# The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions

Jason Murphy  Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA
<EMAIL_ADDRESS>
###### Abstract.

We consider the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger
equation in dimensions $d\geq 5.$ In the spirit of Kenig and Merle [10], we
combine a concentration-compactness approach with the Lin–Strauss Morawetz
inequality to prove that if a solution $u$ is bounded in $\dot{H}^{1/2}$
throughout its lifespan, then $u$ is global and scatters.

##  1\. Introduction

We consider the initial-value problem for the defocusing
$\dot{H}_{x}^{1/2}$-critical nonlinear Schrödinger equation in dimensions
$d\geq 5$:

$\left\\{\begin{array}[]{ll}(i\partial_{t}+\Delta)u=|u|^{\frac{4}{d-1}}u\\\
u(0)=u_{0},\end{array}\right.$ (1.1)

with $u:\mathbb{R}_{t}\times\mathbb{R}_{x}^{d}\to\mathbb{C}$. This equation is
deemed $\dot{H}_{x}^{1/2}$-critical because the rescaling that preserves the
class of solutions to (1.1), that is,
$u(t,x)\mapsto\lambda^{\frac{d-1}{2}}u(\lambda^{2}t,\lambda x),$ leaves
invariant the $\dot{H}_{x}^{1/2}$-norm of the initial data.

In [10], Kenig and Merle considered (1.1) with $d=3$. They proved that if a
solution $u$ stays bounded in $\dot{H}_{x}^{1/2}$ throughout its lifespan,
then $u$ must be global and scatter. The same statement for $d=4$ was proven
as a special case of the results of [20]. In this short note, we establish
this result for all $d\geq 5$.

We begin with some definitions.

######  Definition 1.1 (Solution).

A function $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ on a time interval $I\ni 0$
is a _solution_ to (1.1) if it belongs to
$C_{t}\dot{H}_{x}^{1/2}(K\times\mathbb{R}^{d})\cap
L_{t,x}^{\frac{2(d+2)}{d-1}}(K\times\mathbb{R}^{d})$ for every compact
$K\subset I$ and obeys the Duhamel formula

$u(t)=e^{it\Delta}u_{0}-i\int_{0}^{t}e^{i(t-s)\Delta}(|u|^{\frac{4}{d-1}}u)(s)\,ds$

for all $t\in I$. We call $I$ the _lifespan_ of $u$; we say $u$ is a _maximal-
lifespan solution_ if it cannot be extended to any strictly larger interval.
If $I=\mathbb{R}$, we say $u$ is _global_.

######  Definition 1.2 (Scattering size and blowup).

We define the _scattering size_ of a solution $u$ to (1.1) on a time interval
$I$ by

$S_{I}(u):=\int_{I}\int_{\mathbb{R}^{d}}|u(t,x)|^{\frac{2(d+2)}{d-1}}\,dx\,dt.$
(1.2)

If there exists $t\in I$ such that $S_{[t,\sup I)}(u)=\infty$, then we say $u$
_blows up forward in time_. Similarly, if there exists $t\in I$ such that
$S_{(\inf I,t]}(u)=\infty$, then we say $u$ _blows up backward in time_.

On the other hand, if $u$ is global with $S_{\mathbb{R}}(u)<\infty$, then
standard arguments show that $u$ _scatters_ , that is, there exist unique
$u_{\pm}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$ such that

$\lim_{t\to\pm\infty}\|u(t)-e^{it\Delta}u_{\pm}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}=0.$

Our main result is the following

######  Theorem 1.3.

Let $d\geq 5$ and let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-
lifespan solution to (1.1) such that $u\in
L_{t}^{\infty}\dot{H}_{x}^{1/2}(I\times\mathbb{R}^{d}).$ Then $u$ is global
and scatters, with

$S_{\mathbb{R}}(u)\leq
C\big{(}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}(\mathbb{R}\times\mathbb{R}^{d})}\big{)}\quad\text{for
some function}\quad C:[0,\infty)\to[0,\infty).$

Following the approach of Kenig and Merle [10], we will establish Theorem 1.3
by combining a concentration-compactness argument with the Lin–Strauss
Morawetz inequality of [18]. This estimate is very useful in the study of
(1.1), as it has critical scaling for this problem. In fact, it is the
concentration-compactness component that comprises most of this note; once we
have reduced the problem to the study of almost periodic solutions, we can
quickly bring the argument to a conclusion.

We first need a good local-in-time theory. Building off arguments of Cazenave
and Weissler [3], we can prove the following local well-posedness result (see
Remark 3.8).

######  Theorem 1.4 (Local well-posedness).

Let $d\geq 5$ and $u_{0}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$. Then there
exists a unique maximal-lifespan solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) such that:

$\bullet$ $($Local existence$)$ $I$ is an open neighborhood of $0$.

$\bullet$ $($Blowup criterion$)$ If $\sup I<\infty$, then $u$ blows up forward
in time. If $|\inf I|<\infty$, then $u$ blows up backward in time.

$\bullet$ $($Existence of wave operators$)$ For any
$u_{+}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$, there is a unique solution $u$ to
(1.1) such that $u$ scatters to $u_{+}$, that is,

$\lim_{t\to\infty}\|u(t)-e^{it\Delta}u_{+}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}=0.$

A similar statement holds backward in time.

$\bullet$ $($Small-data global existence$)$ If
$\|u_{0}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}$ is sufficiently small
depending on $d$, then $u$ is global and scatters, with
$S_{\mathbb{R}}(u)\lesssim\|u_{0}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}^{\frac{2(d+2)}{d-1}}.$

###  1.1. Outline of the proof of Theorem 1.3

We argue by contradiction and suppose that Theorem 1.3 fails. Recalling from
Theorem 1.4 that Theorem 1.3 holds for sufficiently small initial data, we
deduce the existence of a threshold size, below which Theorem 1.3 holds, but
above which we can find (almost) counterexamples. We then use a limiting
argument to find blowup solutions _at_ this threshold, and show that such
minimal blowup solutions must possess strong concentration properties.
Finally, in Sections 5 and 6, we show that solutions to (1.1) with such
properties cannot exist.

The main property of these solutions is that of almost periodicity:

######  Definition 1.5 (Almost periodic solutions).

A solution $u$ to (1.1) with lifespan $I$ is said to be _almost periodic_
$($_modulo symmetries $)$_ if $u\in
L_{t}^{\infty}\dot{H}_{x}^{1/2}(I\times\mathbb{R}^{d})$ and there exist
functions $N:I\to\mathbb{R}^{+},$ $x:I\to\mathbb{R}^{d}$, and
$C:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that

$\int_{|x-x(t)|\geq\frac{C(\eta)}{N(t)}}\big{|}|\nabla|^{1/2}u(t,x)\big{|}^{2}\,dx+\int_{|\xi|\geq
C(\eta)N(t)}|\xi|\,|\widehat{u}(t,\xi)|^{2}\,d\xi\leq\eta$

for all $t\in I$ and $\eta>0$. We call $N$ the _frequency scale function_ ,
$x$ the _spatial center function_ , and $C$ the _compactness modulus
function_.

######  Remark 1.6.

Using Arzelà–Ascoli and Sobolev embedding, one can derive the following: for a
nonzero almost periodic solution $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to
(1.1), there exists $C(u)>0$ such that

$\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{\frac{2d}{d-1}}\,dx\gtrsim_{u}1\quad\text{uniformly
for }t\in I.$

We can now describe the first major step in the proof of Theorem 1.3.

######  Theorem 1.7 (Reduction to almost periodic solutions).

If Theorem 1.3 fails, then there exists a maximal-lifespan solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) such that $u$ is almost
periodic and blows up in both time directions.

