$, $q$ and $\tilde{q}$ to have the corresponding values as defined above.

Recall that by Sobolev embedding, one has

[TABLE]

More generally than the above case, $\sigma_{c}$ should satisfy $-\frac{1}{2}\leq\sigma_{c}<\frac{1}{2}$ to apply this Sobolev embedding, that is, the case $\sigma_{c}=0$ (namely $p=3$) is included for this embedding.

First, we recall here the local wellposedness result of (1.1) established in Theorem 1.1 of [11].

Proposition 1.1**.**

Let $p>1$ and $\psi_{0}\in H^{1}.$ Then, there exist $T^{*}>0$ and a solution $\psi(x,t)$ to (1.1) on $[0,T^{*})$ satisfying for $T<T^{*},$

[TABLE]

Here, the derivatives $\partial_{x}\psi(0^{\pm},t):=\lim_{x\to\pm 0}\partial_{x}\psi(x,t)$, exist in the sense of $H^{\frac{1}{4}}_{(0,T)}$ and $\psi$ satisfies

[TABLE]

as an equality of $H_{(0,T)}^{\frac{1}{4}}$ functions (not pointwisely in $t$).

Among all solutions satisfying the above regularity conditions, it is unique. Moreover, the data-to-solution map $\psi_{0}\mapsto\psi$, as a map $H_{x}^{1}\to C^{0}_{[0,T]}H_{x}^{1}$, is continuous, and if $T^{*}<+\infty,$ then $\lim_{t\uparrow T^{*}}\|\partial_{x}\psi(t)\|_{L^{2}_{\mathbb{R}}}=+\infty.$

Hereafter, the solution to (1.1) satisfying the above regularity condition will be referred to as $H^{1}_{x}$ solution to (1.1).

The local virial identity has been also proved in [11]. For any smooth weight function $a(x)$ satisfying $a(0)=\partial_{x}a(0)=\partial_{x}^{(3)}a(0)=0$, the solution $\psi$ to (1.1) satisfies

[TABLE]

Proposition 1.2** ([11, Prop 1.3] sharp Gagliardo-Nirenberg inequality).**

For any $\psi\in H^{1}$,

[TABLE]

Equality is achieved if and only if there exist $\theta\in\mathbb{R}$, $\alpha>0$ and $\beta>0$ such that $\psi(x)=\alpha e^{i\theta}\varphi_{0}(\beta x)$, where $\varphi_{0}=2^{\frac{1}{p-1}}e^{-|x|}$ is the ground state solution to (1.1) (see [11]).

Theorem 1.3** ([11, Prop 1.4] $L^{2}$ supercritical global existence/blow-up dichotomy).**

Suppose that $\psi(t)$ is an $H_{x}^{1}$ solution of (1.1) for $p>3$ satisfying

[TABLE]

Let

[TABLE]

Then

1. (1)

If $\eta(0)<1$, then the solution $\psi(t)$ is global in both time directions and $\eta(t)<1$ for all $t\in\mathbb{R}$. 2. (2)

If $\eta(0)>1$, then the solution $\psi(t)$ blows-up in the negative time direction at some $T_{-}<0$, blows-up in the positive time direction at some $T_{+}>0$, and $\eta(t)>1$ for all $t\in(T_{-},T_{+})$.

Remark that if $E(\psi_{0})<0$, then the condition (1.4) is satisfied, and in that case $\eta(t)>1$ is forced by (1.3), so the condition (2) applies giving the blow-up.

Main result of this paper is the following.

Theorem 1.4**.**

(asymptotic completeness) Let $p>3.$ Let $\psi_{0}\in H^{1}$ and let $\psi(t)$ be a $H^{1}_{x}$ solution of (1.1) satisfying

[TABLE]

and

[TABLE]

Then, there exist $\psi^{+},\psi^{-}\in H^{1}$ such that

[TABLE]

We only consider the focusing nonlinearity, but the scattering for the defocusing case is similarly proved.

This paper is organized as follows: Below in Section 2, we will discuss the local theory, scattering criterion and long-time perturbation theory. Section 2 includes some preliminary and important results which reflect the smoothing properties of the equation (1.1). We will give in Section 3 the profile decomposition in $H^{1}$ in a form well-adapted to our equation. In Section 4, the asymptotic completeness in $H^{1}$ will be established using the results in Sections 2 and 3. We sometimes denote all through the paper by $C_{\theta,...}$ a constant which depends on $\theta$ and so on.

2. Local theory, scattering criterion, and long-time perturbation theory

Write the equation (1.1) in the Duhamel form:

[TABLE]

We remark that the equation (1.1) is completely solved once the one-variable complex function $\psi(0,\cdot)$ is known: indeed, specializing (2.1) to the value $x=0$, one obtains a closed, nonlinear, integral, a Volterra-Abel type equation for $\psi(0,\cdot)$;

[TABLE]

Now, for any $\sigma\in\mathbb{R}$, we define for $f\in\dot{H}^{\sigma},$ $t,s\in\mathbb{R}$ with $t\geq s,$

[TABLE]

Similarly, we define, for $t\in\mathbb{R}$,

[TABLE]

The following smoothing properties of $\mathcal{L}_{s}$ and $\Lambda$ will play important roles in what follows.

Proposition 2.1**.**

Let $\sigma\in\mathbb{R}.$

• (1)

$\|[e^{i(t-s)\partial_{x}^{2}}f](0)\|_{\dot{H}^{\frac{2\sigma+1}{4}}_{t}}\lesssim\|f\|_{\dot{H}^{\sigma}}$, for any $f\in\dot{H}^{\sigma}$ and $t,s\in\mathbb{R}$.

• (2)

Assume $-\frac{1}{2}<\frac{2\sigma-1}{4}<\frac{1}{2}.$ Let $f\in\dot{H}^{\frac{2\sigma-1}{4}}$ and $s\in\mathbb{R}$.

