# Soliton resolution for critical co-rotational wave maps and radial cubic
wave equation

Thomas Duyckaerts1 ,  Carlos Kenig2 ,  Yvan Martel3 and  Frank Merle4

###### Abstract.

In this paper we prove the soliton resolution conjecture for all times, for
all solutions in the energy space, of the co-rotational wave map equation. To
our knowledge this is the first such result for all initial data in the energy
space for a wave-type equation. We also prove the corresponding results for
radial solutions, which remain bounded in the energy norm, of the cubic
(energy-critical) nonlinear wave equation in space dimension 4.

###### Key words and phrases:

Wave maps, focusing wave equation, dynamics, soliton resolution, global
solutions, blow-up

1LAGA (UMR 7539), Université Sorbonne Paris Nord, and Institut Universitaire
de France

2University of Chicago. Partially supported by NSF Grants DMS-14363746 and
DMS-1800082

3CMLS (UMR 7640), Ecole Polytechnique

4AGM (UMR 8088), CY Cergy Paris Université and IHES

###### Contents

  1. 1 Introduction
    1. 1.1 Background on the soliton resolution conjecture
    2. 1.2 Background on rigidity theorems for dispersive equations
    3. 1.3 Main results and ideas of proofs
    4. 1.4 Outline of the paper
  2. 2 Preliminaries
    1. 2.1 Notations
    2. 2.2 Strichartz estimates and well-posedness
    3. 2.3 Solutions outside a wave cone
  3. 3 Exterior energy estimates
    1. 3.1 Exterior energy estimates for the linear equation
    2. 3.2 Exterior energy estimates over channels
    3. 3.3 Exterior energy in space dimension $6$
    4. 3.4 Exterior energy for the linear inhomogeneous wave equation
  4. 4 Rigidity theorem part I: compactly supported pertubations and constant sign solutions
    1. 4.1 From co-rotational wave maps to four dimensional waves
    2. 4.2 Compactly supported perturbation of a stationary solution
    3. 4.3 Constant sign solutions
  5. 5 Rigidity theorem, part II: odd solutions
    1. 5.1 Gain of regularity
    2. 5.2 Approximate non-radiative solution
    3. 5.3 Proof of the rigidity result
  6. 6 Soliton resolution for the wave equation
    1. 6.1 Profile decomposition
    2. 6.2 Finite time blow-up case
      1. 6.2.1 Convergence of the solution
      2. 6.2.2 Analysis along a sequence of times
      3. 6.2.3 End of the proof
    3. 6.3 Comments on the proof in the global case
  7. 7 Soliton resolution for corotional wave maps
    1. 7.1 Preliminaries on wave maps
      1. 7.1.1 Miscellaneous results on wave maps
      2. 7.1.2 Rigidity theorem
      3. 7.1.3 A space-time bound outside the wave cone
      4. 7.1.4 Profile decomposition
    2. 7.2 Proof of the soliton resolution
      1. 7.2.1 Analysis along a sequence of times
      2. 7.2.2 End of the proof of the soliton resolution
  8. A Radiation term for the free wave equation
  9. B Pseudo-ortogonality and channels of energy
    1. B.1 Linear wave equation
    2. B.2 Pythagorean expansion of the energy for wave maps
  10. C Boundedness of integral operators on $L^{p}L^{q}$ spaces

##  1\. Introduction

In this paper we first consider co-rotational wave maps, from Minkowski space
into the two-sphere, which in spherical coordinates for the two-sphere,
corresponds to solutions of the following equation:

(1.1)
$\partial_{t}^{2}\psi-\partial_{r}^{2}\psi-\frac{1}{r}\partial_{r}\psi+\frac{\sin(2\psi)}{2r^{2}}=0,$

where $r>0$ and $t\in\mathbb{R}$, with initial data

(1.2) $\vec{\psi}_{\restriction t=0}=(\psi_{0},\psi_{1})\in\mathbf{E},$

where $\vec{\psi}=(\psi,\partial_{t}\psi)$ (see for example [72] and the
introduction of [8]). Here $\mathbf{E}$ is the space of initial data such that
the conserved energy:

$E_{M}(\psi_{0},\psi_{1})=\frac{1}{2}\int_{0}^{\infty}(\psi_{1})^{2}rdr+\frac{1}{2}\int_{0}^{\infty}(\partial_{r}\psi_{0})^{2}rdr+\frac{1}{2}\int_{0}^{\infty}\frac{\sin^{2}\psi_{0}}{r^{2}}rdr$

is finite.

If $(\psi_{0},\psi_{1})$ in $\mathbf{E}$, there exists
$(\ell,m)\in\mathbb{Z}^{2}$ such that

(1.3) $\lim_{r\to
0}\psi_{0}(r)=\ell\pi,\quad\lim_{r\to\infty}\psi_{0}(r)=m\pi.$

We denote by $\mathbf{H}_{\ell,m}$ the set of $(\psi_{0},\psi_{1})$ in
$\mathbf{E}$ such that (1.3) holds. This is an affine space, parallel to the
Hilbert space $\mathbf{H}=\mathbf{H}_{0,0}$ of functions
$(\psi_{0},\psi_{1})\in\left(L^{2}_{\operatorname{loc}}\big{(}(0,\infty)\big{)}\right)^{2}$
such that

(1.4)
$\|(\psi_{0},\psi_{1})\|^{2}_{\mathbf{H}}=\int_{0}^{\infty}\left((\partial_{r}\psi_{0})^{2}+\frac{1}{r^{2}}(\psi_{0}(r))^{2}\right)rdr+\int_{0}^{\infty}\psi_{1}^{2}rdr<\infty.$

It is well known ([72]) that (1.1) is locally well-posed in $\mathbf{E}$ and
that the spaces $\mathbf{H}_{\ell,m}$ are stable by the flow (see for instance
[8]). The equation (1.1) has the following scaling invariance: if $\psi$ is a
solution of (1.1), then $\psi(\lambda\,\cdot,\lambda\,\cdot)$ is also a
solution of (1.1), with same energy.

The bubble

(1.5) $Q(r)=2\arctan r$

is a stationary solution of (1.1) such that $(Q,0)\in\mathbf{H}_{0,1}$. Other
stationary solutions of (1.1) are given by $\ell\pi\pm Q(\lambda\,\cdot)$,
$\lambda>0$. These are the only finite energy, stationary solutions of (1.1).

The stationary solution $Q$ has several important properties. It is stable up
to the symmetries (including scaling) as a solution of (1.1), but also as a
general wave map from the Minkowski space into the two-sphere (this is, for
example, a simple consequence of Theorem 6.1 in [16] and the conservation laws
for general wave maps). In addition, $Q$ is up to symmetries the static
solution of least energy among all the (general) static wave maps from
Minkowski space into the two-sphere. Thus, $Q$ is the “ground state”.

The equation (1.1) is the case $k=1$ of the equation obtained by considering
$k$-equivariant wave maps from Minkowski space into the two-sphere:

(1.6)
$\partial_{t}^{2}\psi-\partial_{r}^{2}\psi-\frac{1}{r}\partial_{r}\psi+k^{2}\frac{\sin(2\psi)}{2r^{2}}=0,$

The co-rotational case $k=1$ is distinguished among the $k$-equivariant ones,
because it is the one satisfied by the ground state $Q$ (of the whole wave map
equation).

The linearized equation of (1.6) at $\psi=0$ is

$\partial_{t}^{2}\psi_{L}-\partial_{r}^{2}\psi_{L}-\frac{1}{r}\partial_{r}\psi_{L}+\frac{k^{2}}{r^{2}}\psi_{L}=0,$

which can be reduced to the radial wave equation in dimension $2k+2$ by the
change of unknown function $r^{k}u_{L}=\psi_{L}$. The linearized equation for
(1.1) is thus essentially the $4D$ wave equation.

In this paper we will also consider a twin problem, the wave equation on
$\mathbb{R}^{4}$, with the energy-critical focusing nonlinearity:

(1.7) $\partial_{t}^{2}u-\Delta u=u^{3},$

where $t\in\mathbb{R}$ and $x\in\mathbb{R}^{4}$, with initial data

(1.8) $\vec{u}_{\restriction t=0}=(u_{0},u_{1})\in\mathcal{H},$

where $\vec{u}=(u,\partial_{t}u)$, and
$\mathcal{H}=\dot{H}^{1}(\mathbb{R}^{4})\times L^{2}(\mathbb{R}^{4})$. We will
only consider radial initial data, i.e. data depending only on
$r=|x|=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}$.

We denote by

$W(x)=\frac{1}{1+\frac{|x|^{2}}{8}}$

the ground state of (1.7).

The equation (1.7) is a special case of the energy-critical wave equation

(1.9) $\partial_{t}^{2}u-\Delta u=|u|^{\frac{4}{N-2}}u$

in general space dimension $N\geq 3$. In this general case, the ground state
stationary solution is given by
$W(x)=\left(1+\frac{|x|^{2}}{N(N-2)}\right)^{1-\frac{N}{2}}$.

###  1.1. Background on the soliton resolution conjecture

_The main results of this paper are the proofs of soliton resolution, without
size constraints, and for all times, for solutions of ( 1.1) and for radial
solutions of (1.7). _In particular, we will obtain the complete classification
of the solutions of (1.1), which is to our knowledge the first result of
soliton resolution _for all initial data in the energy space_ for a wave-type
equation. In this work we only consider co-rotational wave maps and radial
solutions of (1.7). The general problems, without any symmetry assumption,
seem out of reach by current methods.

To put these results in perspective, we start with a general discussion of the
soliton resolution conjecture for nonlinear dispersive equations. This
conjecture predicts that any global in time solution of this type of equation
evolves asymptotically as a sum of decoupled solitons (traveling wave
solutions, which are well-localized and traveling at a fixed speed), a
radiative term (typically a solution to a linear equation) and a term going to
zero in the energy space. For finite time blow-up solutions, a similar
decomposition should hold, depending on the nature of the blow-up.

This conjecture arose in the 1970’s from numerical simulations and the theory
of integrable equations, in order to explain a “puzzling paradox” stemming
from the birth of scientific computation, in a numerical simulation carried
out at Los Alamos in the early 1950’s (see [32]) by Fermi, Pasta and Ulam. In
1965 Kruskal found that as the spacial mesh in the discretization of the
Fermi-Pasta-Ulam model tends to $0$, the solutions of the Fermi-Pasta-Ulam
problem converge to solutions of the KdV equation (see [53] for an account).
The fact that the KdV equation has soliton solutions provided an explanation
for the “puzzling paradox”. Following this discovery, Zabusky and Kruskal [82]
conducted another influential numerical simulation, which showed numerically
the emergence of solitons and multisolitons in the KdV equation. This
experiment led to the soliton resolution conjecture, and to the theory of
complete integrability, to explain the inelastic collision of solitons that
was observed.

The first theoretical results in the direction of soliton resolution were
obtained for the completely integrable KdV, mKdV and $1$-dimensional cubic
NLS, using the method of inverse scattering ([54], [29],[28], [70], [71],
[61], [3]). Very few complete results exist for equations that are not
completely integrable. For non-integrable nonlinear dispersive equations, such
as nonlinear Schrödinger and Klein-Gordon equations, complete results seem out
of reach. Known results include scattering below a threshold given by the
ground state of the equation (see e.g. [47], [14], [39], [13]), local study
close to the ground state soliton [60]), and in some special cases the
existence of a global compact attractor (e.g. [79]). We refer to the
introduction of [15] for a more complete discussion and more references on the
subject.

_The soliton resolution conjecture is believed to hold unconditionally for
energy-critical wave maps into the two-sphere, and for energy-bounded
solutions of the energy-critical nonlinear wave equation._ For the case of
wave maps, a first result in this direction was that any solution that blows
up in finite time converges locally in space, up to the symmetries of the
equation, for a sequence of times, to a traveling wave solution (see [5], [78]
for the equivariant case, [74], [75] for the general case). Similar results
hold for solutions that exist for infinite time but don’t scatter (see [75]).
In the equivariant case, using techniques developed by the first, second and
fourth authors, one can prove stronger statements, namely that the soliton
resolution holds in the equivariant case with a condition on the energy ruling
out a multisoliton configuration [8], [9]) and that it holds for a sequence of
times without this condition in the co-rotational case [7] and for the $k=2$
equivariant case [45] (and for all equivariant cases, modulo a technical
condition regarding the local well-posedness theory, also in [45]). The
limiting case of a pure two-soliton was treated in [41]), where it is shown,
in particular, that the collision between the two solitons is inelastic.
Subsequent work on the case of two solitons is in [42], [44] and [69], while
in [43], the full resolution is proved in the equivariant case, in the two-
soliton setting. For wave maps into the two-sphere without any symmetry
assumption, the resolution is known only close to the ground state, in work of
the first, second and fourth authors, with Hao Jia [16] (see also [37] for a
weaker version of the decomposition, without size constraint, for a sequence
of times).