The reduction to almost periodic solutions has become a well-known and widely
used technique in the study of dispersive equations at critical regularity.
The existence of such solutions was first established by Keraani [12] in the
context of the mass-critical NLS, while Kenig and Merle [9] were the first to
use them to prove a global well-posedness result (in the energy-critical
setting). These techniques have since been adapted to a variety of settings
(see [7, 10, 13, 14, 15, 16, 20, 23, 24] for some examples in the case of
NLS), and the general approach is well-understood.

The argument, which we will carry out in Section 4, requires three
ingredients: (i) a profile decomposition for the linear propagator, (ii) a
stability result for the nonlinear equation, and (iii) a decoupling statement
for nonlinear profiles. The first profile decompositions established for
$e^{it\Delta}$ were adapted to the mass- and energy-critical settings (see [1,
2, 11, 19]); the case of non-conserved critical regularity was addressed in
[21]. We will be able to import the profile decomposition we need directly
from [21] (see Section 2.4).

Ingredients (ii) and (iii) are closely related, in that the decoupling must be
established in a space that is dictated by the stability result. Stability
results most often require errors to be small in a space with the scaling-
critical number of derivatives (say $s_{c}$). In [11], Keraani showed how to
establish the decoupling in such a space for the energy-critical problem (that
is, $s_{c}=1$). The argument relies on pointwise estimates and hence is also
applicable to the mass-critical problem ($s_{c}=0$). For
$s_{c}\notin\\{0,1\\}$, however, the nonlocal nature of $|\nabla|^{s_{c}}$
prevents the direct use of this argument.

In certain cases for which $s_{c}\notin\\{0,1\\}$ it has nonetheless been
possible to adapt the arguments of [11] to establish the decoupling in a space
with $s_{c}$ derivatives. Kenig and Merle [10] were able to succeed in the
case $s_{c}=1/2$, $d=3$ (for which the nonlinearity is cubic) by exploiting
the polynomial nature of the nonlinearity and making use of a paraproduct
estimate. Killip and Vişan [16] handled some cases for which $s_{c}>1$ by
utilizing a square function of Strichartz that shares estimates with
$|\nabla|^{s_{c}}$. In [20], some cases were treated for which $s_{c}\in(0,1)$
(and the nonlinearity is non-polynomial) by making use of the Littlewood–Paley
square function and working at the level of individual frequencies.

In this paper, we take a simpler approach to (ii) and (iii), inspired by the
work of Holmer and Roudenko [7] on the focusing $\dot{H}_{x}^{1/2}$-critical
NLS in $d=3$. It relies on the observation that for $s_{c}=1/2$, one can
develop a stability theory for NLS that only requires errors to be small in a
space without derivatives. In Section 3, we do exactly this (see Theorem 3.6).
To prove the decoupling in a space without derivatives, we can then rely
simply on pointwise estimates and apply the arguments of [11] directly (see
Lemma 4.3). By proving a more refined stability result, we are thus able to
avoid entirely the technical issues related to fractional differentiation
described above. In this way, we can greatly simplify the analysis needed to
carry out the reduction to almost periodic solutions in our setting.

Continuing from Theorem 1.7, we can make some further reductions to the class
of solutions that we consider. In particular, we can prove the following

######  Theorem 1.8.

If Theorem 1.3 fails, then there exists an almost periodic solution
$u:[0,T_{max})\to\mathbb{C}$ to (1.1) with the following properties:

$(i)$ $u$ blows up forward in time,

$(ii)$ $\inf_{t\in[0,T_{max})}N(t)\geq 1,$

$(iii)$ $|x(t)|\lesssim_{u}\int_{0}^{t}N(s)\,ds$ for all $t\in[0,T_{max}).$

Let us briefly sketch the proof of Theorem 1.8. Beginning with an almost
periodic solution as in Theorem 1.7 and using a rescaling argument (as in [24,
Theorem 3.3], for example), one can deduce the existence of an almost periodic
blowup solution that does not escape to artibrarily low frequencies on at
least half of its maximal lifespan, say $[0,T_{max})$. In this way, we may
find an almost periodic solution such that $(i)$ and $(ii)$ hold.

It remains to see how one can modify an almost periodic solution
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ so that $(iii)$ holds. We may
always translate $u$ so that $x(0)=0$; thus, it suffices to show that we can
modify the modulation parameters of $u$ so that $|\dot{x}(t)|\sim_{u}N(t)$ for
a.e. $t\in[0,T_{max})$. This will follow from the following local constancy
property for the modulation parameters of almost periodic solutions (see [14,
Lemma 5.18], for example):

######  Lemma 1.9 (Local constancy).

Let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a maximal-lifespan almost
periodic solution to (1.1). Then there exists $\delta=\delta(u)>0$ such that
if $t_{0}\in I$, then

$[t_{0}-\delta N(t_{0})^{-2},t_{0}+\delta N(t_{0})^{-2}]\subset I,$

with

$N(t)\sim_{u}N(t_{0}),\quad|x(t)-x(t_{0})|\lesssim_{u}N(t_{0})^{-1}\quad\text{for}\quad|t-t_{0}|\leq\delta
N(t_{0})^{-2}.$

Using Lemma 1.9, we may subdivide $[0,T_{max})$ into _characteristic
subintervals_ $I_{k}$ and set $N(t)$ to be constant and equal to some $N_{k}$
on each $I_{k}$. Note that $|I_{k}|\sim_{u}N_{k}^{-2}$ and that this requires
us to modify the compactness modulus function by a (time-independent)
multiplicative factor. We may then modify $x(t)$ by $O(N(t)^{-1})$ so that
$x(t)$ becomes piecewise linear on each $I_{k}$, with
$|\dot{x}(t)|\sim_{u}N(t)$ for $t\in I_{k}^{\circ}$. Thus, we get
$|\dot{x}(t)|\sim_{u}N(t)$ for a.e. $t\in[0,T_{max}),$ as desired.

To complete the proof of Theorem 1.3, it therefore suffices to rule out the
existence of the almost periodic solutions described in Theorem 1.8.

In Section 5, we preclude the possibility of finite time blowup (i.e.
$T_{max}<\infty$). To do this, we make use of the following ‘reduced’ Duhamel
formula for almost periodic solutions, which one can prove by adapting the
argument in [14, Proposition 5.23]:

######  Proposition 1.10 (Reduced Duhamel formula).

Let $u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ be an almost periodic
solution to (1.1). Then for all $t\in[0,T_{max})$, we have

$u(t)=i\lim_{T\to
T_{max}}\int_{t}^{T}e^{i(t-s)\Delta}(|u|^{\frac{4}{d-1}}u)(s)\,ds,$

where the limits are taken in the weak $\dot{H}_{x}^{1/2}$ topology.

Using Proposition 1.10, Strichartz estimates, and conservation of mass, we can
show that an almost periodic solution that blows up in finite time must have
zero mass, contradicting the fact that the solution blows up in the first
place.

In Section 6, we use the Lin–Strauss Morawetz inequality to rule out the
remaining case, $T_{max}=\infty$. This estimate tells us that solutions to NLS
that are bounded in $\dot{H}_{x}^{1/2}$ cannot remain concentrated near the
origin for too long. However, almost periodic solutions to (1.1) as in Theorem
1.8 with $T_{max}=\infty$ do essentially this; thus we can reach a
contradiction in this case.

### Acknowledgements

I am grateful to my advisors Rowan Killip and Monica Vişan for helpful
discussions and for a careful reading of the manuscript. This work was
supported in part by NSF grant DMS-1001531 (P.I. Rowan Killip).

##  2\. Notation and useful lemmas

###  2.1. Some notation

We write $X\lesssim Y$ or $Y\gtrsim X$ whenever $X\leq CY$ for some
$C=C(d)>0$. If $X\lesssim Y\lesssim X$, we write $X\sim Y$. Dependence on
additional parameters will be indicated with subscripts, for example,
$X\lesssim_{u}Y$.