• (2a)

$\|[\mathcal{L}_{s}f](0,\cdot)\|_{\dot{H}_{t}^{\frac{2\sigma+1}{4}}}\lesssim\|\chi_{[s,+\infty)}f\|_{\dot{H}^{\frac{2\sigma-1}{4}}}\lesssim\|f\|_{\dot{H}^{\frac{2\sigma-1}{4}}}$**

• (2b)

$\|[\Lambda f](0,\cdot)\|_{\dot{H}_{t}^{\frac{2\sigma+1}{4}}}\lesssim\|f\|_{\dot{H}^{\frac{2\sigma-1}{4}}}$**

• (3)

Assume $-\frac{1}{2}<\frac{2\sigma-1}{4}<\frac{1}{2}.$ Let $f\in\dot{H}^{\frac{2\sigma-1}{4}}$ and $s\in\mathbb{R}$.

• (3a)

$\|\mathcal{L}_{s}f\|_{L^{\infty}_{\mathbb{R}_{t}}\dot{H}_{x}^{\sigma}}\lesssim\|f\|_{\dot{H}^{\frac{2\sigma-1}{4}}}.$**

• (3b)

$\|\Lambda f\|_{L^{\infty}_{\mathbb{R}_{t}}\dot{H}_{x}^{\sigma}}\lesssim\|f\|_{\dot{H}^{\frac{2\sigma-1}{4}}}.$**

For the proof of Proposition 2.1, we need some preparations.

Lemma 2.2**.**

For any $-\frac{1}{2}<\mu<\frac{1}{2}$, and any $t>0$, we have

[TABLE]

with implicit constant independent of $t$.

Proof.

First, we claim that it suffices to show

[TABLE]

Indeed, suppose that we have proved (2.4). Since $\chi_{[0,t]}=\chi_{[0,+\infty)}\chi_{(-\infty,t]}$, to prove (2.3) we note

[TABLE]

where $\tilde{f}(s)=f(-s+t)$. In the last step, we have used that

[TABLE]

We continue and apply (2.4) to obtain

[TABLE]

where, in the last step, we used that $\widehat{\tilde{f}}(\tau)=e^{-it\tau}\hat{f}(-\tau)$. This completes the proof of (2.3) assuming (2.4).

To prove (2.4), we note $\hat{\chi}_{[0,+\infty)}(\tau)=\operatorname{pv}\frac{1}{i\tau}+\pi\delta(\tau)$ and thus

[TABLE]

where $H$ denotes the Hilbert transform. Hence

[TABLE]

Since $-\frac{1}{2}<\mu<\frac{1}{2}$, we can apply Corollary of Theorem 2 on page 205 in [18], combined with (6.4) on p. 218 of [18] (for $p=2$, $n=1$, $a=2\mu$) to estimate the above as

[TABLE]

∎

Proof.

(of Proposition 2.1) (1) was already proved in Lemma 1 of [3], but for the sake of completeness we give a proof. We use here the notation $\hat{}$, which means the Fourier transform in space, and $\mathcal{F}$ is in time. It suffices to show the case $s=0$. Since the free Schrödinger group is unitary in $\dot{H}_{x}^{\sigma}$ for any $\sigma\in\mathbb{R}$, We may write

[TABLE]

By a change of variables this equals

[TABLE]

Thus the Fourier transform in time gives

[TABLE]

Therefore

[TABLE]

where, again we changed the variables $\pm\sqrt{\omega}=k$ in the second inequality. For (2a), we may write

[TABLE]

where

[TABLE]

We operate the Fourier transform and obtain

[TABLE]

It thus follows that by Lemma 2.2, for $-\frac{1}{2}<\frac{2\sigma-1}{4}<\frac{1}{2},$

[TABLE]

The proof of (2b) is similar, since

[TABLE]

For (3a), it suffices to prove that for any $g\in\dot{H}_{x}^{-\sigma}(\mathbb{R})$ with $\|g\|_{\dot{H}_{x}^{-\sigma}}=1,$

[TABLE]

The left hand side can be estimated as follows.

[TABLE]

where we have used (1) with the unitary property of free Schrödinger group in $\dot{H}^{s}_{x}$ for any $s\in\mathbb{R}$, and Lemma 2.2 in the last inequality. Since (3b) can be similarly proved, we omit the proof, but we remark that for any $\sigma\in\mathbb{R},$ (that is, without the restriction $-\frac{1}{2}<\frac{2\sigma-1}{4}<\frac{1}{2}$),

[TABLE]

holds. ∎

From now on, we prepare some basic facts in order to prove the asymptotic completeness. For the sake of simplicity we will study the following Propositions 2.3-2.5 only in the case $t>0,$ but we can consider the negative time $t<0$ similarly.

Proposition 2.3** (small data global well-posedness).**

Let $p\geq 3$. There exists $\delta_{\text{sd}}>0$ such that if $\psi_{0}\in\dot{H}^{\sigma_{c}}$ and $\|[e^{it\partial_{x}^{2}}\psi_{0}](0)\|_{L_{t>0}^{q}}\leq\delta_{\text{sd}}$, then $\psi\in\dot{H}^{\sigma_{c}}$ solving (1.1) is global in $\dot{H}^{\sigma_{c}}$ and

[TABLE]

[TABLE]

(Note that by Proposition 2.1 (1) and Sobolev embedding, the smallness assumption $\|[e^{it\partial_{x}^{2}}\psi_{0}](0)\|_{L_{t>0}^{q}}\leq\delta_{\text{sd}}$ is satisfied if $\|\psi_{0}\|_{\dot{H}^{\sigma_{c}}}\leq C\delta_{\text{sd}}$. )

Proof.