For the parabolic analog of wave maps (the harmonic map heat flow) these
questions have been studied earlier (see e.g. [77], [64], [65], [81]). In full
generality, in the parabolic case, the soliton resolution is only known for a
sequence of times. An example of Topping [81] shows that for a general target
manifold, two sequences of times may lead to different decompositions. In the
case when this target is the two-sphere, it is conjectured that the analog of
the soliton resolution holds, but this is still open. For a recent work in the
parabolic analog of (1.9), in dimension greater than or equal to $7$, without
radial symmetry, near the ground state, see [6].

All the results in non-integrable cases and in the absence of size
constraints, mentioned in the last two paragraphs, are proven using
monotonicity laws after time averaging, and hence hold only for well-chosen
sequences of times. The example of Topping just mentioned shows the necessity
of this. Thus, in order to obtain results that hold for all times, new methods
that go beyond monotonic time averages have to be created, which address the
notoriously difficult issue of time oscillations. This, for the case of (1.1)
and the radial case of (1.7), is a key contribution of this work.

For the case of the energy-critical nonlinear wave equation (1.9),
classification results for solutions “below the ground state” were obtained in
[48], [27]. (Corresponding classification results for co-rotational wave maps
are in [11] and in [8], [9]). These results are inspired by earlier results
for the nonlinear Schrödinger equation going back to [59]. (See also the
results for gKdV in [58]).

Decomposition results for (1.9) in the $3D$ radial case, near the ground
state, are in [17], and corresponding results in the non-radial case, near the
ground state are in [19]. Constructions near the ground state of center stable
manifold, in the radial case and non-radial cases, are in [50], [51]. The
soliton resolution for sequences of time in the radial case, for solutions
which are bounded in the energy norm, was proved by [18] in $3$ dimensions,
[68] in all other odd dimensions, in [10] in $4$ dimensions and in [45] in $6$
dimensions. In [15], the first, second and fourth authors, with Hao Jia,
proved the decomposition for sequences of time, in the nonradial case, for
solutions which are bounded in the energy norm, in dimensions $3$, $4$ and
$5$. In the radial case, $W$, defined above, is the only soliton up to sign
change and scaling. The full resolution (for all times), for solutions bounded
in the energy norm, for $N=3$, in the radial case, was proved in [20], by the
first, second and fourth authors. The key fact in the proof is a rigidity
theorem giving a dynamical characterization of the static solutions, in terms
of outer energy lower bounds. The proof of this fact used techniques that are
very specific to radial solutions in $3$ space dimensions. Recently, combining
the papers [22], [23] and [25], the first, second and fourth authors were able
to prove the soliton resolution for solutions of (1.9) that remain bounded in
energy norm, in the radial case, for all odd dimensions. An important tool in
this proof is the odd dimensional linear outer energy inequalities proved in
[46], that hold up to certain finite dimensional subspaces with increasing
dimension as a function of $N$. When $N=3$, the dimension is $1$ and one can
use scaling to deal with this exceptional subspace. For higher odd dimensions,
the dimension is larger than $1$ and this was a stumbling block for many
years. The proofs mentioned above involve modulation analysis to analyse the
collisions of solitons, showing that they are inelastic and hence pure
multisolitons in both time directions are ruled out.

###  1.2. Background on rigidity theorems for dispersive equations

As seen in many recent works, _the proof of rigidity (also called Liouville)
theorems, classifying solutions that are non-dispersive (in a sense to be
specified) is crucial in the understanding of the asymptotic dynamics of
semilinear dispersive equations as ( 1.6) and (1.9)._ A typical statement is
that the only non-dispersive solutions are the stationary solutions (or more
generally the solitons) of the equation.

A first notion of non-dispersive solutions is given by _solutions with the
compactness property_ , that are solutions whose trajectory is precompact up
the modulations of the equation. These solutions are also sometimes called
quasi-periodic or almost periodic. This concept goes back to (at least) the
work of the third and fourth authors [56], in the context of the KdV equation.
See also [47] for the energy-critical NLS, [49], [80] for mass-critical NLS.
We refer to [79] and [26] for works highlighting the importance of solutions
with the compactness property in the asymptotic study of bounded solutions of
nonlinear dispersive equations.

For equation (1.9), solutions with the compactness property were first
considered in [48], where a rigidity theorem with a size constraint is proved.
A general rigidity theorem, without a size constraint (but with an additional
nondegeneracy assumption for solutions without symmetry) is proved in [21]
(see also [17, Theorem 2] for the radial, $3D$ case).

Nevertheless, this notion is not useful to understand the collision of
solitons. Indeed a pure multisoliton in both time directions, that is a
solution that is, asymptotically as $t\to+\infty$ and as $t\to-\infty$, a sum
of decoupled solitons, is not a solution with the compactness property, but it
should be definitively considered as a non-dispersive solution.

It is believed that for completely integrable equations, the collision of
solitons is always elastic, but that, for non-integrable equations such as
(1.6) and (1.9), it is not elastic and should always generate some radiation
(see e.g. [57] in the context of generalized Korteweg-de Vries equations).

To deal with this problem, the first, second and fourth authors have
introduced the concept of _non-radiative_ solutions of (1.9). By definition,
these are solutions of (1.9), defined for $|x|>|t|$, and such that

(1.10) $\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>|t|}\left(|\nabla
u(t,x)|^{2}+(\partial_{t}u(t,x))^{2}\right)dx=0.$

(see [20] where this concept is used without formal definition, and [23] where
the term “non-radiative” is introduced). This definition can be easily adapted
to (1.6).

Note that solitary waves (which always travel at speed $\ell<1$ [26]) are non-
radiative. As shown in [20] using the profile decomposition of [2], a rigidity
theorem stating that all non-radiative solutions are solitons (or in a radial
context, stationary solutions) essentially implies the soliton resolution.

The usefulness of this concept is that, using finite speed of propagation, it
can be applied by first studying solutions in the exterior of a wave cone
$\\{|x|>R+|t|\\}$, for large $R$, thus restricting to small solutions, that
are close to solutions of the linear wave equation. In connection with this,
the study of lower bounds of the form

(1.11)
$C\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R+|t|}|\nabla_{t,x}u_{L}(t,x)|^{2}dx\geq\int_{|x|>R}(u_{1}(x))^{2}+|\nabla
u_{0}(x)|^{2}dx$

for radial solutions of the linear wave equation

(1.12) $\partial_{t}^{2}u_{L}-\Delta
u_{L}=0,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{N},\quad|x|>R+|t|$

with initial data $\vec{u}_{\restriction t=0}=(u_{0},u_{1})$ is crucial to
prove the rigidity theorem for the equations (1.9) and (1.6). The linear
estimate (1.11) depends strongly on the dimension $N$. In the case $R=0$ it
was proved that for $N$ odd, (1.11) holds for any
$(u_{0},u_{1})\in\mathcal{H}$ (see [19]), whereas when $N$ is even, it is not
valid in full generality, but it holds (at least in the radial case) for
initial data of the form $(u_{0},0)$ or $(0,u_{1})$ depending on the
congruence of $N$ modulo $4$ (see [12]). In the case $R>0$, explicit
counterexamples to (1.11) exist and one can hope to prove (1.12) only on a
strict subspace of the energy space $\mathcal{H}$. For $N=3$, (1.11) is valid
for all radial initial data $(u_{0},u_{1})\in\mathcal{H}(\\{r>R\\})$, that are
orthogonal to $\left(\frac{1}{r},0\right)$. This single nondegenerate
direction can be handled with the scaling of the equation, and corresponds to
the stationary solution $W=\left(1+\frac{|x|^{2}}{3}\right)^{-1/2}$. This
leads to the proof of a strong rigidity theorem, as was mentioned earlier, and
the soliton resolution for all radial solutions of (1.9) with $N=3$ that are
bounded in the energy space $\mathcal{H}$ [20].

For $N$ odd, $N\geq 5$, (1.11) holds in the radial case, for all radial data
in an $\frac{N-1}{2}$ co-dimensional subspace of $\mathcal{H}(\\{r>R\\})$,
which does not seem sufficient to deduce a strong rigidity result for (1.9) as
in space dimension $3$, using the scaling invariance of the equation. However,
as mentioned earlier, combining asymptotic estimates on non-radiative
solutions of (1.7) deduced from (1.11) with a careful study of the modulation
equations close to a multisoliton, the soliton resolution for all radial
solutions of (1.7) that are bounded in the energy space, for all times, was
proved by three of the authors in [23, 25, 22].

In even space dimension, up to now, no lower bound of the form (1.11) has been
known. Note that this is relevant for equivariant wave maps (verifying (1.6)),
for which the underlying space dimension, after linearization, is $2k+2$.

###  1.3. Main results and ideas of proofs

In this article, we consider co-rotational wave maps (equation (1.1)), for
which the underlying space dimension is $4$, and (1.7). Besides the difficulty
arising from linear estimates in even space dimensions, these are also special
issues at the nonlinear level, since the modulation equations close to a
multisoliton degenerate, and this makes the method used in [22] for large odd
dimensions ineffective.

Our first step is to prove that (1.11) is valid for $R>0$, when $N=4$, for
radial solutions of (1.12), with initial data of the form $(u_{0},0)$, where
$u_{0}\in\dot{H}^{1}(\\{r>R\\})$ is radial and orthogonal to
$\frac{1}{r^{2}}$. The proof of this is based on the result of [12] and
involved (but elementary) explicit computations, using a different method of
proof than in the corresponding results in [46].

When $N=4$, the inequality (1.11) is not valid for radial solutions of (1.12)
with data of the form $(0,u_{1})$, $u_{1}\in L^{2}(\mathbb{R}^{4})$, even if
$u_{1}$ is taken in a finite co-dimensional subspace of $L^{2}$. This is
essentially due to the resonant solution $\frac{t}{r^{2}}$, whose initial data
barely fails to be in the energy space. In Subsection 5.2 of this paper, we
will construct an approximate solution for each of the equations (1.1) and
(1.7), which is odd in time and whose time derivative is non-radiative. The
initial data for this solution again barely fail to be in the energy space,
but the solution itself satisfies the same global $L^{p}L^{q}$ estimates in
the exterior of wave cones, as the global, finite energy solutions of (1.1) or
(1.7). _Although this solution has similar properties as the linear solution
$\frac{t}{r^{2}}$, it is not a perturbation of this solution, but a truly
nonlinear object, which comes from a cancellation between the linear and
nonlinear parts and depends on the specific form of the nonlinearity in
equations (1.1) and (1.7)._

Using the new estimate (1.11) for the linear problem, and the approximate non-
radiative solution, we obtain in this paper strong rigidity theorems for the
equations (1.1) and (radial) (1.7), which are identical to the one proved in
[20] for radial solutions of (1.9) in dimension $N=3$. The proofs, which are
much deeper than the one in [20], are first based on the decoupling of a non-
radiative solution of (1.1) or (radial) (1.7) into its odd and even parts (in
the time variable), which satisfy similar approximate equations, with a cubic
interaction term. Using this decoupling, we consider two cases.

  * •

If the non-radiative solution has constant sign, one easily sees that the even
part is dominant. Using that (1.11) holds for initial data of the form
$(u_{0},0)$, where $u_{0}$ is taken on a codimension $1$ subspace of
$\dot{H}^{1}(\\{r>R\\})$, we prove that the solution is equal, up to the
invariances of the equation, to one of the nonzero explicit stationary
solutions $\pm Q(\lambda\,\cdot)+p\pi$ for (1.1) or $\pm\lambda
W(\lambda\,\cdot)$ for (1.7) for large $r$. We then extend the result for all
$r>0$. This strategy can be seen as a dispersive analog of Alexandrov’s method
of moving plane for elliptic equations.

  * •

If, on the other hand, $u$ vanishes at some point $(t_{0},r_{0})$, we first
prove using (1.11) and some kind of maximum principle for non-radiative
solutions, that $u(t_{0},r)=0$ for $r>r_{0}$. Thus $u$ is after time
translation odd in time in the exterior of the wave cone $\\{r>|t|+r_{0}\\}$.
In this case we prove by contradiction that $\partial_{t}u(t_{0},r)=0$ for
$r>r_{0}$. If not, using (1.11) on the time derivative of the solution, we
prove that $u$ is asymptotically close to the non-radiative nonlinear solution
mentioned above, which does not have finite energy, giving a contradiction.
Extending the result to $r>0$ as in the previous step, we conclude that
$u(t,r)$ is the zero solution.