For a spacetime slab $I\times\mathbb{R}^{d}$, we define

$\|u\|_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}:=\|\,\|u(t)\|_{L_{x}^{r}(\mathbb{R}^{d})}\,\|_{L_{t}^{q}(I)}.$

If $q=r$, we write $L_{t}^{q}L_{x}^{q}=L_{t,x}^{q}.$ We also sometimes write
$\|f\|_{L_{x}^{r}(\mathbb{R}^{d})}=\|f\|_{L_{x}^{r}}$.

We define the Fourier transform on $\mathbb{R}^{d}$ by

$\widehat{f}(\xi):=(2\pi)^{-d/2}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)\,dx.$

For $s>-d/2$, we then define the fractional differentiation operator
$|\nabla|^{s}$ and the homogeneous Sobolev norm via
$\widehat{|\nabla|^{s}f}(\xi):=|\xi|^{s}\widehat{f}(\xi)$ and
$\|f\|_{\dot{H}_{x}^{s}(\mathbb{R}^{d})}:=\||\nabla|^{s}f\|_{L_{x}^{2}(\mathbb{R}^{d})}.$

###  2.2. Basic harmonic analysis

Let $\varphi$ be a radial bump function supported in the ball
$\\{\xi\in\mathbb{R}^{d}:|\xi|\leq\tfrac{11}{10}\\}$ and equal to 1 on the
ball $\\{\xi\in\mathbb{R}^{d}:|\xi|\leq 1\\}.$ For $N\in 2^{\mathbb{Z}}$, we
define the Littlewood–Paley projection operators via

$\displaystyle\widehat{P_{\leq
N}f}(\xi):=\varphi(\tfrac{\xi}{N})\widehat{f}(\xi),\quad\widehat{P_{N}f}(\xi):=\big{(}\varphi(\tfrac{\xi}{N})-\varphi(\tfrac{2\xi}{N})\big{)}\widehat{f}(\xi),\quad
P_{>N}:=\text{Id}-P_{\leq N}.$

We note that these operators commute with $e^{it\Delta}$ and all differential
operators, as they are Fourier multiplier operators. They also obey the
following

######  Lemma 2.1 (Bernstein estimates).

For $1\leq r\leq q\leq\infty$ and $s\geq 0$,

$\displaystyle\|P_{>N}f\|_{L_{x}^{r}(\mathbb{R}^{d})}\lesssim
N^{-s}\||\nabla|^{s}f\|_{L_{x}^{r}(\mathbb{R}^{d})},\quad\|P_{N}f\|_{L_{x}^{q}(\mathbb{R}^{d})}\lesssim
N^{\frac{d}{r}-\frac{d}{q}}\|f\|_{L_{x}^{r}(\mathbb{R}^{d})}.$

We will also need some fractional calculus estimates.

######  Lemma 2.2 (Fractional chain rule, [5]).

Suppose $G\in C^{1}(\mathbb{C})$ and $s\in(0,1]$. Let $1<r,r_{1}<\infty,$
$1<r_{2}\leq\infty$ be such that
$\tfrac{1}{r}=\tfrac{1}{r_{1}}+\tfrac{1}{r_{2}}.$ Then

$\||\nabla|^{s}G(u)\|_{L_{x}^{r}}\lesssim\|G^{\prime}(u)\|_{L_{x}^{r_{1}}}\||\nabla|^{s}u\|_{L_{x}^{r_{2}}}.$

######  Lemma 2.3 (Derivatives of differences, [17]).

Let $F(u)=|u|^{p}u$ for some $p>0$ and let $0<s<1$. For
$1<r,r_{1},r_{2}<\infty$ such that
$\tfrac{1}{r}=\tfrac{1}{r_{1}}+\tfrac{p}{r_{2}},$ we have

$\||\nabla|^{s}[F(u+v)-F(u)]\|_{L_{x}^{r}}\lesssim\||\nabla|^{s}u\|_{L_{x}^{r_{1}}}\|v\|_{L_{x}^{r_{2}}}^{p}+\||\nabla|^{s}v\|_{L_{x}^{r_{1}}}\|u+v\|_{L_{x}^{r_{2}}}^{p}.$

###  2.3. Strichartz estimates

Let $e^{it\Delta}$ be the free Schrödinger propagator:

$[e^{it\Delta}f](x)=(4\pi
it)^{-d/2}\int_{\mathbb{R}^{d}}e^{i|x-y|^{2}/4t}f(y)\,dy\quad\text{for }t\neq
0.$

For $d\geq 3$, we call a pair of exponents $(q,r)$ _Schrödinger admissible_ if
$2\leq q,r\leq\infty$ and $\tfrac{2}{q}+\tfrac{d}{r}=\tfrac{d}{2}$. For a time
interval $I$ and $s\geq 0$, we define the Strichartz space $\dot{S}^{s}(I)$
via the norm

$\|u\|_{\dot{S}^{s}(I)}=\sup\big{\\{}\||\nabla|^{s}u\|_{L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})}:(q,r)\text{
Schr\"{o}dinger admissible}\big{\\}}.$

We will make frequent use of the following standard estimates for
$e^{it\Delta}$:

######  Lemma 2.4 (Strichartz estimates, [6, 8, 22]).

Let $d\geq 3$, $s\geq 0$ and let $I$ be a compact time interval. Let
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a solution to the forced Schrödinger
equation $(i\partial_{t}+\Delta)u=F$. Then for any $t_{0}\in I$, we have

$\|u\|_{\dot{S}^{s}(I)}\lesssim\|u(t_{0})\|_{\dot{H}_{x}^{s}(\mathbb{R}^{d})}+\min\bigg{\\{}\||\nabla|^{s}F\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}(I\times\mathbb{R}^{d})},\||\nabla|^{s}F\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}(I\times\mathbb{R}^{d})}\bigg{\\}}.$

###  2.4. Concentration-compactness

We record here the linear profile decomposition that we will use in Section 4.
We begin with the following

######  Definition 2.5 (Symmetry group).

For any position $x_{0}\in\mathbb{R}^{d}$ and scaling parameter $\lambda>0$,
we define a unitary transformation
$g_{x_{0},\lambda}:\dot{H}_{x}^{1/2}(\mathbb{R}^{d})\to\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$
by

$[g_{x_{0},\lambda}f](x):=\lambda^{-\frac{d-1}{2}}f(\lambda^{-1}(x-x_{0})).$

We let $G$ denote the group of such transformations.

We now state the linear profile decomposition. For the mass-critical NLS, this
result was originally proven in [1, 2, 19], while for the energy-critical NLS,
it was established in [11]. In the generality we need, a proof can be found in
[21].

######  Lemma 2.6 (Linear profile decomposition, [21]).

Let $\\{u_{n}\\}_{n\geq 1}$ be a bounded sequence in
$\dot{H}_{x}^{1/2}(\mathbb{R}^{d}).$ After passing to a subsequence if
necessary, there exist functions $\\{\phi^{j}\\}_{j\geq
1}\subset\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$, group elements $g_{n}^{j}\in G$
$($with parameters $x_{n}^{j}$ and $\lambda_{n}^{j}$$)$, and times
$t_{n}^{j}\in\mathbb{R}$ such that for all $J\geq 1$, we have the following
decomposition:

$u_{n}=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+w_{n}^{J}.$

This decomposition satisfies the following properties:

$\bullet$ For each $j$, either $t_{n}^{j}\equiv 0$ or $t_{n}^{j}\to\pm\infty$
as $n\to\infty.$

$\bullet$ For $J\geq 1$, we have the following decoupling:

$\lim_{n\to\infty}\bigg{[}\|u_{n}\|_{\dot{H}_{x}^{1/2}}^{2}-\sum_{j=1}^{J}\|\phi^{j}\|_{\dot{H}_{x}^{1/2}}^{2}-\|w_{n}^{J}\|_{\dot{H}_{x}^{1/2}}^{2}\bigg{]}=0.$
(2.1)

$\bullet$ For any $j\neq k$, we have the following asymptotic orthogonality
condition:

$\frac{\lambda_{n}^{j}}{\lambda_{n}^{k}}+\frac{\lambda_{n}^{k}}{\lambda_{n}^{j}}+\frac{|x_{n}^{j}-x_{n}^{k}|^{2}}{\lambda_{n}^{j}\lambda_{n}^{k}}+\frac{|t_{n}^{j}(\lambda_{n}^{j})^{2}-t_{n}^{k}(\lambda_{n}^{k})^{2}|}{\lambda_{n}^{j}\lambda_{n}^{k}}\to\infty\quad\text{as
}n\to\infty.$ (2.2)

$\bullet$ For all $n$ and all $J\geq 1$, we have
$w_{n}^{J}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$, with

$\lim_{J\to\infty}\limsup_{n\to\infty}\|e^{it\Delta}w_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}(\mathbb{R}\times\mathbb{R}^{d})}=0.$
(2.3)

##  3\. Stability theory

In this section, we develop a stability theory for (1.1). The main result of
this section is Theorem 3.6, which will play a key role in the reduction to
almost periodic solutions carried out in Section 4. Throughout this section,
we will denote the nonlinearity $|u|^{\frac{4}{d-1}}u$ by $F(u)$.