Define a map: for a $\psi_{0}\in\dot{H}^{\sigma_{c}}$ given,

[TABLE]

By Proposition 2.1 and Sobolev embedding, we have

[TABLE]

Let

[TABLE]

If $\|[e^{it\partial_{x}^{2}}\psi_{0}](0)\|_{L_{t>0}^{q}}\leq\delta_{\text{sd}}$ then $\mathcal{T}_{\psi_{0}}\psi\in B$ for any $\psi\in B$, taking $\delta_{\text{sd}}$ sufficiently small.

The difference $\|\mathcal{T}_{\psi_{0}}\psi-\mathcal{T}_{\psi_{0}}\tilde{\psi}\|_{L^{q}_{t}}$ is similarly estimated by

[TABLE]

for $\psi,\tilde{\psi}\in B$. Again taking $\delta_{\text{sd}}$ sufficiently small, we conclude that $\mathcal{T}_{\psi_{0}}$ is a contraction on $B$. There thus exists a unique solution $\tilde{\psi}\in B$ such that $\mathcal{T}_{\psi_{0}}\tilde{\psi}=\tilde{\psi}.$

For the last inequality in the proposition, we use Eq. (2.1) for the unique solution $\tilde{\psi}$ obtained above in $B$. Inserting $\tilde{\psi}$ as the value of $\psi(0,t)$ at time $t$ in the RHS of (2.1), The values of $\psi(x,t)$ for any $x$ can be expressed as

[TABLE]

with $\psi(0,\cdot)\in B$. Then, Sobolev embedding and Proposition 2.1 implies

[TABLE]

Since $\psi(0,\cdot)\in B$ with $\|[e^{it\partial_{x}^{2}}\psi_{0}](0,t)\|_{L_{t>0}^{q}}\leq\delta_{\text{sd}}$, by Sobolev embedding and Proposition 2.1(1),

[TABLE]

Taking $\delta_{\text{sd}}$ sufficiently small, the RHS of (2.6) is bounded by $2\|\psi_{0}\|_{\dot{H}^{\sigma_{c}}_{x}}$. Note that the time continuity property follows from the fundamental solution, and this concludes

[TABLE]

. ∎

Proposition 2.4** (scattering criterion).**

Let $p\geq 3$. Suppose that $\psi_{0}\in H^{1}$ and $\psi\in H^{1}_{x}$ solving (1.1) is forward global with

[TABLE]

and with a uniform $H^{1}_{x}$ bound

[TABLE]

Then $\psi(t)$ scatters in $H^{1}_{x}$ as $t\nearrow+\infty$. This means that there exists $\psi^{+}\in H^{1}_{x}$ such that

[TABLE]

Proof.

Using the equation (2.1), we may write

[TABLE]

where

[TABLE]

Therefore,

[TABLE]

Thus we shall estimate $\|\Lambda(|\psi|^{p-1}\psi)(\cdot,t)\|_{L^{2}_{x}}$ and $\|\Lambda(|\psi|^{p-1}\psi)(\cdot,t)\|_{\dot{H}^{1}_{x}}.$ First, $\|\Lambda(|\psi|^{p-1}\psi)(\cdot,t)\|_{L^{2}_{x}}$ is estimated by (3b) of Proposition 2.1 and the Sobolev embedding as follows. For any $t>0$,

[TABLE]

Second, by the Sobolev embedding and fractional chain rule [8], for any $t>0$,

[TABLE]

with $\frac{1}{2}=\frac{1}{r_{1}}+\frac{1}{r_{2}},$ $1<r_{1},r_{2}<+\infty$. Taking $q<r_{1}<+\infty$ and $2<r_{2}<4$, by interpolation,

[TABLE]

where we have used the Sobolev embedding $H^{1}(\mathbb{R}_{x})\subset L^{\infty}(\mathbb{R}_{x})$. Again by interpolation

[TABLE]

where we have used the Sobolev embedding $H^{1}(\mathbb{R}_{t})\subset L^{\infty}(\mathbb{R}_{t})$ in the second inequality. We go back to the equation (2.7), evaluating at $x=0$, to estimate

[TABLE]

and

[TABLE]

Note that we used Lemma 2.2, and Proposition 2.1 (2b). Plugging these results into (2.9), we see that for $t>0$ sufficiently large, $\|\chi_{[t,+\infty)}|\psi|^{p-1}\psi\|_{\dot{H}^{\frac{1}{4}}}$ is small. This completes the proof combining with (2.8). ∎

Proposition 2.5** (long-time perturbation theory).**

Let $p\geq 3$. For each $A\gg 1$, there exists $\epsilon_{0}=\epsilon_{0}(A)\ll 1$ and $c=c(A)\gg 1$ such that the following holds. Let $\psi\in H_{x}^{1}$ for all $t$ solving

[TABLE]

Let $\tilde{\psi}\in H_{x}^{1}$ for all $t$ and suppose that there exists $e\in L_{t>0}^{\tilde{q}}$ such that

[TABLE]

If

[TABLE]

and

[TABLE]

for some $t_{0}\geq 0$, then

[TABLE]

Proof.

Put $w=\psi-\tilde{\psi}$. Then $w$ satisfies

[TABLE]

where

[TABLE]

Since $\|\tilde{\psi}(0,\cdot)\|_{L^{q}_{[t_{0},+\infty)}}\leq A$, there exists a $N=N(A)$ so that the interval $[t_{0},+\infty)$ may be divided into the sum of $N(A)$ intervals. Namely, $[t_{0},+\infty)=\cup_{j=1}^{N(A)}I_{j}$ with $I_{j}=[t_{j},t_{j+1}]$ ($j=0,1,2,..$) so that $\|\tilde{\psi}(0,\cdot)\|_{L^{q}_{I_{j}}}\leq\eta$ ($\eta$ is small to be determined later). Let $t\in I_{j}$. Write the equation (2.10) in the integral form.

[TABLE]

We estimate the time $L^{q}$ norm of $w$ evaluated at $x=0$.

[TABLE]

The last term can be written as, taking into account for the delta potential in $W$,

[TABLE]

and then we estimate as follows.