Precisely, we obtain the following two rigidity theorems for equations (1.1)
and (radial) (1.7).

######  Theorem 1 (Rigidity for co-rotational wave maps).

Let $(\psi_{0},\psi_{1})\in\mathbf{E}$ (and thus to $\mathbf{H}_{\ell,m}$ for
some $(\ell,m)\in\mathbb{Z}^{2}$). Assume that $(\psi_{0},\psi_{1})$ is not
the initial data of a stationary solution of (1.1), i.e. if $\ell=m$,
$(\psi_{0},\psi_{1})\neq(m\pi,0)$ and otherwise, for all $\lambda>0$,
$(\psi_{0}-m\pi,\psi_{1})$ is not equal to $\pm(\pi-Q(\lambda\,\cdot),0)$.
Then there exists $\eta>0$ such that the following holds for all $t>0$ or for
all $t<0$:

(1.13)
$\int_{|t|}^{\infty}\left((\partial_{t}\psi(t,r))^{2}+(\partial_{r}\psi(t,r))^{2}+\frac{1}{r^{2}}\sin^{2}(\psi(t,r))\right)rdr\geq\eta.$

We refer to §2.3 for the definition of solutions outside a wave cone. As
proved there, for any initial data $(\psi_{0},\psi_{1})\in\mathbf{E}$, the
solution $\psi$ with initial data $(\psi_{0},\psi_{1})$ is always well-defined
for $r>|t|$.

######  Theorem 2 (Rigidity for $4D$ radial critical wave).

Let $(u_{0},u_{1})\in\mathcal{H}$, radial. Assume that $(u_{0},u_{1})$ is not
the initial data of a radial stationary solution of (1.7), i.e. that
$\forall\lambda\in\mathbb{R}$, $(u_{0},u_{1})$ is not equal to $(\lambda
W(\lambda\,\cdot),0)$. Then there exists $R>0$ such that the solution $u$ with
initial data $(u_{0},u_{1})$ is defined on $\\{|x|>R+|t|\\}$ and there exists
$\eta>0$ such that the following holds for all $t>0$ or for all $t<0$:

(1.14)
$\int_{|x|>R+|t|}(\partial_{t}u(t,x))^{2}+|\nabla_{x}u(t,x)|^{2}dx\geq\eta.$

Note that Theorems 1 and 2 imply the fact that the collision of two or more
solitons yields dispersion, and thus that there is no pure multisoliton
solution of equation (1.1) and of equation (1.7) in the radial case. This was
previously known only in the case of equation (1.1), for two-solitons (see
[69], and also [41] for two-soliton solutions of equation (1.6) with $k\geq
2$).

As a consequence of the two rigidity theorems, we obtain the soliton
resolution for both equations. We first state the result for co-rotational
wave maps.

If $f(t)$ and $g(t)$ are positive functions defined in a neighborhood of
$a\in\mathbb{R}\cup\\{\pm\infty\\}$, we will write $f(t)\ll g(t)$ as $t\to a$
when $\lim_{t\to a}f(t)/g(t)=0$.

######  Theorem 3 (Soliton resolution for co-rotational wave maps).

Consider any solution $\psi$ in the energy space of (1.1) with maximal time of
existence $T_{+}$. Then one of the following holds:

Blow-up. $T_{+}<\infty$, and there exist $(\phi_{0},\phi_{1})\in\mathbf{E}$,
an integer $J\in\mathbb{N}\setminus\\{0\\}$, and for each
$j\in\\{1,\ldots,J\\}$, a sign $\iota_{j}\in\\{\pm 1\\}$, and a positive
function $\lambda_{j}(t)$ defined for $t$ close to $T_{+}$ such that

(1.15)
$\displaystyle\lambda_{1}(t)\ll\lambda_{2}(t)\ll\ldots\ll\lambda_{J}(t)\ll
T_{+}-t\text{ as }t\to T_{+}$ (1.16)
$\displaystyle\left\|(\psi(t),\partial_{t}\psi(t))-\left(\phi_{0}+\sum_{j=1}^{J}\iota_{j}Q\left(\frac{\cdot}{\lambda_{j}(t)}\right),\phi_{1}\right)\right\|_{\mathbf{H}}\underset{t\to
T_{+}}{\longrightarrow}0.$

Global solution. $T_{+}=+\infty$ and there exist $\ell\in\mathbb{Z}$, a
solution $\psi_{L}$ of

$\partial_{t}^{2}\psi_{L}-\partial_{r}^{2}\psi_{L}-\frac{1}{r}\partial_{r}\psi_{L}-\frac{1}{r^{2}}\psi_{L}=0,$

with initial data in $\mathbf{H}$, an integer $J\in\mathbb{N}$, and for each
$j\in\\{1,\ldots,J\\}$, a sign $\iota_{j}\in\\{\pm 1\\}$, and a positive
function $\lambda_{j}(t)$ defined for large $t$ such that

(1.17)
$\displaystyle\lambda_{1}(t)\ll\lambda_{2}(t)\ll\ldots\ll\lambda_{J}(t)\ll
t\text{ as }t\to+\infty$ (1.18)
$\displaystyle\Bigg{\|}(\psi(t),\partial_{t}\psi(t))-\left(\psi_{L}(t)+\ell\pi+\sum_{j=1}^{J}\iota_{j}Q\left(\frac{\cdot}{\lambda_{j}(t)}\right),\partial_{t}\psi_{L}(t)\right)\Bigg{\|}_{\mathbf{H}}\underset{t\to+\infty}{\longrightarrow}0.$

######  Remark 1.1.

In the finite time-blow-up case, if
$(\psi_{0},\psi_{1})\in\mathbf{H}_{\ell,m}$,
$(\phi_{0},\phi_{1})\in\mathbf{H}_{\ell^{\prime},m^{\prime}}$, the expansion
(1.16) yields the following constraints:

$\ell=\ell^{\prime},\quad m=m^{\prime}+\sum_{j=1}^{J}\iota_{j}.$

Similarly, in the global case, $(\psi_{0},\psi_{1})\in\mathbf{H}_{\ell,m}$,
where $\ell$ is as in (1.18) and $m=\ell+\sum_{j=1}^{J}\iota_{j}$.

######  Remark 1.2.

We highlight here the fact that Theorem 3 is a soliton resolution for _any
finite energy initial data_ of equation (1.1), which should be compared with
the results for radial solutions of (1.7), or (1.9), where we must assume that
the solution is bounded in energy norm or global in time, or sometimes both.

The strategy of the proof of the soliton resolution from the rigidity theorem
is the same as in [20]. However, since the exterior energy bound is not valid
for data of the form $(0,u_{1})$, we need the additional fact, proved in [7],
that there exists a sequence $\\{t_{n}\\}_{n}\to T_{+}$ such that
$\partial_{t}\psi(t_{n})-\phi_{1}$ in the finite time blow-up case
(respectively $\partial_{t}\psi(t_{n})-\partial_{t}\psi_{L}(t_{n})$ in the
global case) goes to $0$ in $L^{2}$. Moreover, the proof uses a “nonlinear
profile decomposition” outside wave cones which was not known earlier for
equivariant wave maps (see Proposition 7.8). We prove this nonlinear profile
decomposition using a new space-time bound outside wave cones, involving the
nonlinearity (see Lemma 7.5). These results should be of independent interest
for further study of equivariant wave maps.

######  Remark 1.3.

Solutions of (1.1) satisfying (1.16) with $J=1$ are known: see [52], [66] and
[34]. See also [67] for solutions of (1.6) with $J=1$, $k\geq 4$. We do not
know any construction of solutions of (1.1) satisfying (1.20) or (1.21) with
$J\geq 2$, known examples of solutions satisfying (1.21) with $J=1$ are the
stationary solutions and the recently constructed solutions in [62], [63]. For
construction of solutions of (1.6) with $k\geq 3$, satisfying (1.21) with
$J=2$, see [40].

######  Theorem 4 (Soliton resolution for radial $4D$ critical wave).

Let $u$ be a radial solution of (1.7) and $T_{+}$ its maximal time of
existence. Then one of the following holds:

Type I blow-up. $T_{+}<\infty$ and

(1.19) $\lim_{t\to T_{+}}\|(u(t),\partial_{t}u(t)\|_{\mathcal{H}}=+\infty.$

Type II blow-up. $T_{+}<\infty$, and there exist
$(v_{0},v_{1})\in\mathcal{H}$, an integer $J\in\mathbb{N}\setminus\\{0\\}$,
and for each $j\in\\{1,\ldots,J\\}$, a sign $\iota_{j}\in\\{\pm 1\\}$, and a
positive function $\lambda_{j}(t)$ defined for $t$ close to $T_{+}$ such that
(1.15) holds and

(1.20)
$\displaystyle\left\|(u(t),\partial_{t}u(t))-\left(v_{0}+\sum_{j=1}^{J}\frac{\iota_{j}}{\lambda_{j}(t)}W\left(\frac{x}{\lambda_{j}(t)}\right),v_{1}\right)\right\|_{\mathcal{H}}\underset{t\to
T_{+}}{\longrightarrow}0.$

Global solution. $T_{+}=+\infty$ and there exist a solution $v_{L}$ of the
linear wave equation, an integer $J\in\mathbb{N}$, and for each
$j\in\\{1,\ldots,J\\}$, a sign $\iota_{j}\in\\{\pm 1\\}$, and a positive
function $\lambda_{j}(t)$ defined for large $t$ such that (1.17) holds and

(1.21)
$\Bigg{\|}(u(t),\partial_{t}u(t))-\left(v_{L}(t)+\sum_{j=1}^{J}\frac{\iota_{j}}{\lambda_{j}(t)}W\left(\frac{x}{\lambda_{j}(t)}\right),\partial_{t}v_{L}(t)\right)\Bigg{\|}_{\mathcal{H}}\underset{t\to+\infty}{\longrightarrow}0.$

As in the wave maps case, Theorem 4 follows from the rigidity theorem, Theorem
2, using the strategy of [20] and the extra-fact, proved in [10], that there
exists a sequence $\\{t_{n}\\}_{n}\to T_{+}$ such that
$\partial_{t}u(t_{n})-v_{1}$ (in the finite time blow-up case) or
$\partial_{t}u(t_{n})-\partial_{t}v_{L}((t_{n})$ (in the global case) goes to
$0$ with $n$ going to infinity.

######  Remark 1.4.

Except for the scattering solutions ((1.21) with $J=0$) and the stationary
solutions $\pm\lambda W(\lambda\,\cdot)$, the only constructed radial
solutions of (1.7), which are bounded in $\mathcal{H}$, are the blow-up
solutions constructed in [38], satisfying (1.20) with $J=1$. Note however that
the construction in [51] of center-stable manifolds for (1.9), with $N=3$ or
$N=5$, should be adaptable to the case $N=4$ in the radial case, using Theorem
4. This scarcity of contructions seems to be the result of the very strong
interaction between solitons, due to their slow decay, and the fact that the
modulation equations for (1.9) degenerate when $N=4$, as was mentioned
earlier.

######  Remark 1.5.

The results in Theorems 3 and 4 naturally suggest the possibility of proving
analogous results for solutions of (1.6) with $k\geq 2$, for the radial Yang-
Mills equation, and for radial solutions of (1.9) in the even dimensional
case, $N\geq 6$. We expect that the ideas in this paper, together with the
work in [22], [23] and [25] will set the path to solve these problems.

###  1.4. Outline of the paper

We conclude the introduction with a very brief outline of the paper. Section 2
introduces notations, reviews the well-posedness theory and relevant space-
times estimates (Strichartz estimates) and reviews solutions outside a wave
cone. Section 3 is devoted to the proof of our new linear estimate (1.11).
Sections 4 and 5 prove the rigidity theorems, with Section 4 treating constant
sign solutions and Section 5 treating odd solutions in time. In Section 6 we
prove the soliton resolution for radial solutions of the non-linear wave
equation (1.7), while in Section 7 we prove the necessary results on co-
rotational wave maps, and then prove the soliton resolution for them. The
Appendices A, B, C collect technical results that are needed in the proofs.