We begin by recording a local well-posedness result of Cazenave and Weissler
[3]. This result requires the data to belong to the inhomogeneous Sobolev
space, so that a contraction mapping argument may be run in mass-critical
spaces.

######  Theorem 3.1 (Standard local well-posedness [3]).

Let $d\geq 5$ and $u_{0}\in{H}_{x}^{1/2}(\mathbb{R}^{d})$. If $I\ni 0$ is a
time interval such that

$\||\nabla|^{1/2}e^{it\Delta}u_{0}\|_{L_{t}^{\frac{2(d+1)}{d-1}}L_{x}^{\frac{2d(d+1)}{d^{2}-d+2}}(I\times\mathbb{R}^{d})}$

is sufficiently small, then we may find a unique solution
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1).

Next, we turn to the stability results. We will make use of function spaces
that are critical with respect to scaling, but do not involve any derivatives.
In particular, for a time interval $I$, we define the following norms:

$\displaystyle\|u\|_{X(I)}:=\|u\|_{L_{t}^{\frac{4(d+1)}{d-1}}L_{x}^{\frac{2(d+1)}{d-1}}(I\times\mathbb{R}^{d})},\quad\|F\|_{Y(I)}:=\|F\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}(I\times\mathbb{R}^{d})}.$

We first relate the $X$-norm to the usual Strichartz norms. By Sobolev
embedding, we get $\|u\|_{X(I)}\lesssim\|u\|_{\dot{S}^{1/2}(I)},$ while Hölder
and Sobolev embedding together imply

$\|u\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}(I\times\mathbb{R}^{d})}\lesssim\|u\|_{X(I)}^{c}\|u\|_{\dot{S}^{1/2}(I)}^{1-c}\quad\text{for
some}\quad 0<c(d)<1.$ (3.1)

Next, we record a Strichartz estimate, which one can prove via the standard
approach (namely, by applying the dispersive estimate and
Hardy–Littlewood–Sobolev).

######  Lemma 3.2.

Let $I$ be a compact time interval and $t_{0}\in I$. Then for all $t\in I$,

$\bigg{\|}\int_{t_{0}}^{t}e^{i(t-s)\Delta}F(s)\,ds\bigg{\|}_{X(I)}\lesssim\|F\|_{Y(I)}.$
(3.2)

Finally, we collect some estimates that will allow us to control the
nonlinearity.

######  Lemma 3.3.

Fix $d\geq 5$. Then, with spacetime norms over $I\times\mathbb{R}^{d}$, we
have

$\displaystyle\|F(u)\|_{Y(I)}$
$\displaystyle\lesssim\|u\|_{X(I)}^{\frac{d+3}{d-1}}$ (3.3)
$\displaystyle\|F(u)-F(\tilde{u})\|_{Y(I)}$
$\displaystyle\lesssim\big{\\{}\|u\|_{X(I)}^{\frac{4}{d-1}}+\|\tilde{u}\|_{X(I)}^{\frac{4}{d-1}}\big{\\}}\|u-\tilde{u}\|_{X(I)}$
(3.4) $\displaystyle\||\nabla|^{1/2}F(u)\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\|u\|_{X(I)}^{\frac{4}{d-1}}\|u\|_{\dot{S}^{1/2}(I)}$
(3.5)
$\displaystyle\||\nabla|^{1/2}[F(u)-F(\tilde{u})]\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim\|u-\tilde{u}\|_{X(I)}^{\frac{4}{d-1}}\|\tilde{u}\|_{\dot{S}^{1/2}(I)}+\|u\|_{X(I)}^{\frac{4}{d-1}}\|u-\tilde{u}\|_{\dot{S}^{1/2}(I)}.$
(3.6)

###### Proof.

We first note that (3.3) follows from Hölder, while (3.4) follows from the
fundamental theorem of calculus followed by Hölder.

Next, we see that (3.5) follows from Hölder and the fractional chain rule.
Indeed,

$\||\nabla|^{1/2}F(u)\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}\lesssim\|u\|_{L_{t}^{\frac{4(d+1)}{d-1}}L_{x}^{\frac{2(d+1)}{d-1}}}^{\frac{4}{d-1}}\||\nabla|^{1/2}u\|_{L_{t}^{\frac{2(d+1)}{d-1}}L_{x}^{\frac{2d(d+1)}{d^{2}-d+2}}}.$

Using these same exponents with Lemma 2.3, we deduce (3.6). ∎

We may now state our first stability result.

######  Lemma 3.4 (Short-time perturbations).

Let $d\geq 5$ and let $I$ be a compact time interval, with $t_{0}\in I$. Let
$\tilde{u}:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a solution to
$(i\partial_{t}+\Delta)\tilde{u}=F(\tilde{u})+e$ with
$\tilde{u}(t_{0})=\tilde{u}_{0}\in\dot{H}_{x}^{1/2}$. Suppose

$\|\tilde{u}\|_{\dot{S}^{1/2}(I)}\leq
E\quad\text{and}\quad\||\nabla|^{1/2}e\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}(I\times\mathbb{R}^{d})}\leq
E$ (3.7)

for some $E>0$. Let $u_{0}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$ satisfy

$\displaystyle\|u_{0}-\tilde{u}_{0}\|_{\dot{H}_{x}^{1/2}}$ $\displaystyle\leq
E,$ (3.8)

and suppose that we have the smallness conditions

$\displaystyle\|\tilde{u}\|_{X(I)}$ $\displaystyle\leq\delta,$ (3.9)
$\displaystyle\|e^{i(t-t_{0})\Delta}(u_{0}-\tilde{u}_{0})\|_{X(I)}+\|e\|_{Y(I)}$
$\displaystyle\leq\varepsilon,$ (3.10)

for some small $0<\delta=\delta(E)$ and $0<\varepsilon<\varepsilon_{0}(E).$
Then there exists $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ solving (1.1) with
$u(t_{0})=u_{0}$ such that

$\displaystyle\|u-\tilde{u}\|_{X(I)}+\|F(u)-F(\tilde{u})\|_{Y(I)}$
$\displaystyle\lesssim\varepsilon,$ (3.11)
$\displaystyle\|u-\tilde{u}\|_{\dot{S}^{1/2}(I)}+\||\nabla|^{1/2}[F(u)-F(\tilde{u})]\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim_{E}1.$ (3.12)

###### Proof.

We first suppose $u_{0}\in L_{x}^{2}$, so that Theorem 3.1 provides the
solution $u$. We will then prove (3.11) and (3.12) as _a priori_ estimates.
After the lemma is proven for $u_{0}\in H_{x}^{1/2}$, we can use approximation
by $H_{x}^{1/2}$-functions to see that the lemma holds for
$u_{0}\in\dot{H}_{x}^{1/2}$. Throughout the proof, spacetime norms will be
over $I\times\mathbb{R}^{d}$.