[TABLE]

where, in the first inequality, we have used, by density of $C_{0}^{\infty}(I_{j})\subset L^{\tilde{q}}(I_{j})$, Sobolev embedding, and Proposition 2.1 (2a).

The first term of RHS is estimated by Hölder inequality as follows.

[TABLE]

Thus, we have

[TABLE]

We then obtain

[TABLE]

provided

[TABLE]

and

[TABLE]

Now take $t=t_{j+1}$ in (2.11), apply $e^{i(t-t_{j+1})\partial_{x}^{2}}$ to both hands,

[TABLE]

and we take $L^{q}(\mathbb{R}_{t})$ norm of this equation after evaluating at $x=0$,

[TABLE]

Thus, by (2.12),

[TABLE]

Iterating this inequalty starting from $j=0$, we have

[TABLE]

To satisfy (2.13) for all $I_{j}$ with $0\leq j\leq N-1,$ we require $\epsilon_{0}=\epsilon_{0}(N)$ to be sufficiently small such that $2^{N+2}C\epsilon_{0}<\left(\frac{1}{2C}\right)^{\frac{1}{p-1}}$ (i.e. $\epsilon_{0}$ needs to be taken in terms of $A$), and we obtain

[TABLE]

∎

3. Profile decomposition

Proposition 3.1** (profile decomposition).**

Let $p\geq 3.$ Suppose that $\{\psi_{n}\}$ is a uniformly bounded sequence in $H^{1}_{x}$. Then for each $M$, there exists a subsequence of $\{\psi_{n}\}$, also denoted $\{\psi_{n}\}$ and

1. (1)

for each $1\leq j\leq M$, there exists a (fixed in $n$) profile $\phi^{j}\in H^{1}$ 2. (2)

for each $1\leq j\leq M$, there exists a sequence (in $n$) of time shifts $t_{n}^{j}$ 3. (3)

there exists a sequence (in $n$) of remainders $w_{n}^{M}(x)$ in $H^{1}$ such that

[TABLE]

The time sequences have a pairwise divergence property: for $1\leq i\neq j\leq M$, we have

[TABLE]

The remainder sequence $\{w_{n}^{M}\}_{n}$ has the following asymptotic smallness property

[TABLE]

For fixed $M$ and any $0\leq\sigma_{c}\leq 1$, we have the asymptotic $\dot{H}^{\sigma_{c}}$ decoupling

[TABLE]

also we have

[TABLE]

Proof.

For $R>0$, let $\chi_{R}(\xi)$ be a smooth cutoff to $R^{-1}<|\xi|<R$. Let $A=\limsup_{n\to\infty}\|\psi_{n}\|_{H^{1}_{x}}$ and $B_{1}=\lim_{n\to\infty}\|[e^{it\partial_{x}^{2}}\psi_{n}](0)\|_{L_{\mathbb{R}_{t}}^{q}}$. If $B_{1}=0$, the proof is done. Let $B_{1}>0$. Since for $0\leq\sigma_{c}\leq 1$,

[TABLE]

[TABLE]

We may take a $R_{1}$ large enough so that $AR_{1}^{-\sigma_{c}}\leq B_{1}/2$ and $AR_{1}^{\sigma_{c}-1}\leq B_{1}/2$, specifically $R_{1}=\langle 2AB_{1}^{-1}\rangle^{\max\{\frac{1}{\sigma_{c}},\frac{1}{1-\sigma_{c}}\}}$ so that

[TABLE]

It thus follows, using Proposition 2.1(1),

[TABLE]

For the factor $\|[\check{\chi}_{R_{1}}*e^{it\partial_{x}^{2}}\psi_{n}](0)\|_{L_{t>0}^{2}}^{2}$, we use again the smoothing estimate of Proposition 2.1(1) to bound by

[TABLE]

Thus, we see $\lim_{n\to\infty}\|[\check{\chi}_{R_{1}}*e^{it\partial_{x}^{2}}\psi_{n}](0)\|_{L_{\mathbb{R}_{t}}^{\infty}}>(R_{1}A^{2})^{-\frac{1}{q-2}}(B_{1}/2)^{\frac{q}{q-2}}$, and we take a sequence $\{t_{n}^{1}\}_{n}$ such that

[TABLE]

and

[TABLE]

Consider the sequence $\{e^{it_{n}^{1}\partial_{x}^{2}}\psi_{n}\}_{n}$, which is uniformly bounded in $H^{1}_{x}$, and pass to subsequence such that $e^{it_{n}^{1}\partial_{x}^{2}}\psi_{n}$ converges weakly in $H^{1}_{x}$ to some $\phi^{1}\in H^{1}$. By Cauchy-Schwarz inequality, using that $\|\check{\chi}_{R_{1}}\|_{\dot{H}^{-\sigma_{c}}}\lesssim R_{1}^{\frac{1}{2}-\sigma_{c}}$ and (3.3),

[TABLE]

Then for any $0\leq\sigma_{c}\leq 1$

[TABLE]

If $|t_{n}^{1}|\to+\infty,$ since $\|[e^{-it\partial_{x}^{2}}\phi^{1}](0)\|_{L^{q}_{\mathbb{R}_{t}}}\leq\|\phi^{1}\|_{\dot{H}_{x}^{\sigma_{c}}}$, possibly taking a subsequence, we have $|[e^{-it_{n}^{1}\partial_{x}^{2}}\phi^{1}](0)|^{q}\to 0$ as $n\to+\infty$. On the other hand, since $\psi_{n}$ is uniformly bounded in $H^{1}_{x}$, there is a weak limit $\tilde{\psi}\in H^{1}_{x}$ and $\psi_{n}(0)\to\tilde{\psi}(0)$ as $n\to\infty$ by Proposition 4.1 of [11]. Then, we have

[TABLE]

i.e.