##  2\. Preliminaries

###  2.1. Notations

We will reduce co-rotational wave maps to a radial wave equation in space
dimension $4$, and work in most of the article with functions that are defined
on $\mathbb{R}^{4}$ or $\mathbb{R}_{t}\times\mathbb{R}^{4}_{x}$ and that are
radial in the space variable, i.e. depending only on $r=|x|$,
$x\in\mathbb{R}^{4}$. Unless explicitly mentioned, the notation $L^{p}$ will
always denote the space $L^{p}(\mathbb{R}^{4})$. We normalize the $L^{p}$ norm
as follows:

$\|f\|_{L^{p}}^{p}=\int_{0}^{\infty}|f(r)|^{p}r^{3}dr,\quad f\in L^{p},\text{
radial.}$

We will denote by $L^{p}L^{q}$ the space
$L^{p}(\mathbb{R}_{t},L^{q}_{x}(\mathbb{R}^{4}))$. If $A$ is a space of
functions on $\mathbb{R}^{4}$, we will denote by $A(R)$ the space of
restrictions of radial elements of $A$ to $[R,\infty)$. If $B$ is a space of
functions on $\mathbb{R}_{t}\times\mathbb{R}_{x}^{4}$, we denote by $B(R)$ the
space of restriction of radial elements of $B$ in the space variable $x$ to
$\\{(t,x)\;|x|>R+|t|\\}$. We will endow this space with the usual norms, e.g.

$\displaystyle\|\varphi\|_{L^{p}(R)}^{p}=\int_{R}^{+\infty}|\varphi(r)|^{p}r^{3}dr,\quad\|\varphi\|_{\dot{H}^{1}(R)}^{2}=\int_{R}^{\infty}\left|\frac{d\varphi}{dr}(r)\right|^{2}r^{3}dr$
$\displaystyle\|f\|_{(L^{p}L^{q})(R)}^{p}=\int_{\mathbb{R}}\bigg{(}\int_{R+|t|}^{+\infty}|f(r)|^{q}r^{3}dr\bigg{)}^{\frac{p}{q}}dt.$

If $f$ is a function of space and time, we write
$\nabla_{t,x}f=(\partial_{t}f,\nabla_{x}f)$, and (if $f$ is radial),
$\partial_{t,r}f=(\partial_{t}f,\partial_{r}f)$. The notation $\mathcal{H}$
denotes the energy space for the wave equation
$\dot{H}^{1}(\mathbb{R}^{4})\times L^{2}(\mathbb{R}^{4})$. The notation
$\mathbf{E}$ stands for the space of finite energy data for the co-rotational
wave maps equation (1.1). We recall that $\mathbf{E}$ is the disjoint union

$\mathbf{E}=\bigcup_{\ell,m}\mathbf{H}_{\ell,m}$

where the affine spaces $\mathbf{H}_{\ell,m}$ parallel to the vector space
$\mathbf{H}:=\mathbf{H}_{0,0}$, are defined in the introduction. Note that
$\mathbf{H}=H\times(L^{2}((0,\infty),rdr))$, where $H$ is the space of radial
$L^{1}_{\operatorname{loc}}$ functions $\psi_{0}$ such that:

(2.1)
$\|\psi_{0}\|_{H}^{2}:=\int_{0}^{\infty}(\partial_{r}\psi_{0})^{2}rdr+\int_{0}^{\infty}\frac{1}{r^{2}}\psi_{0}^{2}rdr<\infty.$

As above, we will use the notations $\mathcal{H}(R)$ to denote the space of
restriction to $r>R$ of pair of radial functions in $\mathcal{H}$ and
$\mathbf{H}(R)$ for the space of restrictions to $r>R$ of pair of functions in
$\mathbf{H}$, endowed with the following norm

$\|(\psi_{0},\psi_{1})\|_{\mathbf{H}(R)}^{2}=\int_{R}^{\infty}(\psi_{1})^{2}rdr+\int_{R}^{\infty}(\partial_{r}\psi_{0})^{2}rdr+\int_{R}^{\infty}\frac{1}{r^{2}}\psi_{0}^{2}rdr.$

If $u$ is a function of space and time, $\vec{u}$ denotes the pair
$(u,\partial_{t}u)$.

In all the article (except in Appendix A),
$\Delta=\frac{\partial^{2}}{\partial
r^{2}}+\frac{3}{r^{2}}\frac{\partial}{\partial r}$ will denote the radial
Laplace operator in dimension $4$.

###  2.2. Strichartz estimates and well-posedness

We first recall some results on Strichartz estimates and local well-posedness.
We refer to [35], [36], [55] and [48] for the details.

Let $I$ be an interval, $t_{0}\in I$, $(u_{0},u_{1})\in\mathcal{H}$, $f\in
L^{1}(I,L^{2}(\mathbb{R}^{4})))$. By definition, the solution of the wave
equation

(2.2) $\partial_{t}^{2}u-\Delta u=f$

with initial data

(2.3) $\vec{u}_{\restriction t=t_{0}}=(u_{0},u_{1})$

is given by the Duhamel formulation:

$u(t,x)=\cos((t-t_{0})\sqrt{-\Delta})u_{0}+\frac{\sin((t-t_{0})\sqrt{-\Delta})}{\sqrt{-\Delta}}u_{1}+\int_{t_{0}}^{t}\frac{\sin\left((t-s)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f(s)ds.$

Note that $\vec{u}\in C^{0}(I,\mathcal{H})$. Furthermore, $u$ satisfies the
following Strichartz estimates (see [76], [36]):

(2.4) $\|u\|_{L^{2}(I,L^{8})}+\|u\|_{L^{3}(I,L^{6})}+\sup_{t\in
I}\|\vec{u}(t)\|_{\mathcal{H}}\lesssim\|(u_{0},u_{1})\|_{\mathcal{H}}+\|f\|_{L^{1}(I,L^{2})}.$

Radial solutions of (2.2) in the case where $I=\mathbb{R}$ and $f=0$ also
satisfy the following dispersive property:

(2.5) $\lim_{t\to\pm\infty}\|ru(t)\|_{L^{\infty}}=0.$

This can be proved for smooth compactly supported initial data using the
explicit formula for the solution. The general case can be deduced by a
density argument and the radial Sobolev inequality:

$\|rU\|_{L^{\infty}}\lesssim\|U\|_{\dot{H}^{1}},\quad
U\in\dot{H}^{1}_{\operatorname{rad}}(\mathbb{R}^{4}).$

If $(u_{0},u_{1})\in\mathcal{H}$, a solution $u$ of the nonlinear wave
equation (1.7) on $I$ with initial data (2.3) is a function $u\in
L^{3}_{\operatorname{loc}}(I,L^{6})$ which is a solution of (1.7) in the
preceding Duhamel sense, on every compact subinterval of $I$. Using Strichartz
estimates and a standard fixed point argument, one can prove that for any
initial data $(u_{0},u_{1})\in\mathcal{H}$, there exists a unique _maximal_
solution $u$ such that $\vec{u}\in
C^{0}\left((T_{-}(u),T_{+}(u)),\mathcal{H}\right)$ satisfying the blow-up
criterion

$T_{+}<\infty\;(\text{ respectively }|T_{-}|<\infty)\\\
\Longrightarrow\|u\|_{L^{3}\left((t_{0},T_{+}),L^{6}\right)}=+\infty\;(\text{respectively
}\|u\|_{L^{3}\left((T_{-},t_{0}),L^{6}\right)}=+\infty),$

where we have denoted $T_{\pm}=T_{\pm}(u)$.

We say that a solution $u$ of (1.7) _scatters_ forward in time if and only if
$T_{+}(u)=\infty$ and there exists a solution $u_{L}$ of the free wave
equation $\partial_{t}^{2}u_{L}=\Delta u_{L}$ such that

$\lim_{t\to\infty}\|\vec{u}(t)-\vec{u}_{L}(t)\|_{\mathcal{H}}=0,$

or equivalently, $u\in L^{3}((t_{0},T_{+}),L^{6})$.

###  2.3. Solutions outside a wave cone

We will now restrict to radial functions and define solutions of wave
equations outside wave cones. Let us mention that it is possible to construct
a local well-posedness theory in this context (see Subsection 2.3 of [22]) and
also the space-time maximal domain of influence of a given solution (in the
spirit of [1]). We will not need these notions here.

To simplify notations, we will restrict to initial data at $t_{0}=0$. Recall
from §2.1 above the notations $\mathcal{H}(R)=\dot{H}^{1}(R)\times L^{2}(R)$
and $(L^{p}L^{q})(R)$. If $f\in(L^{1}L^{2})(R)$ and
$(u_{0},u_{1})\in\mathcal{H}(R)$, we define the solution of (2.2), (1.8) on
$\\{r>R+|t|\\}$ as the restriction to $\\{r>R+|t|\\}$ of a solution
$\tilde{u}$ of the equation

$\partial_{t}^{2}\tilde{u}-\Delta\tilde{u}=\tilde{f},\quad\vec{\tilde{u}}_{\restriction
t=0}=(\tilde{u}_{0},\tilde{u}_{1}),$

where $(\tilde{u}_{0},\tilde{u}_{1})\in\mathcal{H}$, $\tilde{f}\in
L^{1}L^{2}$, $(u_{0},u_{1})(r)=(\tilde{u}_{0},\tilde{u}_{1})(r)$ for $r>R$,
and $\tilde{f}(t,r)=f(t,r)$ for $r>|t|+R$. By finite speed of propagation, $u$
does not depend on the choice of the extensions $\tilde{u}_{0},\tilde{u}_{1}$
and $\tilde{f}$. Furthermore, (2.4) and finite speed of propagation implies
the Strichartz estimates outside the wave cone:

(2.6)
$\|u\|_{(L^{2}L^{8})(R)}+\|u\|_{(L^{3}L^{6})(R)}+\sup_{t\in\mathbb{R}}\|\vec{u}(t)\|_{\mathcal{H}(R+|t|)}\\\
\lesssim\|(u_{0},u_{1})\|_{\mathcal{H}(R)}+\|f\|_{(L^{1}L^{2})(R)}.$

Let $(u_{0},u_{1})\in\mathcal{H}(R)$. We say that $u$ is a solution of (1.7)
in $\\{r>R+|t|\\}$ if $u\in(L^{3}L^{6})(R)$ and $u$ is a solution of (2.2),
(1.8) with $f=u^{3}$ on $\\{r>R+|t|\\}$. Using the usual small data theory and
finite speed of propagation, one sees that there exists $\delta_{0}$
(independent of $R$) such that if
$\|(u_{0},u_{1})\|_{\mathcal{H}(R)}<\delta_{0}$, then there exists a (unique)
solution $u$ on $\\{|x|>R+|t|\\}$ with initial data $(u_{0},u_{1})$. As a
consequence, for any $(u_{0},u_{1})\in\mathcal{H}$, there exists $R\geq 0$ and
a solution of (1.7) in $\\{|x|>R+|t|\\}$ with initial data $(u_{0},u_{1})$.
One also has the following uniqueness properties:

  * •

If $(u_{0},u_{1})\in\mathcal{H}$, $R\geq 0$, and there exists a solution of
(1.7) on $\\{|x|>R+|t|\\}$ with initial data $(u_{0},u_{1})$, then this
solution coincides with the solution of (1.7) defined in §2.2 in
$\\{t\in(T_{-}(u_{0},u_{1}),T_{+}(u_{0},u_{1})\\}\cap\\{|x|>R+|t|\\}$. We will
consider $u$ as a solution of (1.7) with initial data on the domain
$\\{t\in(T_{-}(u_{0},u_{1}),T_{+}(u_{0},u_{1})\\}\cup\\{|x|>R+|t|\\}$, with
obvious modifications of the definitions above.

  * •

If $(u_{0},u_{1})\in\mathcal{H}$ and $0\leq R\leq R^{\prime}$, and there
exists a solution of (1.7) with initial data $(u_{0},u_{1})$ in
$\\{|x|>R+|t|\\}$, then the restriction of this solution to
$\\{|x|>R^{\prime}+|t|\\}$ is the solution of (1.7) with initial data
$(u_{0},u_{1})_{\restriction\\{r>R^{\prime}\\}}$ in
$\\{|x|>R^{\prime}+|t|\\}$.

We can define the same way solutions of the co-rotational wave maps equation
(1.1) outside wave cones. However in this case the solution is always global
in time:

######  Claim 2.1.

For any initial data $(\psi_{0},\psi_{1})\in\mathbf{E}$, the solution $\psi$
of (1.1) is well-defined on $\\{r>|t|\\}$, and satisfies
$\vec{\psi}(t)\in\mathbf{H}(|t|)+(m\pi,0)$ for all $t$, where
$m\pi=\lim_{r\to\infty}\psi_{0}(r)$.