We will first show

$\|u\|_{X(I)}\lesssim\delta.$ (3.13)

By the triangle inequality, (3.2), (3.3), (3.9), and (3.10), we get

$\displaystyle\|e^{i(t-t_{0})\Delta}\tilde{u}_{0}\|_{X(I)}\lesssim\|\tilde{u}\|_{X(I)}+\|F(\tilde{u})\|_{Y(I)}+\|e\|_{Y(I)}\lesssim\delta+\delta^{\frac{d+3}{d-1}}+\varepsilon.$

Combining this estimate with (3.10) and using the triangle inequality then
gives

$\|e^{i(t-t_{0})\Delta}u_{0}\|_{X(I)}\lesssim\delta$

for $\delta$ and $\varepsilon\lesssim\delta$ sufficiently small. Thus, by
(3.2) and (3.3), we get

$\displaystyle\|u\|_{X(I)}\lesssim\delta+\|F(u)\|_{Y(I)}\lesssim\delta+\|u\|_{X(I)}^{\frac{d+3}{d-1}},$

which (taking $\delta$ sufficiently small) implies (3.13).

We now turn to proving the desired estimates for $w:=u-\tilde{u}$. Note first
that $w$ is a solution to $(i\partial_{t}+\Delta)w=F(u)-F(\tilde{u})-e$, with
$w(t_{0})=u_{0}-\tilde{u}_{0}$; thus, we can use (3.2), (3.4), (3.9), (3.10),
and (3.13) to see

$\displaystyle\|w\|_{X(I)}$
$\displaystyle\lesssim\|e^{i(t-t_{0})\Delta}(u_{0}-\tilde{u}_{0})\|_{X(I)}+\|e\|_{Y(I)}+\|F(u)-F(\tilde{u})\|_{Y(I)}$
$\displaystyle\lesssim\varepsilon+\big{\\{}\|u\|_{X(I)}^{\frac{4}{d-1}}+\|\tilde{u}\|_{X(I)}^{\frac{4}{d-1}}\big{\\}}\|w\|_{X(I)}$
$\displaystyle\lesssim\varepsilon+\delta^{\frac{4}{d-1}}\|w\|_{X(I)}.$

Taking $\delta$ sufficiently small, we see that the first estimate in (3.11)
holds. Using the first estimate in (3.11), along with (3.4), (3.9), and
(3.13), we see that the remaining estimate in (3.11) holds, as well.

Next, by Strichartz, (3.6), (3.7), (3.8), (3.9), (3.11), and (3.13), we get

$\displaystyle\|w\|_{\dot{S}^{1/2}(I)}$
$\displaystyle\lesssim\|u_{0}-\tilde{u}_{0}\|_{\dot{H}_{x}^{1/2}}+\||\nabla|^{1/2}e\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}+\||\nabla|^{1/2}[F(u)-F(\tilde{u})]\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}}$
$\displaystyle\lesssim_{E}1+\|\tilde{u}\|_{\dot{S}^{1/2}(I)}\|w\|_{X(I)}^{\frac{4}{d-1}}+\|w\|_{\dot{S}^{1/2}(I)}\|u\|_{X(I)}^{\frac{4}{d-1}}$
$\displaystyle\lesssim_{E}1+\delta^{\frac{4}{d-1}}\|w\|_{\dot{S}^{1/2}(I)}.$

Taking $\delta=\delta(E)$ sufficiently small then gives the first estimate in
(3.12). We get the remaining estimate in (3.12) by using (3.6) with (3.7),
(3.11), (3.13), and the first estimate in (3.12). ∎

######  Remark 3.5.

As mentioned in the introduction, the error $e$ is only required to be small
in a space without derivatives (see (3.10)); it merely needs to be _bounded_
in a space with derivatives (see (3.7)). This will also be the case in Theorem
3.6 below (see (3.14) and (3.15)). We will see the benefit of this refinement
when we carry out the proof of Theorem 1.7 in Section 4 (see Remark 4.4).

We continue to the main result of this section:

######  Theorem 3.6 (Stability).

Let $d\geq 5$, and let $I$ be a compact time interval, with $t_{0}\in I$.
Suppose $\tilde{u}$ is a solution to
$(i\partial_{t}+\Delta)\tilde{u}=F(\tilde{u})+e$, with
$\tilde{u}(t_{0})=\tilde{u}_{0}$. Suppose

$\displaystyle\|\tilde{u}\|_{\dot{S}^{1/2}(I)}\leq
E\quad\text{and}\quad\||\nabla|^{1/2}e\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}(I\times\mathbb{R}^{d})}$
$\displaystyle\leq E$ (3.14)

for some $E>0$. Let $u_{0}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{d})$, and suppose
we have the smallness conditions

$\displaystyle\|u_{0}-\tilde{u}_{0}\|_{\dot{H}_{x}^{1/2}(\mathbb{R}^{d})}+\|e\|_{Y(I)}$
$\displaystyle\leq\varepsilon$ (3.15)

for some small $0<\varepsilon<\varepsilon_{1}(E)$. Then, there exists
$u:I\times\mathbb{R}^{d}\to\mathbb{C}$ solving (1.1) with $u(t_{0})=u_{0}$,
and there exists $0<c(d)<1$ such that

$\displaystyle\|u-\tilde{u}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}(I\times\mathbb{R}^{d})}$
$\displaystyle\lesssim_{E}\varepsilon^{c}.$ (3.16)

One can derive Theorem 3.6 from Lemma 3.4 in the standard fashion, namely, by
applying Lemma 3.4 inductively (see [14], for example). We omit these details,
but pause to point out the following: this induction will actually yield the
bounds

$\|u-\tilde{u}\|_{X(I)}\lesssim\varepsilon\quad\text{and}\quad\|u-\tilde{u}\|_{\dot{S}^{1/2}(I)}\lesssim_{E}1.$

With these bounds in hand, we then use (3.1) to see that (3.16) holds.

######  Remark 3.7.

The smallness condition on $u_{0}-\tilde{u}_{0}$ appearing in (3.15) may
actually be relaxed to the condition appearing in (3.10). In our setting, it
will not be difficult to prove the stronger condition (see Lemma 4.3).

######  Remark 3.8.

Using arguments from [4, 5], one can establish Theorem 1.4 for data in the
imhomogeneous Sobolev space $H_{x}^{1/2}$. Using Theorem 3.6, one can then
remove the assumption $u_{0}\in L_{x}^{2}$ _a posteriori_ (by approximating
$u_{0}\in\dot{H}_{x}^{1/2}$ by ${H}_{x}^{1/2}$\- functions). We omit the
standard details.

##  4\. Reduction to almost periodic solutions

In this section, we sketch a proof of Theorem 1.7. As described in the
introduction, the key ideas come from [11, 12] and are well-known. Thus, we
will merely outline the argument, providing full details only when our
approach deviates from the usual one. We model our presentation primarily
after [15, Section 3]. Throughout this section, we denote the nonlinearity
$|u|^{\frac{4}{d-1}}u$ by $F(u)$.

We suppose that Theorem 1.3 fails. We then define $L:[0,\infty)\to[0,\infty]$
by

$L(E):=\sup\big{\\{}S_{I}(u)\ \big{|}\
u:I\times\mathbb{R}^{d}\to\mathbb{C}\text{ solving }\eqref{nls}\text{ with
}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}(I\times\mathbb{R}^{d})}^{2}\leq
E\big{\\}}.$

We note that $L$ is non-decreasing, with $L(E)\lesssim E^{\frac{d+2}{d-1}}$
for $E$ sufficiently small (cf. Theorem 1.4). Thus, there exists a unique
critical threshold $E_{c}\in(0,\infty]$ such that $L(E)<\infty$ for $E<E_{c}$
and $L(E)=\infty$ for $E>E_{c}$. The failure of Theorem 1.3 implies that
$0<E_{c}<\infty.$

The key to proving Theorem 1.7 is the following convergence result. With this
result in hand, establishing Theorem 1.7 is a straightforward exercise (see
[15, Section 3.2]).