[TABLE]

If $t_{n}^{1}\to t^{*}$ for some finite $t^{*}$, by the time continuity of free Schrödinger group, $\lim_{n\to\infty}\psi_{n}(0)=\tilde{\psi}(0)=[e^{-it^{*}\partial_{x}^{2}}\phi^{1}](0)$. Thus we may write

[TABLE]

which again gives (3.4).

Repeat the process, keeping the same $A$ but switching to $B_{2}$ obtaining $R_{2}$ in terms of $B_{2}$. Basically this amounts to replacing $\psi_{n}$ by $\psi_{n}-e^{-it_{n}^{1}\partial_{x}^{2}}\phi^{1}$ and rewriting the above to obtain $t_{n}^{2}$ and $\phi^{2}$ where

[TABLE]

As a result,

[TABLE]

and same for

[TABLE]

If $t_{n}^{2}-t_{n}^{1}$ converged to something finite (say $t^{*}$), then $\phi^{2}$ would be the weak limit of $e^{it^{*}\partial_{x}^{2}}[e^{it_{n}^{1}\partial_{x}^{2}}\psi_{n}-\phi^{1}]$, which is zero, contradicting the lower bound. Hence $|t_{n}^{1}-t_{n}^{2}|\to\infty$ and thus

[TABLE]

Again repeat this process, we have

[TABLE]

Let $B_{M+1}:=\lim_{n\to+\infty}\|[e^{it\partial_{x}^{2}}w_{n}^{M}](0)\|_{L^{q}_{\mathbb{R}_{t}}}$ and we wish to show that $B_{M+1}\to 0.$ Note that from the above equality and the lower bound for $\|\phi^{M}\|_{\dot{H}^{\sigma_{c}}}$, we obtain

[TABLE]

whose LHS diverges if $B_{M}$ does not converge to [math]. ∎

Lemma 3.2**.**

With $w_{n}^{M}$ as defined in Proposition 3.1 (in particular, $w_{n}^{0}=\psi_{n}$), let

[TABLE]

Then

[TABLE]

Proof.

We will write the argument for $M=1$ (the general case is analogous). As in the proof of Proposition 3.1, let

[TABLE]

and

[TABLE]

and $\chi_{R_{1}}(\xi)$ be a cutoff to $R_{1}^{-1}\leq|\xi|\leq R_{1}$. As in the beginning of the proof of Proposition 3.1,

[TABLE]

This, and the similar estimates at the beginning of the proof of Proposition 3.1, show that it suffices to prove

[TABLE]

and this can be seen as follows. By the translation invariance of $L^{q}_{\mathbb{R}_{t}}$ norm,

[TABLE]

and by Sobolev embedding and Proposition 2.1, we have,

[TABLE]

∎

4. Minimal non scattering solution

In this section we will prove that there exists a minimal non scattering solution. For this purpose we prepare the following lemma which gives additional estimates under the situation (1) of Theorem 1.3. We recall that $\varphi_{0}$ is the ground state to (1.1). It is known that $\varphi_{0}(x)=2^{\frac{1}{p-1}}e^{-|x|}$ (see (1.9) of [11]).

Lemma 4.1**.**

Let $p>3$ and $\psi_{0}\in H^{1}_{x}.$ Assume $(\ref{E:ME})$ and $\eta(0)<1.$ If $\psi$ is a $H^{1}_{x}$ solution to (1.1), then for all $t\in\mathbb{R},$

[TABLE]

Furthermore, if we take $\delta>0$ such that $M(\psi_{0})^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\psi_{0})\leq(1-\delta)M(\varphi_{0})^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\varphi_{0}),$ then there exists $c_{\delta}>0$ such that for all $t\in\mathbb{R}$,

[TABLE]

Proof.

The upper bound of the energy in (4.1) follows by the definition of Energy $E$ and the focusing nonlinearity. Use the sharp Gagliardo-Nirenberg inequality and $\eta(t)<1$ for the lower bound, i.e.,

[TABLE]

where we have used the fact $\|\partial_{x}\varphi_{0}\|_{L^{2}}=\|\varphi_{0}\|_{L^{2}}=2^{\frac{1}{p-1}}$ in the last equality (see [11]). Next, we show (4.2). We may take $\delta_{1}=\delta_{1}(\delta)>0$ such that

[TABLE]

for all $t\in\mathbb{R}.$ Let

[TABLE]

By Gagliardo-Nirenberg inequality,

[TABLE]

where $g(y):=4(y^{2}-y^{\frac{p+1}{2}}).$ The inequality (4.3) implies the variable $y$ of $g(y)$ is in the interval $0\leq y\leq 1-\delta_{1}$ and then we see that there exists a constant $c=c_{\delta_{1}}>0$ such that $g(y)\geq cy^{2}$ if $0\leq y\leq 1-\delta_{1}$. ∎

Lemma 4.2**.**

(Existence of wave operator) Let $p>3$. Suppose $\psi^{+}\in H^{1}_{x}$ and

[TABLE]

There exists $\psi_{0}\in H^{1}_{x}$ such that $\psi$ solving (1.1) with initial data $\psi_{0}$ is global in $H^{1}_{x}$, with

[TABLE]

[TABLE]

and

[TABLE]

Moreover, if $\|[e^{it\partial_{x}^{2}}\psi^{+}](0)\|_{L^{q}_{t>0}}\leq\delta_{\text{sd}},$ then

[TABLE]

The statement above is for the case $t>0$, but the case $t<0$ can be similarly proved.

Proof.

It suffices to solve the integral equation:

[TABLE]

for $t\geq T$ with $T$ large. Since

[TABLE]

there exists a large $T>0$ such that $\|[e^{it\partial_{x}^{2}}\psi^{+}](0)\|_{L^{q}_{[T,\infty)}}\leq\delta_{\text{sd}}$. Thus we may solve as in the proof of Proposition 2.3.