###### Proof.

We assume $(\psi_{0},\psi_{1})\in\mathbf{H}_{\ell,m}$. By the standard well-
posedness theory for (1.1) (see [72]), there exist $t_{0}>0$ and a solution of
(1.1) defined on $[-t_{0},t_{0}]\times(0,\infty)$ which satisfies
$\vec{\psi}(t)\in\mathcal{H}_{\ell,m}$ for all $t\in[-t_{0},t_{0}]$. We are
thus reduced to prove the existence of a solution of (1.1) with initial data
$\vec{\psi}(t_{0})$ at $t=t_{0}$, defined for $r>t>t_{0}$, and a solution of
(1.1) with initial data $\vec{\psi}(-t_{0})$ at $t=-t_{0}$, defined for
$r>-t>t_{0}$. Translating in time, we see that the conclusion of the claim
will follow from the fact that for all $R>0$ and $m\in\mathbb{Z}$, for all
initial data $(\psi_{0},\psi_{1})\in\mathbf{H}(R)+(m\pi,0)$, there exists a
solution $\psi$ of (1.1) defined for $r>R+|t|$, with initial data
$(\psi_{0},\psi_{1})$ at $t=0$.

Let $T$ be the positive maximal time of existence of $\psi$ as a solution of
(1.1) in $\\{r>R+|t|\\}$, and assume that $T<\infty$. Let $R_{1}>T+R$. We have
$1\\!\\!1_{|T|+R<r<R_{1}}\frac{\sin(2\psi)}{2r^{2}}\in L^{1}((0,T),L^{2})$.
Thus by standard energy estimates, $\vec{\psi}(t)$ has a limit, as $t\to T$,
in $(H^{1}\times L^{2})(\\{R+T<r<R_{1}\\})$. Taking $R_{1}$ large, and
combining with small data theory and finite speed of propagation to obtain
convergence for $r>R_{1}$, we obtain that $\vec{\psi}(t)$ converges in
$\mathbf{H}(R+T)+(m\pi,0)$ as $t\to T$, a contradiction. ∎

##  3\. Exterior energy estimates

This section concerns lower bounds on the exterior energy for radial solutions
of the free wave equation

(3.1) $\left\\{\begin{aligned} \partial_{t}^{2}u-\Delta
u&=0,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{4}\\\ \vec{u}_{\restriction
t=0}&=(u_{0},u_{1})\in\mathcal{H}\end{aligned}\right.$

and of its inhomogeneous counter part

(3.2) $\left\\{\begin{aligned} \partial_{t}^{2}u-\Delta
u&=f,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{4}\\\ \vec{u}_{\restriction
t=0}&=(u_{0},u_{1})\in\mathcal{H}.\end{aligned}\right.$

Exterior energy estimates were first etablished for radial solutions of the 3D
linear wave equation in [17] where the following typical result was obtained
(see [17, Lemma 4.2 and proof of Corollary 4.3]): _for any
$(u_{0},v_{0})\in\dot{H}^{1}(\mathbb{R}^{3})\times L^{2}(\mathbb{R}^{3})$ with
radial symmetry, the solution of the $3$D linear wave equation $u_{L}$
satisfies, either for all $t\geq 0$ or for all $t\leq 0$,_

(3.3) $\|\nabla
u_{L}(t)\|_{L^{2}(|x|>t)}^{2}+\|\partial_{t}u_{L}(t)\|_{L^{2}(|x|>t)}^{2}\geq\frac{1}{2}\left(\|\nabla
u_{0}\|_{L^{2}}^{2}+\|u_{1}\|_{L^{2}}^{2}\right).$

This result was later extended to any odd dimension $d\geq 3$, see [19] and
references therein. Moreover, [17, 46] proved suitable variants of such
estimates over _channels_ , _i.e._ in space time regions of the form
$|x|>t+R$, for any $R>0$: see [46, Corollary 1] and Proposition 3.2 below.

The possiblity of extending estimates of the form (3.3) to even dimensions
$d\geq 2$ is thoroughly discussed in [12]. While estimate (3.3) is proved to
_fail_ in any even dimension for general data $(u_{0},u_{1})$ (even after
replacing $\frac{1}{2}$ by any constant $C>0$), it is proved to hold for any
data of the form $(u_{0},0)$, if $d=0$ mod $4$ and for any data of the form
$(0,u_{1})$, if $d=2$ mod $4$. More specifically, in dimension $4$, [12,
Corollary 2] first proved the existence of a constant $C>0$ such that any
radial solution of the free wave equation (3.1) satisfies either for all
$t\geq 0$ or for all $t\leq 0$,

(3.4) $\|\nabla
u_{L}(t)\|_{L^{2}(|x|>t)}^{2}+\|\partial_{t}u_{L}(t)\|_{L^{2}(|x|>t)}^{2}\geq\frac{1}{C}\|\nabla
u_{0}\|^{2}_{L^{2}}.$

Note that by decomposing the solution $u_{L}$ in even and odd parts (in the
time variable), it is equivalent to state an estimate of the type (3.3) for
initial data of the form $(u_{0},0)$ for both $t\geq 0$ and $t\leq 0$, or to
state (3.4) for any general initial data $(u_{0},u_{1})$, either for all
$t\geq 0$ or for all $t\leq 0$. We also recall that the exterior energy being
a nonincreasing function of $|t|$, it is also equivalent to formulate an
asymptotic estimate, _i.e._ in terms of the limits
$\lim_{\pm\infty}\big{\\{}\|\nabla
u_{L}(t)\|_{L^{2}(|x|>t)}^{2}+\|\partial_{t}u_{L}(t)\|_{L^{2}(|x|>t)}^{2}\big{\\}}$.
In the sequel, we will restrict ourselves to initial data of the form
$(u_{0},0)$ and we will focus on limits of the exterior energy as
$t\to\infty$.

The main goal of this section is to obtain lower bounds on the exterior energy
over channels $\\{|x|>t+R\\}$ in space dimension $4$, in the same spirit as in
[17, 46] for odd dimensions. We will also deduce, by a simple trick, an
analogous estimate in space dimension $6$ (see Subsection 3.3). This will
leave open the same question for dimensions $d=4n$, $n\geq 2$ integer, and the
analogue estimate for initial data of the form $(0,u_{1})$ for dimension
$d=4n+2$, $n\geq 2$ integer.

###  3.1. Exterior energy estimates for the linear equation

Here, we discuss a slightly more precise version of (3.4) which follows from
combining [12] and [24].

######  Proposition 3.1 ([12],[24]).

Let $u_{0}\in\dot{H}^{1}(\mathbb{R}^{4})$ have radial symmetry. Let $u_{L}$ be
the solution of (3.1) with initial data $(u_{0},0)$. Then

(3.5) $\lim_{t\to\infty}\|\nabla
u_{L}(t)\|_{L^{2}(|x|>t)}^{2}=\lim_{t\to\infty}\|\partial_{t}u_{L}(t)\|_{L^{2}(|x|>t)}^{2}\geq\frac{1}{4}\|\nabla
u_{0}\|_{L^{2}}^{2}.$

For the reader’s convenience, we provide a simplified proof of (3.5) compared
to [12], using computations from [12, 46] and the notion of _radiation
profile_ (see [24, Theorem 2.1] and references therein).

###### Proof.

Let $u_{0}\in\dot{H}^{1}(\mathbb{R}^{4})$ with radial symmetry and let $u_{L}$
be the solution of (3.1) with initial data $(u_{0},0)$. From Proposition A.1
in the case $f=0$, there exists a function $G\in L^{2}(\mathbb{R})$, called
the radiation profile, such that

(3.6)
$\displaystyle\lim_{t\to\infty}\int_{0}^{\infty}\left|r^{\frac{3}{2}}\partial_{r}u_{L}(t,r)-G(r-t)\right|^{2}dr=0,$
$\displaystyle\lim_{t\to\infty}\int_{0}^{\infty}\left|r^{\frac{3}{2}}\partial_{t}u_{L}(t,r)+G(r-t)\right|^{2}dr=0,$

with $c_{3}\|G\|_{L^{2}}^{2}=\frac{1}{2}\int|\nabla u_{0}|^{2}$, where $c_{3}$
is the measure of the unit sphere of $\mathbb{R}^{4}$. We observe that the
equality of the two limits in (3.5) is a direct consequence of (3.6). This
being known, the result of Proposition 3.1 follows from [12, Corollary 2],
since looking carefully at the proof of this result, one sees that the
constant $c(d)$ appearing there is exactly $c(d)=\frac{1}{2}$. As in [12], the
proof extends to any space dimension $d=4n$, where $n\geq 1$ is integer.

_Notation._ We use the following notation for the Fourier transform of a
Schwartz function $f$

$\hat{f}(\xi)=\int_{\mathbb{R}^{4}}e^{-ix\xi}f(x)dx,\quad
f(x)=(2\pi)^{-4}\int_{\mathbb{R}^{4}}e^{ix\xi}\hat{f}(\xi)d\xi.$

In particular,

(Plancherel) $\displaystyle\|\hat{f}\|_{L^{2}(\mathbb{R}^{4})}^{2}$
$\displaystyle=(2\pi)^{4}\|f\|_{L^{2}(\mathbb{R}^{4})}^{2},$ (Parseval)
$\displaystyle\int_{\mathbb{R}^{4}}\hat{f}(\xi)\hat{g}(\xi)d\xi$
$\displaystyle=(2\pi)^{4}\int_{\mathbb{R}^{4}}f(x)\bar{g}(x)dx.$

Recall that if $f$ has the radial symmetry, then $\hat{f}$ also has the radial
symmetry and the Plancherel identity yields

$\int_{0}^{\infty}|\hat{f}(\rho)|^{2}\rho^{3}d\rho=(2\pi)^{4}\int_{0}^{\infty}|f(r)|^{2}r^{3}dr.$

Moreover,

(3.7)
$f(r)=(2\pi)^{-2}\int_{0}^{\infty}\hat{f}(\rho)J_{1}(r\rho)(r\rho)^{-1}\rho^{3}d\rho$

where $J_{1}$ is the Bessel function of first type of order one.

_Reduction of the proof._ We assume without loss of generality that
$u_{0}:\mathbb{R}^{4}\to\mathbb{R}$ is a Schwartz function with radial
symmetry such that for some $0<\rho_{*}<\rho^{*}$, it holds

$\operatorname{supp}\hat{u}_{0}\subset B_{\rho^{*}}(0)\setminus
B_{\rho_{*}}(0).$

Using the formulas (12), (14) of [12] with $f=u_{0}$ and $g\equiv 0$, we have

(3.8)
$(2\pi)^{4}\lim_{t\to\infty}\int_{t}^{\infty}|\partial_{t}u_{L}(t,r)|^{2}r^{3}dr=\frac{2}{\pi}\lim_{t\to\infty}\lim_{\varepsilon\downarrow
0}A_{\varepsilon}(t)$

where

$A_{\varepsilon}(t)=\int_{t}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\hat{u}_{0}(\rho_{1})\overline{\hat{u}}_{0}(\rho_{2})k(t,\rho_{1},\rho_{2})\rho_{1}^{\frac{5}{2}}\rho_{2}^{\frac{5}{2}}e^{-\varepsilon
r}d\rho_{1}\,d\rho_{2}\,dr$

the function $k$ being defined by

$k(t,\rho_{1},\rho_{2})=\sin(t\rho_{1})\sin(t\rho_{2})\cos\left(r\rho_{1}-\frac{3\pi}{4}\right)\cos\left(r\rho_{2}-\frac{3\pi}{4}\right).$

Recall that (3.8) follows from the Fourier inversion formula for radial
functions (3.7), and the asymptotic behavior of the Bessel functions, see [12,
(10)]. Now, we use standard trigonometry formulas to obtain

$\displaystyle
2\cos\left(r\rho_{1}-\frac{3\pi}{4}\right)\cos\left(r\rho_{2}-\frac{3\pi}{4}\right)$
$\displaystyle=\cos
r(\rho_{1}-\rho_{2})+\cos\left(r(\rho_{1}+\rho_{2})-\frac{3\pi}{2}\right)$
$\displaystyle=\cos r(\rho_{1}-\rho_{2})-\sin r(\rho_{1}+\rho_{2}).$

Thus,

$k(t,\rho_{1},\rho_{2})=\frac{1}{2}\left[\cos r(\rho_{1}-\rho_{2})-\sin
r(\rho_{1}+\rho_{2})\right]\sin(t\rho_{1})\sin(t\rho_{2})$