######  Proposition 4.1 (Palais–Smale condition modulo symmetries).

Let $d\geq 5$ and let $u_{n}:I_{n}\times\mathbb{R}^{d}\to\mathbb{C}$ be a
sequence of solutions to (1.1) such that

$\limsup_{n\to\infty}\|u_{n}\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}(I_{n}\times\mathbb{R}^{d})}^{2}=E_{c}.$

Suppose $t_{n}\in I_{n}$ are such that

$\lim_{n\to\infty}S_{[t_{n},\sup I_{n})}(u_{n})=\lim_{n\to\infty}S_{(\inf
I_{n},t_{n}]}(u_{n})=\infty.$ (4.1)

Then, $\\{u_{n}(t_{n})\\}$ converges along a subsequence in
$\dot{H}_{x}^{1/2}(\mathbb{R}^{d})/G$.

###### Proof.

We first translate so that each $t_{n}=0$ and apply Lemma 2.6 to write

$u_{n}(0)=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+w_{n}^{J}$ (4.2)

along some subsequence. Recall that for each $j$, either $t_{n}^{j}\equiv 0$
or $t_{n}^{j}\to\pm\infty$. To prove Proposition 4.1, we need to show that
there is exactly one profile $\phi^{1}$, with $t_{n}^{1}\equiv 0$ and
$\|w_{n}^{1}\|_{\dot{H}_{x}^{1/2}}\to 0$.

First, using Theorem 1.4, for each $j$ we define
$v^{j}:I^{j}\times\mathbb{R}^{d}\to\mathbb{C}$ to be the maximal-lifespan
solution to (1.1) such that

$\left\\{\begin{array}[]{ll}v^{j}(0)=\phi^{j}&\text{if }t_{n}^{j}\equiv 0,\\\
v^{j}\text{ scatters to }\phi^{j}\text{ as }t\to\pm\infty&\text{if
}t_{n}^{j}\to\pm\infty.\end{array}\right.$

Next, we define nonlinear profiles
$v_{n}^{j}:I_{n}^{j}\times\mathbb{R}^{d}\to\mathbb{C}$ by

$v_{n}^{j}(t)=g_{n}^{j}v^{j}\big{(}(\lambda_{n}^{j})^{-2}t+t_{n}^{j}\big{)},\quad\text{where}\quad
I_{n}^{j}=\\{t:(\lambda_{n}^{j})^{-2}t+t_{n}^{j}\in I^{j}\\}.$

To complete the proof, we need the following three claims:

(i) There is at least one ‘bad’ profile $\phi^{j}$, in the sense that

$\limsup_{n\to\infty}S_{[0,\sup I_{n}^{j})}(v_{n}^{j})=\infty.$ (4.3)

(ii) There can then be at _most_ one profile (which we label $\phi^{1}$), and
$\|w_{n}^{1}\|_{\dot{H}_{x}^{1/2}}\to 0$.

(iii) We have $t_{n}^{1}\equiv 0$.

We will provide a proof of (i) below. The proofs of (ii) and (iii) require
only small variations of the analysis given for (i), so we will merely outline
the arguments here. For (ii), one can adapt the argument of [15, Lemma 3.3] to
show that the decoupling (2.1) persists in time (this is not obvious, as the
$\dot{H}_{x}^{1/2}$-norm is not a conserved quantity for (1.1)). The critical
nature of $E_{c}$ may then be used to preclude the possibility of multiple
profiles (and to show $\|w_{n}^{1}\|_{\dot{H}_{x}^{1/2}}\to 0$). For (iii), we
only need to rule out the cases $t_{n}^{1}\to\pm\infty.$ To do this, one can
argue by contradiction: if $t_{n}^{1}\to\pm\infty$, one can use the stability
result Theorem 3.6 (comparing $u_{n}$ to $e^{it\Delta}u_{n}(0)$) to contradict
(4.1). See [15, p. 391] for more details.

We now turn to the proof of (i). We first note that the decoupling (2.1)
implies that the $v_{n}^{j}$ are global and scatter for $j$ sufficiently
large, say for $j\geq J_{0}$; indeed, for $j$ sufficiently large, the
$\dot{H}_{x}^{1/2}$-norm of $\phi^{j}$ must be below the small-data threshold
described in Theorem 1.4. Thus, we need to show that there is at least one bad
profile $\phi^{j}$ (in the sense of (4.3)) in the range $1\leq j<J_{0}$.

Suppose towards a contradiction that there are no bad profiles. By the blowup
criterion of Theorem 1.4, this immediately implies that $\sup
I_{n}^{j}=\infty$ for all $j$ and for all $n$ sufficiently large. In fact, we
claim that we have the following:

$\limsup_{J\to\infty}\limsup_{n\to\infty}\sum_{j=1}^{J}\|v_{n}^{j}\|_{\dot{S}^{1/2}([0,\infty))}^{2}\lesssim_{E_{c}}1.$
(4.4)

Indeed, for $\eta>0$, the decoupling (2.1) implies the existence of
$J_{1}=J_{1}(\eta)$ such that

$\sum_{j>J_{1}}\|\phi^{j}\|_{\dot{H}_{x}^{1/2}}^{2}\lesssim\eta.$

Thus, choosing $\eta$ smaller than the small-data threshold, Strichartz and a
standard bootstrap argument give

$\sum_{j>J_{1}}\|v_{n}^{j}\|_{\dot{S}^{1/2}([0,\infty))}^{2}\lesssim\sum_{j>J_{1}}\|\phi^{j}\|_{\dot{H}_{x}^{1/2}}^{2}\lesssim\eta.$

As the $v_{n}^{j}$ satisfy $S_{[0,\infty)}(v_{n}^{j})\lesssim 1$ for $n$
large, we may use Strichartz and another bootstrap argument to see
$\|v_{n}^{j}\|_{\dot{S}^{1/2}}\lesssim 1$ for $1\leq j\leq J_{1}$ and $n$
large. Thus, we conclude that (4.4) holds.

Next, using the fact that there are no bad profiles, together with the
orthogonality condition (2.2), one can use the arguments of [11] to arrive at
the following

######  Lemma 4.2 (Orthogonality).

For $j\neq k$, we have

$\displaystyle\bigg{[}\|v_{n}^{j}v_{n}^{k}$
$\displaystyle\|_{L_{t,x}^{\frac{d+2}{d-1}}}+\|v_{n}^{j}v_{n}^{k}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2d(d+1)}{2d^{2}-d-5}}}+\|(|\nabla|^{1/2}v_{n}^{j})(|\nabla|^{1/2}v_{n}^{k})\|_{L_{t,x}^{\frac{d+2}{d}}}$
$\displaystyle+\|\big{(}|\nabla|^{1/2}F(v_{n}^{j})\big{)}\big{(}|\nabla|^{1/2}F(v_{n}^{k})\big{)}^{\frac{d}{d+4}}\|_{L_{t,x}^{1}}\bigg{]}\to
0\quad\text{as}\quad n\to\infty,$ (4.5)

where all spacetime norms are taken over $[0,\infty)\times\mathbb{R}^{d}$.