[TABLE]

If $T$ is sufficiently large, we have $\|\psi(0,\cdot)\|_{L^{q}_{[T,+\infty)}}<2\|[e^{it\partial_{x}^{2}}\psi^{+}](0)\|_{L^{q}_{[T,+\infty)}}.$ Using this, similarly as in the proof of Proposition 2.4, we obtain if $t\geq T$,

[TABLE]

[TABLE]

which are small if $T$ is sufficiently large. Thus, $\psi(t)-e^{it\partial_{x}^{2}}\psi^{+}\to 0$ in $H^{1}_{x}$ as $t\to+\infty$. Note that $\|\partial_{x}e^{it\partial_{x}^{2}}\psi^{+}\|_{L^{2}_{x}}=\|\partial_{x}\psi^{+}\|_{L^{2}}$. On the other hand, since $[e^{it\partial_{x}^{2}}\psi^{+}](0)$ is uniformly bounded in $L^{q}_{t>0}$, there exists a sequence $\{t_{n}\}_{n}\to+\infty$ such that [$e^{it_{n}\partial_{x}^{2}}\psi^{+}](0)\to 0$ as $n\to+\infty$. Together with all these facts, we have

[TABLE]

Similarly, $M(\psi(t))=\|\psi^{+}\|_{L^{2}_{x}}^{2}$. It now follows from (4.4) that

[TABLE]

and

[TABLE]

We can take a large $T$ such that $\|\partial_{x}\psi(T)\|_{L^{2}_{x}}\|\psi(T)\|_{L^{2}_{x}}^{\frac{1-\sigma_{c}}{\sigma_{c}}}<\|\partial_{x}\varphi_{0}\|_{L^{2}_{x}}\|\varphi_{0}\|_{L^{2}_{x}}^{\frac{1-\sigma_{c}}{\sigma_{c}}}$. Then, applying Theorem 1.3 we evolve $\psi(t)$ from $T$ back to the time [math]. ∎

We are now in position to enter in the main subject of this section. If the initial data $\psi_{0}$ to (1.1) satisfies $M(\psi_{0})^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\psi_{0})\leq\frac{p-1}{2(p+1)}\delta_{sd}$ and $\eta(0)<1$, we have

[TABLE]

and the scattering holds by the small data scattering, Proposition 2.3. Now let $A$ be the infimum of $M(\psi)^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\psi)$, taken over all evolution of $\psi$ which does not scatter. In what follows $\mathrm{NLS}(t)\psi$ denotes the solution to (1.1) with initial data $\psi$. By the above argument, $0<\frac{p-1}{2(p+1)}\delta_{sd}\leq A$, and moreover due to Proposition 2.4, $A$ satisfies

1. (1)

For any $\psi$ such that $M(\psi)^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\psi)<A$, it holds $\|[\mathrm{NLS}(t)\psi](0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}<\infty$, 2. (2)

For any $A^{\prime}>A$, there exists a non scattering $\mathrm{NLS}(t)\psi$ for which

[TABLE]

If $A\geq M(\varphi_{0})^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\varphi_{0})$, Theorem 1.4 is true. We therefore proceed with the proof by assuming $A<M(\varphi_{0})^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\varphi_{0})$.

The first task is to apply the profile decomposition to show that there exists $\psi$ such that $M(\psi)^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\psi)=A$ and $\mathrm{NLS}(t)\psi$ does not scatter. We will call such a solution a minimal non scattering solution. Take a sequence of initial data $\psi_{0,n}$, with $1>\eta_{n}(0):=\|\psi_{0,n}\|_{L^{2}}^{\frac{1-\sigma_{c}}{\sigma_{c}}}\|\partial_{x}\psi_{0,n}\|_{L^{2}}/\|\varphi_{0}\|_{L^{2}}^{\frac{1-\sigma_{c}}{\sigma_{c}}}\|\partial_{x}\varphi_{0}\|_{L^{2}}$, each evolving to non scattering solutions, for which $M(\psi_{0,n})=1$, $E(\psi_{0,n})\geq A$ and $E(\psi_{0,n})\to A$. Apply the profile decomposition to $\psi_{0,n}$ which is uniformly bounded in $H^{1}$ to obtain, extracting a subsequence,

[TABLE]

where $M$ will be taken large later. Remark that each term in (4.6) is non negative by the same reason for (4.1), using the decompositions (3.1) and (3.2) in $\eta_{n}(0)<1$. Taking the limit $n\to\infty$ in both hand sides,

[TABLE]

for all $j$. Also, by $\sigma_{c}=0$ in (3.1), we have

[TABLE]

Here we consider two cases.

• Case 1

There are at least two indexes $j$ such that $\phi^{j}$ is not zero.

• Case 2

Only one profile is non zero, i.e. without loss of generality $\phi^{1}\neq 0$, and $\phi^{j}=0$ for all $j\geq 2.$

We begin with Case 1. By (4.8), we necessarily have $0\leq M(\phi^{j})<1$ for each $j$ which, by (4.7), implies that for $n$ sufficiently large

[TABLE]

with each $A_{j}<A$. For a given $j$, there are two possibilities. Case a) $|t_{n}^{j}|\to\infty$ as $n\to\infty$ and Case b) there is a finite limit $t_{*}$ such that $t_{n}^{j}\to t_{*}$ as $n\to\infty$. Both cases allow us to ensure the existence of a new profile $\tilde{\phi^{j}}\in H^{1}$ associated to $\phi^{j}$ such that

[TABLE]

indeed, if Case a) occurs, by the uniform $L^{q}$ integrability in time of $[e^{-it\partial_{x}^{2}}\phi^{j}](0)$ (cf. the same argument in Proposition 3.1), passing to a subsequence of $t_{n}^{j}$,

[TABLE]

and thus

[TABLE]

Since $A<M(\varphi_{0})^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\varphi_{0})$, $\phi^{j}$ satisfies the assumption of Lemma 4.2. Namely, there exists $\tilde{\phi^{j}}\in H^{1}$ such that

[TABLE]

with

[TABLE]