Now, applying formulas (15) and (16) from [12] with

$\phi(\rho_{1})=\hat{u}_{0}(\rho_{1})\sin(t\rho_{1})\rho_{1}^{\frac{5}{2}},\quad\psi(\rho_{2})=\overline{\hat{u}_{0}(\rho_{2})}\sin(t\rho_{2})\rho_{2}^{\frac{5}{2}},$

we obtain

$\lim_{\varepsilon\downarrow 0}A_{\varepsilon}(t)=A_{1}(t)+A_{2}(t)+A_{3}(t)$

where

$\displaystyle A_{1}(t)$
$\displaystyle=\frac{\pi}{2}\int_{0}^{\infty}|\hat{u}_{0}(\rho)|^{2}\sin^{2}(t\rho)\rho^{5}d\rho,$
$\displaystyle A_{2}(t)$
$\displaystyle=-\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}\hat{u}_{0}(\rho_{1})\overline{\hat{u}}_{0}(\rho_{2})k_{2}(t,\rho_{1},\rho_{2})\rho_{1}^{\frac{5}{2}}\rho_{2}^{\frac{5}{2}}d\rho_{1}\,d\rho_{2},$
$\displaystyle A_{3}(t)$
$\displaystyle=-\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}\hat{u}_{0}(\rho_{1})\overline{\hat{u}}_{0}(\rho_{2})k_{3}(t,\rho_{1},\rho_{2})\rho_{1}^{\frac{5}{2}}\rho_{2}^{\frac{5}{2}}d\rho_{1}\,d\rho_{2},$

the functions $k_{2}$ and $k_{3}$ being defined by

$\displaystyle k_{2}(t,\rho_{1},\rho_{2})$ $\displaystyle=\frac{\sin
t(\rho_{1}-\rho_{2})}{\rho_{1}-\rho_{2}}\sin(t\rho_{1})\sin(t\rho_{2}),$
$\displaystyle k_{3}(t,\rho_{1},\rho_{2})$ $\displaystyle=\frac{\cos
t(\rho_{1}+\rho_{2})}{\rho_{1}+\rho_{2}}\sin(t\rho_{1})\sin(t\rho_{2}).$

_Calculation of $\lim_{\infty}A_{1}$._ We use
$\sin^{2}(t\rho)=\frac{1}{2}(1-\cos(2t\rho))$ and we remark that

$\lim_{t\to\infty}\int_{0}^{\infty}|\hat{u}_{0}(\rho)|^{2}\cos(2t\rho)\rho^{5}d\rho=0,$

by integration by parts. Thus, we obtain

(3.9)
$\lim_{t\to\infty}A_{1}(t)=\frac{\pi}{4}\int_{0}^{\infty}|\hat{u}_{0}(\rho)|^{2}\rho^{5}d\rho.$

_Calculation of $\lim_{\infty}A_{2}$._ Applying the formula

$2\sin(t\rho_{1})\sin(t\rho_{2})=\cos t(\rho_{1}-\rho_{2})-\cos
t(\rho_{1}+\rho_{2}),$

and

$\sin t(\rho_{1}-\rho_{2})\cos t(\rho_{1}-\rho_{2})=\frac{1}{2}\sin
2t(\rho_{1}-\rho_{2}),$

we find

$k_{2}(t,\rho_{1},\rho_{2})=\frac{1}{4(\rho_{1}-\rho_{2})}\Big{(}\sin
2t(\rho_{1}-\rho_{2})-2\sin t(\rho_{1}-\rho_{2})\cos
t(\rho_{1}+\rho_{2})\Big{)}.$

For the first term in the above expression of $k_{2}$, we use the computation
after (19) in [46], which yields

$\lim_{t\to\infty}\int_{0}^{\infty}\int_{0}^{\infty}\hat{u}_{0}(\rho_{1})\overline{\hat{u}}_{0}(\rho_{2})\frac{\sin
2t(\rho_{1}-\rho_{2})}{\rho_{1}-\rho_{2}}\rho_{1}^{\frac{5}{2}}\rho_{2}^{\frac{5}{2}}d\rho_{1}\,d\rho_{2}=\pi\int_{0}^{\infty}|\hat{u}_{0}(\rho)|^{2}\rho^{5}d\rho.$

For the second term in the above expression of $k_{2}$, we rewrite

$\int_{0}^{\infty}\int_{0}^{\infty}\hat{u}_{0}(\rho_{1})\overline{\hat{u}}_{0}(\rho_{2})\frac{\sin
t(\rho_{1}-\rho_{2})}{\rho_{1}-\rho_{2}}\cos
t(\rho_{1}+\rho_{2})\,\rho_{1}^{\frac{5}{2}}\rho_{2}^{\frac{5}{2}}d\rho_{1}\,d\rho_{2}\\\
=\int_{0}^{\infty}\int_{0}^{\infty}t\operatorname{sinc}t(\rho_{1}-\rho_{2})\cos
t(\rho_{1}+\rho_{2})\phi(\rho_{1}+\rho_{2})d\rho_{1}d\rho_{2},$

where $\operatorname{sinc}\theta=\frac{\sin\theta}{\theta}$,
$\phi(\rho_{1},\rho_{2})=\hat{u}_{0}(\rho_{1})\overline{\hat{u}}_{0}(\rho_{2})\rho_{1}^{\frac{5}{2}}\rho_{2}^{\frac{5}{2}}$.
By the assumptions on $u_{0}$ and $\hat{u}_{0}$, the function $\phi$ is smooth
and compactly supported on $(0,\infty)^{2}$. By integration by parts, the
above expression is equal to

$-\frac{1}{4t}\int_{0}^{\infty}\int_{0}^{\infty}\operatorname{sinc}t(\rho_{1}-\rho_{2})\left(\frac{d}{d\rho_{1}}+\frac{d}{d\rho_{2}}\right)^{2}\Big{(}\cos
t(\rho_{1}+\rho_{2})\Big{)}\phi(\rho_{1},\rho_{2})d\rho_{1}d\rho_{2}\\\
=-\frac{1}{4t}\int_{0}^{\infty}\int_{0}^{\infty}\operatorname{sinc}t(\rho_{1}-\rho_{2})\cos
t(\rho_{1}+\rho_{2})\left(\frac{d}{d\rho_{1}}+\frac{d}{d\rho_{2}}\right)^{2}\Big{(}\phi(\rho_{1},\rho_{2})\Big{)}d\rho_{1}d\rho_{2},$

which goes to $0$ as $t\to\infty$. Therefore, we obtain

(3.10)
$\lim_{t\to\infty}A_{2}(t)=-\frac{\pi}{8}\int_{0}^{\infty}|\hat{u}_{0}(\rho)|^{2}\rho^{5}d\rho.$

_Estimate for $\lim_{\infty}A_{3}$._ Using trigonometry, we compute

$\displaystyle-k_{3}(t,\rho_{1},\rho_{2})$
$\displaystyle=\frac{1}{2(\rho_{1}+\rho_{2})}\cos
t(\rho_{1}+\rho_{2})\left(\cos t(\rho_{1}+\rho_{2})-\cos
t(\rho_{1}-\rho_{2})\right)$
$\displaystyle=\frac{1}{4(\rho_{1}+\rho_{2})}\left(1+\cos
2t(\rho_{1}+\rho_{2})-\cos 2t\rho_{1}-\cos 2t\rho_{2}\right).$

By integration by parts, the contribution of the last three terms to the limit
of $A_{3}$ is zero. Therefore, we obtain

$\displaystyle\lim_{t\to\infty}A_{3}(t)$
$\displaystyle=\frac{1}{8}\int_{0}^{\infty}\int_{0}^{\infty}\frac{1}{\rho_{1}+\rho_{2}}\hat{u}_{0}(\rho_{1})\overline{\hat{u}}_{0}(\rho_{2})\rho_{1}^{\frac{5}{2}}\rho_{2}^{\frac{5}{2}}d\rho_{1}\,d\rho_{2}$
$\displaystyle=\frac{1}{8}\int_{0}^{\infty}\hat{u}_{0}(\rho_{1})\rho_{1}^{\frac{5}{2}}H\big{(}\rho^{\frac{5}{2}}\overline{\hat{u}}_{0}\big{)}(\rho_{1})d\rho_{1},$

where $H$ denotes the Hankel operator, defined by

$(Hu)(\rho_{1})=\int_{0}^{\infty}\frac{u(\rho)}{\rho+\rho_{1}}d\rho.$

Since $H$ is a positive operator (see references in the Introduction of [12]),
and since $\hat{u}_{0}$ is real-valued ($u_{0}$ has radial symmetry), we
obtain

(3.11) $\lim_{t\to\infty}A_{3}(t)\geq 0.$

_Conclusion._ Gathering (3.8), (3.9), (3.10), (3.11) and using

$\int_{0}^{\infty}|\hat{u}_{0}(\rho)|^{2}\rho^{5}d\rho=(2\pi)^{4}\int_{0}^{\infty}|\partial_{r}u_{0}(r)|^{2}r^{3}dr,$

we have proved

$\lim_{t\to\infty}\int_{t}^{\infty}|\partial_{t}u_{L}(t,r)|^{2}r^{3}dr\geq\frac{1}{4}\int_{0}^{\infty}|\partial_{r}u_{0}(r)|^{2}r^{3}dr.$

Since

$\lim_{t\to\infty}\int_{t}^{\infty}|\partial_{r}u_{L}(t,r)|^{2}r^{3}dr=\lim_{t\to\infty}\int_{t}^{\infty}|\partial_{t}u_{L}(t,r)|^{2}r^{3}dr$

as a direct consequence of (3.6), the proof of (3.5) is complete. ∎

###  3.2. Exterior energy estimates over channels

We prove a lower bound of the asymptotic energy over channels of the form
$\\{|x|>|t|+R\\}$ for any $R>0$. We recall that $1/r^{2}$ is a solution of
(3.1) on the wave cone $\\{|x|>R+|t|\\}$, with initial data
$(1/r^{2},0)\in\dot{H}^{1}(R)\times L^{2}(R)$. We denote by
$\operatorname{span}_{R}(1/r^{2})$ the one dimensional subspace of
$\dot{H}^{1}(R)$ spanned by the function $1/r^{2}$ and by $\pi_{R}$ the
orthogonal projection on $\operatorname{span}_{R}(1/r^{2})$ for the
$\dot{H}^{1}(R)$ scalar product, explicitly given for any function
$f\in\dot{H}^{1}(R)$ by

$\pi_{R}(f)(r)=\frac{R^{2}f(R)}{r^{2}},\quad r>R.$

The projection onto the orthogonal complement of
$\operatorname{span}_{R}(1/r^{2})$ is given by

$\pi_{R}^{\perp}(f)(r)=f(r)-\frac{R^{2}f(R)}{r^{2}},\quad r>R.$

We have the following result.

######  Proposition 3.2.

Let $R>0$ and let $u_{0}\in\dot{H}^{1}(R)$ have radial symmetry. Let $u_{L}$
be the solution of (3.1) with initial data $(u_{0},0)$ in the region
$\\{(t,x)\in[0,\infty)\times\mathbb{R}^{4}:|x|>R+|t|\\}$. Then

$\lim_{t\to\infty}\|\nabla
u_{L}(t)\|_{L^{2}(|x|>t+R)}^{2}=\lim_{t\to\infty}\|\partial_{t}u_{L}(t)\|_{L^{2}(|x|>t+R)}^{2}\geq\frac{3}{20}\|\nabla\pi_{R}^{\perp}u_{0}\|^{2}_{L^{2}(|x|>R)}.$

To prove Proposition 3.2, we will need the following estimate.

######  Lemma 3.3.

Let $v_{0}:\mathbb{R}^{4}\to\mathbb{R}$ be a smooth function with radial
symmetry such that for some $A>1$,

(3.12) $\operatorname{supp}(v_{0})\subset B_{A}(0)\setminus B_{1}(0).$

Let $v_{L}$ be the solution of (3.1) with initial data $(v_{0},0)$. Then,

(3.13)
$\limsup_{t\to\infty}\|\partial_{r}v_{L}(t)\|_{L^{2}(t<|x|<t+1)}^{2}\leq\frac{1}{10}\|\partial_{r}v_{0}\|_{L^{2}}^{2}.$

###### Proof of Proposition 3.2 assuming Lemma 3.3.

We start with the case $R=1$. We define the space-time region
$\Sigma_{1}=\\{(t,x)\in[0,\infty)\times\mathbb{R}^{4}:|x|>t+1\\}$. First,
consider a smooth initial data $(w_{0},0)$ with radial symmetry such that

(3.14) $\operatorname{supp}(w_{0})\subset B_{A}(0)\setminus B_{1}(0),$

for some $A>0$, and denote by $w_{L}$ the corresponding solution of (3.1) on
$\Sigma_{1}$. Applying Lemma 3.3 to $w_{L}$, we have

$\lim_{t\to\infty}\|\nabla
w_{L}(t)\|_{L^{2}(t<|x|<t+1)}^{2}\leq\frac{1}{10}\|\nabla
w_{0}\|_{L^{2}}^{2}.$

Thus, by Proposition 3.1 and $\|\nabla w_{0}\|_{L^{2}}=\|\nabla
w_{0}\|_{L^{2}(|x|>1)}$, we obtain

(3.15) $\lim_{t\to\infty}\|\nabla
w_{L}(t)\|_{L^{2}(|x|>t+1)}^{2}\geq\frac{3}{20}\|\nabla
w_{0}\|_{L^{2}(|x|>1)}^{2}.$

Second, consider a radial function $w_{0}\in\dot{H}^{1}(\mathbb{R}^{4})$ that
vanishes at $r=1$. By finite speed of propagation, we can assume that
$w_{0}(r)$ is zero on $[0,1]$ without changing the solution $w_{L}$ on
$\Sigma_{1}$. Moreover, such a function $w_{0}$ can be approximated in
$\dot{H}^{1}(\mathbb{R}^{4})$ by a sequence of smooth functions
$(w_{0,n})_{n}$ satisfying (3.14), with some $A_{n}\to\infty$. By this density
argument, (3.15) also holds for any such $w_{0}$.