We now wish to use (4.4) and (4.5), together with Theorem 3.6, to deduce a
bound on the scattering size of the $u_{n}$, thus contradicting (4.1). To this
end, we define approximate solutions to (1.1) and collect the information we
need about them in the following

######  Lemma 4.3.

The approximate solutions
$u_{n}^{J}(t):=\sum_{j=1}^{J}v_{n}^{j}(t)+e^{it\Delta}w_{n}^{J}$ satisfy

$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}\|u_{n}(0)-u_{n}^{J}(0)\|_{\dot{H}_{x}^{1/2}}=0,$
(4.6)
$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}S_{[0,\infty)}(u_{n}^{J})\lesssim_{E_{c}}1,$
(4.7)
$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}\|u_{n}^{J}\|_{\dot{S}^{1/2}([0,\infty))}\lesssim_{E_{c}}1.$
(4.8)

The errors
$e_{n}^{J}:=(i\partial_{t}+\Delta)u_{n}^{J}-F(u_{n}^{J})=\sum_{j=1}^{J}F(v_{n}^{j})-F(u_{n}^{J})$
satisfy

$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}\||\nabla|^{1/2}e_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}([0,\infty)\times\mathbb{R}^{d})}\lesssim_{E_{c}}1,$
(4.9)
$\displaystyle\limsup_{J\to\infty}\limsup_{n\to\infty}\|e_{n}^{J}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}([0,\infty)\times\mathbb{R}^{d})}=0.$
(4.10)

######  Remark 4.4.

It is here that we see the benefit of the refined stability result Theorem
3.6. In particular, to apply Theorem 3.6 we only need to exhibit smallness of
the $e_{n}^{J}$ in the space appearing in (4.10). As this space contains no
derivatives, we can achieve this simply by relying on pointwise estimates.

###### Proof.

We first note that (4.6) follows from the construction of the $v^{j}$.

Next, we turn to (4.7). To begin, we notice that by Sobolev embedding and the
fact that $\frac{2(d+2)}{d-1}\geq 2$, we may deduce from (4.4) that

$\sum_{j\geq 1}S_{[0,\infty)}(v_{n}^{j})\lesssim_{E_{c}}1.$

Thus, recalling (2.3), we see that to prove (4.7) it will suffice to show

$\limsup_{J\to\infty}\limsup_{n\to\infty}\bigg{|}S_{[0,\infty)}\bigg{(}\sum_{j=1}^{J}v_{n}^{j}\bigg{)}-\sum_{j=1}^{J}S_{[0,\infty)}(v_{n}^{j})\bigg{|}=0.$
(4.11)

To this end, we fix $J$ and use Hölder’s inequality, (4.4), and (4.5) to see

$\displaystyle\bigg{|}S_{[0,\infty)}\bigg{(}\sum_{j=1}^{J}v_{n}^{j}\bigg{)}-\sum_{j=1}^{J}S_{[0,\infty)}(v_{n}^{j})\bigg{|}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\|v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}}^{\frac{6}{d-1}}\|v_{n}^{j}v_{n}^{k}\|_{L_{t,x}^{\frac{d+2}{d-1}}}\to
0$

as $n\to\infty$. This establishes (4.11) and completes the proof of (4.7).

Let us next turn to (4.9) (which we will later use in the proof of (4.8)). To
begin, we will derive the bound

$\limsup_{J\to\infty}\limsup_{n\to\infty}\||\nabla|^{1/2}u_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}\lesssim_{E_{c}}1.$
(4.12)

As $w_{n}^{J}\in\dot{H}_{x}^{1/2},$ it will suffice to show

$\limsup_{J\to\infty}\limsup_{n\to\infty}\|\sum_{j=1}^{J}|\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}\lesssim_{E_{c}}1.$
(4.13)

To this end, we first note that as $\frac{2(d+2)}{d}\geq 2$, we may use (4.4)
to see

$\limsup_{J\to\infty}\limsup_{n\to\infty}\sum_{j=1}^{J}\||\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}\lesssim_{E_{c}}1.$
(4.14)

On the other hand, for fixed $J$, we can use (4.4) and (4.5) to see

$\displaystyle\bigg{|}\|$
$\displaystyle\sum_{j=1}^{J}|\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}-\sum_{j=1}^{J}\||\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{2(d+2)}{d}}\bigg{|}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\||\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}^{\frac{4}{d}}\|(|\nabla|^{1/2}v_{n}^{j})(|\nabla|^{1/2}v_{n}^{k})\|_{L_{t,x}^{\frac{d+2}{d}}}\to
0\quad\text{as}\quad n\to\infty.$

Then (4.14) implies (4.13), which in turn gives (4.12).

Next, by the fractional chain rule, (4.7), and (4.12), we get

$\displaystyle\||\nabla|^{1/2}F(u_{n}^{J})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}\lesssim\|u_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}}^{\frac{4}{d-1}}\||\nabla|^{1/2}u_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}\lesssim_{E_{c}}1$
(4.15)

as $n,J\to\infty$, which handles one of the terms appearing in (4.9).

To complete the proof of (4.9), it remains to show

$\limsup_{J\to\infty}\limsup_{n\to\infty}\|\sum_{j=1}^{J}|\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}\lesssim_{E_{c}}1.$

We claim it will suffice to establish

$\lim_{J\to\infty}\limsup_{n\to\infty}\sum_{j=1}^{J}\||\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}\lesssim_{E_{c}}1.$
(4.16)

Indeed, for fixed $J$, we have by (4.5)

$\displaystyle\bigg{|}\|\sum_{j=1}^{J}$
$\displaystyle|\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}-\sum_{j=1}^{J}\||\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}\bigg{|}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\||\nabla|^{1/2}F(v_{n}^{j})\,||\nabla|^{1/2}F(v_{n}^{k})|^{\frac{d}{d+4}}\|_{L_{t,x}^{1}}\to
0\quad\text{as }n\to\infty.$

To establish (4.16) and thereby complete the proof of (4.9), we use the
fractional chain rule and Sobolev embedding to see

$\displaystyle\sum_{j=1}^{J}\||\nabla|^{1/2}F(v_{n}^{j})\|_{L_{t,x}^{\frac{2(d+2)}{d+4}}}^{\frac{2(d+2)}{d+4}}$
$\displaystyle\lesssim\sum_{j=1}^{J}\big{(}\|v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}}^{\frac{4}{d-1}}\||\nabla|^{1/2}v_{n}^{j}\|_{L_{t,x}^{\frac{2(d+2)}{d}}}\big{)}^{\frac{2(d+2)}{d+4}}$
$\displaystyle\lesssim\sum_{j=1}^{J}\|v_{n}^{j}\|_{\dot{S}^{1/2}}^{\frac{2(d+2)(d+3)}{(d+4)(d-1)}}.$

Then (4.16) follows from (4.4) and the fact that
$\frac{2(d+2)(d+3)}{(d+4)(d-1)}\geq 2$.

Now (4.8) follows from an application of Strichartz, (4.9) and (4.15).

It remains to establish (4.10). We begin by rewriting

$e_{n}^{J}=\bigg{[}\sum_{j=1}^{J}F(v_{n}^{j})-F\bigg{(}\sum_{j=1}^{J}v_{n}^{j}\bigg{)}\bigg{]}+\big{[}F(u_{n}^{J}-e^{it\Delta}w_{n}^{J})-F(u_{n}^{J})\big{]}=:(e_{n}^{J})_{1}+(e_{n}^{J})_{2}.$

We first fix $J$ and use Hölder, Sobolev embedding, (4.4), and (4.5) to see

$\displaystyle\|(e_{n}^{J})_{1}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\|\,|v_{n}^{j}v_{n}^{k}|^{\frac{4}{d-1}}|v_{n}^{j}|^{\frac{d-5}{d-1}}\,\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}}$
$\displaystyle\lesssim_{J}\sum_{j\neq
k}\|v_{n}^{j}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2d(d+1)}{d^{2}-d-4}}}^{\frac{d-5}{d-1}}\|v_{n}^{j}v_{n}^{k}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2d(d+1)}{2d^{2}-d-5}}}^{\frac{4}{d-1}}\to
0$

as $n\to\infty$. Next, we note that we have the pointwise estimate

$|(e_{n}^{J})_{2}|\lesssim|e^{it\Delta}w_{n}^{J}||f_{n}^{J}|^{\frac{4}{d-1}},$

where $f_{n}^{J}:=u_{n}^{J}+e^{it\Delta}w_{n}^{J}$ satisfies
$\|f_{n}^{J}\|_{\dot{S}^{1/2}}\lesssim_{E_{c}}1$ as $n,J\to\infty$ (cf. (4.8)
and the fact that $w_{n}^{J}\in\dot{H}_{x}^{1/2}$). Thus, we can use Hölder,
Strichartz, Sobolev embedding, $w_{n}^{J}\in\dot{H}_{x}^{1/2}$, and (2.3) to
see