[TABLE]

and thus

[TABLE]

Therefore by the definition of threshold $A$, we have

[TABLE]

If the Case b), by the time continuity in $H^{1}_{x}$ norm of the linear flow, we know

[TABLE]

Thus it suffices to put $\tilde{\phi^{j}}:=\mathrm{NLS}(t_{*})[e^{-it_{*}\partial_{x}^{2}}\phi^{j}]$. Then this $\tilde{\phi}^{j}$ again satisfies (4.10). To see this, note first that by the $H^{1}$ continuity of the flow, sending $n\to\infty$ in (4.9) gives

[TABLE]

By (3.1) applied for $\sigma_{c}=0$ and $\sigma_{c}=1$, and the assumption that $\eta_{n}(0)<1$ for every $n$, we obtain that

[TABLE]

By the defining property of the threshold $A$, we have that the NLS flow with initial data $e^{-it_{*}\partial_{x}^{2}}\phi^{j}$ scatters, i.e.

[TABLE]

Now replace $e^{-it_{n}^{j}\partial_{x}^{2}}\phi^{j}$ by $\mathrm{NLS}(-t_{n}^{j})\tilde{\phi}^{j}$ in (4.5), and we have

[TABLE]

with

[TABLE]

Note that by Sobolev embedding and Proposition 2.1 (1),

[TABLE]

Thus we obtain,

[TABLE]

From this way of writing we might approximately see

[TABLE]

However, from (4.10), the RHS is finite in $L^{q}_{\mathbb{R}_{t}}$ norm, while the LHS cannot scatter by assumption, and so a contradiction could be deduced. We shall justify this argument by Proposition 2.5.

Let $v^{j}(t):=\mathrm{NLS}(t)\tilde{\phi}^{j},$ $\psi_{n}:=\mathrm{NLS}(t)\psi_{0,n},$ and $\tilde{\psi}_{n}=\sum_{j=1}^{M}v^{j}(t-t_{n}^{j}).$ Then, $\tilde{\psi}_{n}$ satisfies

[TABLE]

Here,

[TABLE]

We are going to show that

• 1

there exists a large constant $A$ independent of $M$ satisfying the following property: for any $M$ there is $n_{0}=n_{0}(M)$ such that if $n>n_{0},$ $\|\tilde{\psi}_{n}(0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}\leq A$.

• 2

For each $M$ and $\varepsilon>0$ there exists $n_{1}=n_{1}(M,\varepsilon)$ such that for $n>n_{1}$, $\|e_{n}\|_{L^{\tilde{q}}_{\mathbb{R}_{t}}}\leq\varepsilon.$

Remark that there exists $M_{1}=M_{1}(\varepsilon)$ such that for each $M>M_{1}$, there exists $n_{2}=n_{2}(M)$ such that if $n>n_{2}$, $\|[e^{it\partial_{x}^{2}}(\tilde{\psi}_{n}(0)-\psi_{n}(0))](0)\|_{L^{q}_{\mathbb{R}_{t}}}\leq\varepsilon.$ Thus, if the above 1 and 2 hold, it follows from Proposition 2.5 that for $n$ and $M$ sufficiently large, $\|\psi_{n}\|_{L^{q}_{\mathbb{R}_{t}}}<\infty,$ which gives a contradiction. Therefore it is enough to prove the above claims 1 and 2. First we prove the claim 1. Take $M_{0}$ large enough so that

[TABLE]

Then, by Lemma 3.2, for each $j>M_{0}$, we have $\|[e^{i(t-t_{n}^{j})\partial_{x}^{2}}\phi^{j}](0)\|_{L^{q}_{\mathbb{R}_{t}}}\leq\delta_{\text{sd}}.$ Thus by Lemma 4.2 we obtain, for each $j>M_{0}$, and for large $n$,

[TABLE]

By Minkowski inequality (since $p>3$),

[TABLE]

where we have used (4.11). The last terms $\sum_{j\neq m}\|v^{j}v^{m}\|_{L^{q/2}_{t}}$ can be made small if $n$ is large (see the argument below for the claim 2). On the other hand, using (4.5), the same argument for (3.2) allows us to obtain

[TABLE]

thus, integrating in time,

[TABLE]

which shows that $\sum_{j=M_{0}+1}^{M}\|e^{i(t-t_{n}^{j})\partial_{x}^{2}}\phi^{j}\|_{L^{q}_{\mathbb{R}_{t}}}^{2}$ is bounded independently of $M$ if $n>n_{0}$ since $\|[e^{it\partial_{x}^{2}}\psi_{0,n}](0)\|_{L^{q}_{\mathbb{R}_{t}}}\leq\|\psi_{0,n}\|_{\dot{H}^{\sigma_{c}}}$. Recall that $\|v^{j}(0,\cdot-t_{n}^{j})\|_{L^{q}_{\mathbb{R}_{t}}}=\|\mathrm{NLS}(t)\tilde{\phi}^{j}(0)\|_{L^{q}_{\mathbb{R}_{t}}}<\infty$. Therefore $\|\tilde{\psi_{n}}(0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}^{q}$ is bounded independently of $M$ provided $n>n_{0}$.

We next prove the claim 2. We see that $e_{n}$ is estimated using Hölder inequality with $\frac{1}{\tilde{q}}=\frac{p-2}{q}+\frac{2}{q}$ as follows.

[TABLE]

where we abbreviated $v^{j}(0,t-t_{n}^{j})$ as $v^{j}$. Here, note that by (4.10), for any $\varepsilon>0$, there exists a large $R>0$ such that

[TABLE]

Thus, taking large $n$ such that $|t_{n}^{j}-t_{n}^{k}|>2R$ with $j\neq k$ for such a $R>0$, we can estimate $\|v^{j}v^{k}\|_{L^{q/2}_{\mathbb{R}_{t}}}$ as follows:

[TABLE]

This shows that there exists $n_{1}$ such that the $L^{\tilde{q}}$ norm of $e_{n}$ is small if $n>n_{1}(M,\varepsilon)$.