Third, consider a general initial data $u_{0}\in\dot{H}^{1}(\mathbb{R}^{4})$
with radial symmetry. Let

$w_{0}(x)=\pi_{1}^{\perp}(u_{0})(x)=u_{0}(x)-\frac{u_{0}(1)}{|x|^{2}}.$

Since $w_{0}$ vanishes at $r=1$, if we extend it by $0$ for $r\in[0,1]$, the
property (3.15) holds for $w_{L}$, and $w_{L}=u_{L}-\frac{u_{0}(1)}{|x|^{2}}$
on $\Sigma_{1}$ since $\frac{1}{|x|^{2}}$ is a solution of (3.1) on
$\Sigma_{1}$, and uniqueness holds on $\Sigma_{1}$. Moreover,

$\lim_{t\to\infty}\|\nabla
u_{L}(t)\|_{L^{2}(|x|>t+1)}^{2}=\lim_{t\to\infty}\|\nabla
w_{L}(t)\|_{L^{2}(|x|>t+1)}^{2},$

which implies that

$\lim_{t\to\infty}\|\nabla
u_{L}(t)\|_{L^{2}(|x|>t+1)}^{2}\geq\frac{3}{20}\|\nabla\pi_{1}^{\perp}u_{0}\|^{2}_{L^{2}(|x|>1)}.$

By (3.6), we have

$\lim_{t\to\infty}\|\partial_{t}u_{L}(t)\|_{L^{2}(|x|>t+1)}^{2}=\lim_{t\to\infty}\|\nabla
u_{L}(t)\|_{L^{2}(|x|>t+1)}^{2},$

which proves the result for $R=1$.

We complete the proof by a scaling argument. Let $R_{0}>0$ and let $u_{L}$ be
a solution of (3.1). Applying the result for $R=1$ to the solution
$v_{L}(t,x)=u_{L}(R_{0}t,R_{0}x)$ implies the result for the solution $u_{L}$
for $R=R_{0}$. ∎

###### Proof of Lemma 3.3.

Recall that $v_{L}$ is given explicitly by the representation formula

$v_{L}(t,x^{\prime})=\frac{1}{4\pi^{2}}\left(\frac{\partial}{\partial
t}\right)\left(\frac{1}{t}\frac{\partial}{\partial
t}\right)\left(\int_{|x-x^{\prime}|<t}\frac{v_{0}(x)}{\left(t^{2}-|x-x^{\prime}|^{2}\right)^{\frac{1}{2}}}dx\right)$

(see [30, (38) of §2.4] for $n=4$, and then use the fact that the volume of
the four dimensional unit ball is $\frac{\pi^{2}}{2}$).

Denote by ${\mathbf{e}_{1}}$ the first vector of the canonical basis of
$\mathbb{R}^{4}$. For any $t,s>0$, set

$V(t,s)=\frac{1}{4\pi^{2}t^{3}}\int_{|x-s{\mathbf{e}_{1}}|<t}\frac{v_{0}(x)}{\left(t^{2}-|x-s{\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx,$

so that

$\displaystyle v_{L}(t,s{\mathbf{e}_{1}})$
$\displaystyle=\left(\frac{\partial}{\partial
t}\right)\left(\frac{1}{t}\frac{\partial}{\partial
t}\right)\left(t^{3}V\right)(t,s)$
$\displaystyle=3V(t,s)+5t\partial_{t}V(t,s)+t^{2}\partial_{t}^{2}V(t,s)$
$\displaystyle=(V_{1}+V_{2}+V_{3})(t,s).$

We focus on $V_{3}=t^{2}\partial_{t}^{2}V$. By the change of variable

(3.16) $y=\frac{x-s{\mathbf{e}_{1}}}{t},\quad x=s{\mathbf{e}_{1}}+ty$

we rewrite $V$ as

(3.17)
$V(t,s)=\frac{1}{4\pi^{2}}\int_{|y|<1}\frac{v_{0}(s{\mathbf{e}_{1}}+ty)}{\left(1-|y|^{2}\right)^{\frac{1}{2}}}dy.$

Thus,

$\displaystyle V_{3}(t,s)$
$\displaystyle=\frac{t^{2}}{4\pi^{2}}\partial_{t}^{2}\left(\int_{|y|<1}\frac{v_{0}(s{\mathbf{e}_{1}}+ty)}{\left(1-|y|^{2}\right)^{\frac{1}{2}}}dy\right)$
$\displaystyle=\frac{t^{2}}{4\pi^{2}}\int_{|y|<1}\frac{\sum_{j,k=1}^{4}y_{j}y_{k}\partial_{x_{j}}\partial_{x_{k}}v_{0}(s{\mathbf{e}_{1}}+ty)}{\left(1-|y|^{2}\right)^{\frac{1}{2}}}dy.$

Turning back to the variable $x$ using

$y=-{\mathbf{e}_{1}}+\frac{1}{t}(x-(s-t){\mathbf{e}_{1}}),$

we obtain

$\displaystyle V_{3}=V_{4}+V_{5},$

where

(3.18) $\displaystyle V_{4}(t,s)$
$\displaystyle=-\frac{1}{2\pi^{2}t^{2}}\int_{|x-s{\mathbf{e}_{1}}|<t}\frac{(x_{1}-(s-t))\partial_{x_{1}}^{2}v_{0}(x)}{\left(t^{2}-|x-s{\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx$
$\displaystyle\quad+\frac{1}{4\pi^{2}t^{3}}\int_{|x-s{\mathbf{e}_{1}}|<t}\frac{(x_{1}-(s-t))^{2}\partial_{x_{1}}^{2}v_{0}(x)}{\left(t^{2}-|x-s{\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx$
$\displaystyle\quad-\frac{1}{2\pi^{2}t^{2}}\int_{|x-s{\mathbf{e}_{1}}|<t}\frac{\sum_{k=2}^{4}x_{k}\partial_{x_{1}}\partial_{x_{k}}v_{0}(x)}{\left(t^{2}-|x-s{\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx$
$\displaystyle\quad+\frac{1}{2\pi^{2}t^{3}}\int_{|x-s{\mathbf{e}_{1}}|<t}\frac{\sum_{k=2}^{4}(x_{1}-(s-t))x_{k}\partial_{x_{1}}\partial_{x_{k}}v_{0}(x)}{\left(t^{2}-|x-s{\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx$
$\displaystyle\quad+\frac{1}{4\pi^{2}t^{3}}\int_{|x-s{\mathbf{e}_{1}}|<t}\frac{\sum_{j,k=2}^{4}x_{j}x_{k}\partial_{x_{j}}\partial_{x_{k}}v_{0}(x)}{\left(t^{2}-|x-s{\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx$

and

$V_{5}(t,s)=\frac{1}{4\pi^{2}t}\int_{|x-s{\mathbf{e}_{1}}|<t}\frac{\partial_{x_{1}}^{2}v_{0}(x)}{\left(t^{2}-|x-s{\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx.$

We focus on $V_{5}$. We are interested in the regime $t\gg 1$, for
$s\in[t,t+1]$. Thus, we set

$\eta=s-t\in[0,1],\quad s=t+\eta.$

In particular,

$V_{5}(t,s)=\frac{1}{4\pi^{2}t}\int_{|x-(t+\eta){\mathbf{e}_{1}}|<t}\frac{\partial_{x_{1}}^{2}v_{0}(x)}{\left(t^{2}-|x-(t+\eta){\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx.$

We compute

$t^{2}-|x-(t+\eta){\mathbf{e}_{1}}|^{2}=2t(x_{1}-\eta)-|x-\eta{\mathbf{e}_{1}}|^{2}.$

Since $\eta\in[0,1]$, $|x|\leq A$ and $t\gg 1$, we have
$t^{2}-|x-(t+\eta){\mathbf{e}_{1}}|^{2}=2t(x_{1}-\eta)+\mathcal{O}(1)$ and the
domain of integration defined by $t^{2}-|x-(t+\eta){\mathbf{e}_{1}}|^{2}>0$
converges asymptotically as $t\to\infty$ to the domain
$\\{x\in\mathbb{R}^{4}:x_{1}>\eta\\}$. This motivates the following
decomposition

$V_{5}=V_{6}+V_{7}$

with

$V_{6}(t,s)=V_{5}(t,s)-t^{-\frac{3}{2}}G(s-t)$

and

$V_{7}(t,s)=t^{-\frac{3}{2}}G(s-t)$

where the function $G:[0,1]\to\mathbb{R}$ is defined by

$G(\eta)=\frac{1}{4\sqrt{2}\pi^{2}}\int_{x_{1}>\eta}\frac{\partial_{x_{1}}^{2}v_{0}(x)}{\left(x_{1}-\eta\right)^{\frac{1}{2}}}dx$

(the notation $\int_{x_{1}>\eta}$ means integration over the half-space
$\\{x\in\mathbb{R}^{4}:x_{1}>\eta\\}$). Summarising, we have decomposed
$v_{L}(t,s{\mathbf{e}_{1}})$ as

$v_{L}(t,s{\mathbf{e}_{1}})=\left(V_{1}+V_{2}+V_{4}+V_{6}+V_{7}\right)(t,s).$

To estimate the main term $V_{7}$, we start with the following lemma.

######  Lemma 3.4.

Let $f:\mathbb{R}^{4}\to\mathbb{R}$ be a continuous, compactly supported
function with radial symmetry. For any $\eta>0$,

$\int_{x_{1}>\eta}\frac{f(x)}{\left(x_{1}-\eta\right)^{\frac{1}{2}}}dx=\eta^{-\frac{1}{2}}\int_{\eta}^{\infty}f(\rho)h\left(\frac{\rho}{\eta}\right)\rho^{3}d\rho$

where

(3.19) $h(r)=\frac{4\pi}{r^{2}}\int_{1}^{r}\sqrt{\frac{r^{2}-z^{2}}{z-1}}dz.$

###### Proof.

We use the hyperspherical coordinates in dimension four:

$\displaystyle x_{1}$ $\displaystyle=\rho\cos\psi$ $\displaystyle x_{2}$
$\displaystyle=\rho\sin\psi\cos\theta$ $\displaystyle x_{3}$
$\displaystyle=\rho\sin\psi\sin\theta\cos\varphi$ $\displaystyle x_{4}$
$\displaystyle=\rho\sin\psi\sin\theta\sin\varphi$

with $\rho>0$, $\psi\in(0,\pi)$, $\theta\in(0,\pi)$ and $\varphi\in(0,2\pi)$.
Using

$dx=\rho^{3}\sin^{2}\psi\sin\theta\,d\rho\,d\psi\,d\theta\,d\varphi,$

the change of variable yields

$\int_{x_{1}>\eta}\frac{f(x)}{\left(x_{1}-\eta\right)^{\frac{1}{2}}}dx=4\pi\int_{\eta}^{\infty}f(\rho)\left(\int_{0}^{\arccos\eta/\rho}\frac{\sin^{2}\psi}{(\rho\cos\psi-\eta)^{\frac{1}{2}}}d\psi\right)\rho^{3}d\rho$

(the constant $4\pi$ comes from the integral of
$(\theta,\varphi)\mapsto\sin\theta$ over $(0,\pi)\times(0,2\pi)$). Next,
changing variable $z=(\rho/\eta)\cos\psi$,