$\displaystyle\|(e_{n}^{J})_{2}\|_{L_{t}^{\frac{4(d+1)}{d+3}}L_{x}^{\frac{2(d+1)}{d+3}}}$
$\displaystyle\lesssim\|e^{it\Delta}w_{n}^{J}\|_{L_{t}^{\frac{4(d+1)}{d-1}}L_{x}^{\frac{2(d+1)}{d-1}}}\|f_{n}^{J}\|_{L_{t}^{\frac{4(d+1)}{d-1}}L_{x}^{\frac{2(d+1)}{d-1}}}^{\frac{4}{d-1}}$
$\displaystyle\lesssim\|e^{it\Delta}w_{n}^{J}\|_{L_{t,x}^{\frac{2(d+2)}{d-1}}}^{\frac{d+2}{2(d+1)}}\|w_{n}^{J}\|_{\dot{H}_{x}^{1/2}}^{\frac{d}{2(d+1)}}\|f_{n}^{J}\|_{\dot{S}^{1/2}}^{\frac{4}{d-1}}\to
0\quad\text{as }n,J\to\infty.$

Combining the estimates for $(e_{n}^{J})_{1}$ and $(e_{n}^{J})_{2}$, we
conclude that (4.10) holds. ∎

Using Lemma 4.3, we may apply Theorem 3.6 to deduce that
$S_{[0,\infty)}(u_{n})\lesssim_{E_{c}}1$ for $n$ large, contradicting (4.1).
We conclude that there is at least one bad profile, that is, claim (i) holds.
This completes the proof of Proposition 4.1 and Theorem 1.7. ∎

##  5\. Finite time blowup

In this section, we use Proposition 1.10, Strichartz estimates, and
conservation of mass to preclude the existence of almost periodic solutions as
in Theorem 1.8 with $T_{max}<\infty$.

######  Theorem 5.1 (No finite time blowup).

Let $d\geq 5$. There are no almost periodic solutions
$u:[0,T_{max})\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) with
$T_{max}<\infty$ and $S_{[0,T_{max})}=\infty.$

###### Proof.

Suppose that $u$ were such a solution. Then, for $t\in[0,T_{max})$ and $N>0$,
Proposition 1.10, Strichartz, Hölder, Bernstein, and Sobolev embedding give

$\displaystyle\|P_{N}u(t)\|_{L_{x}^{2}}$
$\displaystyle\lesssim\|P_{N}(|u|^{\frac{4}{d-1}}u)\|_{L_{t}^{2}L_{x}^{\frac{2d}{d+2}}([t,T_{max})\times\mathbb{R}^{d})}$
$\displaystyle\lesssim(T_{max}-t)^{1/2}N^{1/2}\||u|^{\frac{4}{d-1}}u\|_{L_{t}^{\infty}L_{x}^{\frac{2d}{d+3}}}$
$\displaystyle\lesssim(T_{max}-t)^{1/2}N^{1/2}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}}^{\frac{d+3}{d-1}}.$

As $u\in L_{t}^{\infty}\dot{H}_{x}^{1/2}$, we deduce

$\displaystyle\|P_{\leq
N}u(t)\|_{L_{x}^{2}}\lesssim_{u}(T_{max}-t)^{1/2}N^{1/2}\quad\text{for all
}t\in I\text{ and }N>0.$ (5.1)

On the other hand, an application of Bernstein gives

$\displaystyle\|P_{>N}u\|_{L_{t}^{\infty}L_{x}^{2}}\lesssim
N^{-1/2}\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}}\lesssim_{u}N^{-1/2}\quad\text{for
all }N>0.$ (5.2)

We now let $\eta>0$. We choose $N$ large enough that $N^{-1/2}<\eta$, and
subsequently choose $t$ close enough to $T_{max}$ that
$(T_{max}-t)^{1/2}N^{1/2}<\eta.$ Combining (5.1) and (5.2), we then get
$\|u(t)\|_{L_{x}^{2}}\lesssim_{u}\eta.$

As $\eta$ was arbitrary and mass is conserved, we conclude
$\|u(t)\|_{L_{x}^{2}}=0$ for all $t\in[0,T_{max})$. Thus $u\equiv 0$, which
contradicts the fact that $u$ blows up. ∎

##  6\. The Lin–Strauss Morawetz inequality

In this section, we use the Lin–Strauss Morawetz inequality to preclude the
existence of almost periodic solutions as in Theorem 1.8 such that
$T_{max}=\infty.$

######  Proposition 6.1 (Lin–Strauss Morawetz inequality, [18]).

Let $d\geq 3$ and let $u:I\times\mathbb{R}^{d}\to\mathbb{C}$ be a solution to
$(i\partial_{t}+\Delta)u=|u|^{p}u$. Then

$\int_{I}\int_{\mathbb{R}^{d}}\frac{|u(t,x)|^{p+2}}{|x|}\,dx\,dt\lesssim\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}(I\times\mathbb{R}^{d})}^{2}.$
(6.1)

As in [10], we will use this estimate to establish the following

######  Theorem 6.2.

Let $d\geq 5$. There are no almost periodic solutions
$u:[0,\infty)\times\mathbb{R}^{d}\to\mathbb{C}$ to (1.1) such that $u$ blows
up forward in time, $\inf_{t\in[0,\infty)}N(t)\geq 1$, and
$|x(t)|\lesssim_{u}\int_{0}^{t}N(s)\,ds$ for all $t\geq 0$.

###### Proof.

Suppose $u$ were such a solution. In particular $u$ is nonzero, so that by
Remark 1.6 we may find $C(u)>0$ such that

$\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{\frac{2d}{d-1}}\,dx\gtrsim_{u}1\quad\text{uniformly
for }t\in[0,\infty).$

Applying Hölder and rearranging, this implies

$\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}|u(t,x)|^{\frac{2(d+1)}{d-1}}\,dx\gtrsim_{u}N(t)\quad\text{uniformly
for }t\in[0,\infty).$ (6.2)

We now let $T>1$ and use $u\in L_{t}^{\infty}\dot{H}_{x}^{1/2}$, (6.1), and
(6.2) to see

$\displaystyle
1\gtrsim_{u}\int_{1}^{T}\int_{|x-x(t)|\leq\frac{C(u)}{N(t)}}\frac{|u(t,x)|^{\frac{2(d+1)}{d-1}}}{|x|}\,dx\,dt\gtrsim_{u}\int_{1}^{T}\frac{N(t)}{|x(t)|+N(t)^{-1}}\,dt.$

As $\inf_{t\in[1,\infty)}N(t)\geq 1$, to derive a contradiction it will
suffice to show that

$\lim_{T\to\infty}\int_{1}^{T}\frac{N(t)}{1+|x(t)|}\,dt=\infty.$ (6.3)

Recalling that $|x(t)|\lesssim_{u}\int_{0}^{t}N(s)\,ds$ for all $t\geq 0$, we
get

$\displaystyle\int_{1}^{T}\frac{N(t)}{1+|x(t)|}\,dt\gtrsim_{u}\int_{1}^{T}\frac{d}{dt}\log\bigg{(}1+\int_{0}^{t}N(s)\,ds\bigg{)}\,dt\gtrsim_{u}\log\bigg{(}\frac{1+\int_{0}^{T}N(s)\,ds}{1+\int_{0}^{1}N(s)\,ds}\bigg{)}.$

As $\inf_{t\in[1,\infty)}N(t)\geq 1$, we conclude that (6.3) holds, as needed.
∎

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