Now we consider Case 2. In this case, we have $M(\phi^{1})\leq 1$ and $\lim_{n\to\infty}E(e^{-it_{n}^{1}\partial_{x}^{2}}\phi^{1})\leq A.$ As in the Case 1, by the existence of wave operator, there is $\tilde{\phi}^{1}\in H^{1}_{x}$ such that

[TABLE]

Put

[TABLE]

Then we can write

[TABLE]

with

[TABLE]

Let $\psi_{c}$ be the solution to (1.1) with initial data $\psi_{c}(0)=\tilde{\phi}^{1}$. Now we claim that $\|\psi_{c}(0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}=+\infty$ (and thus $M(\psi_{c})^{\frac{1-\sigma_{c}}{\sigma_{c}}}E(\psi_{c})=A$). We proceed as in the Case 1. Suppose $A:=\|\psi_{c}(0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}<\infty.$ By definition, $\|\mathrm{NLS}(t)\tilde{\phi}^{1}(0)\|_{L^{q}_{\mathbb{R}_{t}}}=\|\psi_{c}(0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}=A.$ For any shift $t^{\prime}$, we can say $\|\mathrm{NLS}(t-t^{\prime})\tilde{\phi}^{1}(0)\|_{L^{q}_{\mathbb{R}_{t}}}=\|\mathrm{NLS}(t)\tilde{\phi}^{1}(0)\|_{L^{q}_{\mathbb{R}_{t}}},$ thus we take in particular $t^{\prime}=t_{n}^{1}$ and operate $\mathrm{NLS}(t)$ to $\psi_{0,n}=\mathrm{NLS}(-t_{n}^{1})\tilde{\phi}^{1}+\tilde{w}_{n}^{M}.$ We apply the perturbation argument by Proposition 2.5 to

[TABLE]

with $\tilde{\psi}_{n}=\mathrm{NLS}(t-t_{n}^{1})\tilde{\phi}^{1}$ and $\|\tilde{\psi_{n}}(0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}=A<+\infty$. For $n$ and $M$ sufficiently large, we have

[TABLE]

and also the $L^{\tilde{q}}_{t}$ norm of the corresponding error term is estimated by $\epsilon_{0}$, where $\epsilon_{0}=\epsilon_{0}(A)$ is obtained in Proposition 2.5. Then, by Proposition 2.5, we have $\|\psi_{n}(0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}<\infty$, and this is a contradiction to non scattering assumption on $\psi_{n}.$

On the other hand, the proof of Lemma 5.6 in [10] allows us to have also,

Lemma 4.3**.**

Suppose $\{\psi(t,x),t\geq 0\}$ is precompact in $H^{1}_{x}$. Then for any $\varepsilon>0$, there exists $R_{\varepsilon}>0$ such that

[TABLE]

Using this Lemma and the local viriel identity (1.2), we conclude the following proposition.

Proposition 4.4**.**

Let $p>3$. Assume $\psi_{0}\in H^{1}$ satisfies (1.4) and $\eta(0)<1.$ Let $\psi(t,x)$ be the global solution to (1.1) with the initial data $\psi_{0}$ satisfying the precompactness: for any $\varepsilon>0$, there exists $R_{\varepsilon}>0$ such that

[TABLE]

Then $\psi_{0}\equiv 0$.

Proof.

Take $a(x)$ in the localized virial (1.2), as, for $R>0$ (which will be determined later), and for all $x\in\mathbb{R}$,

[TABLE]

where $\chi\in C_{0}^{\infty}(\mathbb{R}^{+})$, $\chi(r)=r^{2}$ for $r\leq 1$, and $\chi(r)=0$ for $r\geq 2$. Put $z_{R}(t):=\int_{\mathbb{R}}a(x)|\psi|^{2}dx$, then we have

[TABLE]

and

[TABLE]

with a constant $C_{0}=C_{0}(\|\chi^{\prime\prime}\|_{L^{\infty}},\|\chi^{(4)}\|_{L^{\infty}})$ uniform in $R$.

Take $0<\delta<1$ such that

[TABLE]

then by (4.2), there exists $c_{\delta}>0$ such that for any $t\in\mathbb{R}$

[TABLE]

Now, we choose $\varepsilon=\frac{c_{\delta}}{8+C_{0}}\|\partial_{x}\psi_{0}\|_{L^{2}}^{2}$ in (4.12), then for sufficiently large $R_{1}>\max\{1,R_{\varepsilon}\}$,

[TABLE]

Thus, by the choice of $R=R_{1}$, we have $(\ref{eq:pre})\geq c_{\delta}\|\partial_{x}\psi_{0}\|_{L^{2}}^{2}-4\varepsilon$ and so

[TABLE]

Integration in time then implies

[TABLE]

On the other hand,

[TABLE]

where $C$ depends on $p,\|\psi_{0}\|_{L^{2}}$, and $\|\partial_{x}\psi_{0}\|_{L^{2}}.$ This is absurd except the case $\psi_{0}\equiv 0$. ∎

Finally we complete our arguments with

Proposition 4.5**.**

[TABLE]

with $\psi_{c}$ obtained above as the minimal non scattering solution, is precompact in $H^{1}_{x}$.

The proof for this proposition is similar to the proof for the existence of $\psi_{c}$, and we omit it. We apply Proposition 4.4 to $\psi_{c}$, and we have $\psi_{c}(0)\equiv 0$, which contradicts the fact that $\|\psi_{c}(0,\cdot)\|_{L^{q}_{\mathbb{R}_{t}}}=+\infty$. This concludes the statement of Theorem 1.4. ∎

Acknowledgment The work described in this paper is a result of a collaboration made possible by the IMA’s annual program workshop ”Mathematical and Physical Models of Nonlinear Optics.” This work was supported by JSPS KAKENHI Grant Number 15K04944.