$4\pi\int_{0}^{\arccos\eta/\rho}\frac{\sin^{2}\psi}{(\rho\cos\psi-\eta)^{\frac{1}{2}}}d\psi=4\pi\frac{\eta^{\frac{3}{2}}}{\rho^{2}}\int_{1}^{\rho/\eta}\sqrt{\frac{(\rho/\eta)^{2}-z^{2}}{z-1}}dz=\eta^{-\frac{1}{2}}h\left(\frac{\rho}{\eta}\right),$

where the function $h$ is defined in (3.19). ∎

By integration by parts, it holds for $j=2,3,4$,

$\int_{x_{1}>\eta}\frac{\partial_{x_{j}}^{2}v_{0}(x)}{\left(x_{1}-\eta\right)^{\frac{1}{2}}}dx=0.$

This implies that the function $G$ rewrites as

$G(\eta)=\frac{1}{4\sqrt{2}\pi^{2}}\int_{x_{1}>\eta}\frac{\Delta
v_{0}(x)}{\left(x_{1}-\eta\right)^{\frac{1}{2}}}dx.$

The function $\Delta v_{0}$ has radial symmetry and is equal to
$r^{-3}\left(r^{3}v_{0}^{\prime}\right)^{\prime}$. Thus, using Lemma 3.4,

$G(\eta)=\frac{1}{4\sqrt{2}\pi^{2}}\eta^{-\frac{1}{2}}\int_{\eta}^{\infty}\left(\rho^{3}v_{0}^{\prime}\right)^{\prime}(\rho)h\left(\frac{\rho}{\eta}\right)d\rho.$

Thus, integrating by parts twice and using (3.12),

$G(\eta)=\frac{1}{4\sqrt{2}\pi^{2}}\int_{1}^{\infty}\rho^{\frac{1}{2}}v_{0}(\rho)H_{1}\left(\frac{\rho}{\eta}\right)d\rho$

where we have defined

$H_{1}(r)=r^{-\frac{1}{2}}\left(r^{3}h^{\prime}\right)^{\prime}.$

Differentiating in $\eta$, we find

$G^{\prime}(\eta)=-\frac{1}{4\sqrt{2}\pi^{2}}\frac{1}{\eta}\int_{1}^{\infty}\rho^{\frac{1}{2}}v_{0}(\rho)H\left(\frac{\rho}{\eta}\right)d\rho$

where

(3.20)
$H(r)=rH_{1}^{\prime}(r)=r\left(r^{-\frac{1}{2}}\left(r^{3}h^{\prime}\right)^{\prime}\right)^{\prime}.$

The Cauchy-Schwarz inequality and then a change a variable $\rho=r\eta$ yield

$\displaystyle(G^{\prime})^{2}(\eta)$
$\displaystyle\leq\frac{1}{32\pi^{4}}\left(\frac{1}{\eta^{2}}\int_{1}^{\infty}H^{2}\left(\frac{\rho}{\eta}\right)d\rho\right)\left(\int_{1}^{\infty}v_{0}^{2}(r)rdr\right)$
$\displaystyle\leq\frac{1}{32\pi^{4}}\left(\frac{1}{\eta}\int_{1/\eta}^{\infty}H^{2}(r)dr\right)\left(\int_{1}^{\infty}v_{0}^{2}(r)rdr\right).$

Integrating on $[0,1]$ in the variable $\eta$ and then using Tonelli’s
theorem,

$\displaystyle\int_{0}^{1}(G^{\prime})^{2}(\eta)d\eta$
$\displaystyle\leq\frac{1}{32\pi^{4}}\left(\int_{0}^{1}\frac{1}{\eta}\int_{1/\eta}^{\infty}H^{2}(r)dr\,d\eta\right)\left(\int_{1}^{\infty}v_{0}^{2}(r)rdr\right)$
$\displaystyle\leq\frac{1}{32\pi^{4}}\left(\int_{1}^{\infty}H^{2}(r)\log
rdr\right)\left(\int_{1}^{\infty}v_{0}^{2}(r)rdr\right).$

######  Lemma 3.5 (Estimate on $I$).

Let

$I=\frac{1}{32\pi^{4}}\int_{1}^{\infty}H^{2}(r)\log r\,dr.$

It holds

$I\leq\frac{1}{10}.$

###### Proof.

We change variable in the expression of $h$ in (3.19)

$\displaystyle z=1+\zeta(r-1),\quad z-1=\zeta(r-1),$ $\displaystyle
r^{2}-z^{2}=(r-z)(r+z)=(1-\zeta)(r-1)(r+1+\zeta(r-1)),$

and we obtain

$\displaystyle h(r)$
$\displaystyle=4\pi\left(\frac{1}{r}-\frac{1}{r^{2}}\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K(\zeta,r)d\zeta,$
$\displaystyle K(\zeta,r)$ $\displaystyle=(r(1+\zeta)+1-\zeta)^{\frac{1}{2}}.$

Denote

(3.21) $\displaystyle K^{\prime}$ $\displaystyle=\frac{\partial K}{\partial
r}=\frac{1}{2}(1+\zeta)K^{-1},$ (3.22) $\displaystyle K^{\prime\prime}$
$\displaystyle=\frac{\partial^{2}K}{\partial
r^{2}}=-\frac{1}{4}(1+\zeta)^{2}K^{-3},$ (3.23) $\displaystyle
K^{\prime\prime\prime}$ $\displaystyle=\frac{\partial^{3}K}{\partial
r^{3}}=\frac{3}{8}(1+\zeta)^{3}K^{-5}.$

We compute $H(r)$. First,

$\displaystyle
h^{\prime}=4\pi\left(-\frac{1}{r^{2}}+\frac{2}{r^{3}}\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}Kd\zeta+4\pi\left(\frac{1}{r}-\frac{1}{r^{2}}\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime}d\zeta$

and thus

$\displaystyle
r^{3}h^{\prime}=4\pi\left(-r+2\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}Kd\zeta+4\pi\left(r^{2}-r\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime}d\zeta.$

Second,

$\displaystyle(r^{3}h^{\prime})^{\prime}$
$\displaystyle=-4\pi\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}Kd\zeta+4\pi\left(r+1\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime}d\zeta$
$\displaystyle\quad+4\pi\left(r^{2}-r\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime\prime}d\zeta$

and so

$\displaystyle r^{-\frac{1}{2}}\left(r^{3}h^{\prime}\right)^{\prime}$
$\displaystyle=-4\pi
r^{-\frac{1}{2}}\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}Kd\zeta+4\pi\left(r^{\frac{1}{2}}+r^{-\frac{1}{2}}\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime}d\zeta$
$\displaystyle\quad+4\pi\left(r^{\frac{3}{2}}-r^{\frac{1}{2}}\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime\prime}d\zeta.$

Third,

$\displaystyle\left(r^{-\frac{1}{2}}\left(r^{3}h^{\prime}\right)^{\prime}\right)^{\prime}$
$\displaystyle=2\pi
r^{-\frac{3}{2}}\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}Kd\zeta$
$\displaystyle\quad-2\pi\left(r^{-\frac{1}{2}}+r^{-\frac{3}{2}}\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime}d\zeta$
$\displaystyle\quad+2\pi\left(5r^{\frac{1}{2}}+r^{-\frac{1}{2}}\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime\prime}d\zeta$
$\displaystyle\quad+4\pi\left(r^{\frac{3}{2}}-r^{\frac{1}{2}}\right)\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}K^{\prime\prime\prime}d\zeta.$

Thus, we have obtained

$H(r)=2\pi
r^{-\frac{1}{2}}\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}Z(\zeta,r)d\zeta$

where

$Z(\zeta,r)=K-(r+1)K^{\prime}+(5r^{2}+r)K^{\prime\prime}+2(r^{3}-r^{2})K^{\prime\prime\prime}.$

We replace $K^{\prime}$, $K^{\prime\prime}$ and $K^{\prime\prime\prime}$ by
their expressions in (3.21), (3.22) and (3.23),

$Z(\zeta,r)=K^{-5}\bigg{[}K^{6}-\frac{1}{2}(r+1)(1+\zeta)K^{4}\\\
-\frac{1}{4}(5r^{2}+r)(1+\zeta)^{2}K^{2}+\frac{3}{4}(1+\zeta)^{3}(r^{3}-r^{2})\bigg{]}.$

Next, we insert

$K^{2}(\zeta,r)=(r(1+\zeta)+1-\zeta)$

and we expand in powers of $r$

$\displaystyle Z(\zeta,r)$
$\displaystyle=K^{-5}\bigg{[}r^{3}\left\\{(1+\zeta)^{3}-\frac{1}{2}(1+\zeta)^{3}-\frac{5}{4}(1+\zeta)^{3}+\frac{3}{4}(1+\zeta)^{3}\right\\}$
$\displaystyle\qquad+r^{2}\biggl{\\{}3(1+\zeta)^{2}(1-\zeta)-(1+\zeta)^{2}(1-\zeta)-\frac{1}{2}(1+\zeta)^{3}$
$\displaystyle\qquad\qquad-\frac{5}{4}(1+\zeta)^{2}(1-\zeta)-\frac{1}{4}(1+\zeta)^{3}-\frac{3}{4}(1+\zeta)^{3}\biggr{\\}}$
$\displaystyle\qquad+r\biggl{\\{}3(1+\zeta)(1-\zeta)^{2}-\frac{1}{2}(1+\zeta)(1-\zeta)^{2}$
$\displaystyle\qquad\qquad-(1+\zeta)^{2}(1-\zeta)-\frac{1}{4}(1+\zeta)^{2}(1-\zeta)\biggr{\\}}$
$\displaystyle\qquad+\left\\{(1-\zeta)^{3}-\frac{1}{2}(1+\zeta)(1-\zeta)^{2}\right\\}\bigg{]}.$

Note that in the first line, the terms in $r^{3}$ vanish. This expression
simplifies into

$Z(\zeta,r)=\frac{1}{4}K^{-5}\Bigr{[}-3r^{2}(1+\zeta)^{2}(1+3\zeta)+5r(1+\zeta)(1-\zeta)(1-3\zeta)+2(1-\zeta)^{2}(1-3\zeta)\Bigl{]}.$

We observe that for all $\zeta\in[0,1]$, $|1-3\zeta|\leq 1+\zeta$. Indeed, for
$\zeta\in[0,\frac{1}{3}]$, $|1-3\zeta|=1-3\zeta\leq 1$, while for
$\zeta\in[\frac{1}{3},1]$, $|1-3\zeta|=3\zeta-1=1+\zeta-2(1-\zeta)\leq
1+\zeta$. It follows that

$|Z(\zeta,r)|\leq\frac{1}{4}K^{-5}\left[3r^{2}(1+\zeta)^{2}(1+3\zeta)+5r(1+\zeta)^{2}(1-\zeta)+2(1-\zeta)^{2}(1+\zeta)\right].$

Therefore,

$\displaystyle|H(r)|$
$\displaystyle\leq\frac{3\pi}{2}r^{\frac{3}{2}}\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}(1+\zeta)^{2}(1+3\zeta)K^{-5}(\zeta,r)d\zeta$
$\displaystyle\quad+\frac{5\pi}{2}r^{\frac{1}{2}}\int_{0}^{1}(1-\zeta)^{\frac{3}{2}}\zeta^{-\frac{1}{2}}(1+\zeta)^{2}K^{-5}(\zeta,r)d\zeta$
$\displaystyle\quad+\pi
r^{-\frac{1}{2}}\int_{0}^{1}(1-\zeta)^{\frac{5}{2}}\zeta^{-\frac{1}{2}}(1+\zeta)K^{-5}(\zeta,r)d\zeta$

Now, we estimate $K^{-5}$ to reduce to simple integrals. Since

$K^{-5}(\zeta,r)=r^{-\frac{5}{2}}(1+\zeta)^{-\frac{5}{2}}\left(1+\frac{1-\zeta}{r(1+\zeta)}\right)^{-\frac{5}{2}}\leq
r^{-\frac{5}{2}}(1+\zeta)^{-\frac{5}{2}},$

we have

$|H(r)|\leq ar^{-1}+br^{-2}+cr^{-3}$

where

$\displaystyle a$
$\displaystyle=\frac{3\pi}{2}\int_{0}^{1}(1-\zeta)^{\frac{1}{2}}\zeta^{-\frac{1}{2}}(1+\zeta)^{-\frac{1}{2}}(1+3\zeta)d\zeta$
$\displaystyle b$
$\displaystyle=\frac{5\pi}{2}\int_{0}^{1}(1-\zeta)^{\frac{3}{2}}\zeta^{-\frac{1}{2}}(1+\zeta)^{-\frac{1}{2}}d\zeta$
$\displaystyle c$
$\displaystyle=\pi\int_{0}^{1}(1-\zeta)^{\frac{5}{2}}\zeta^{-\frac{1}{2}}(1+\zeta)^{-\frac{3}{2}}d\zeta.$

We estimate $a$, $b$ and $c$. For $a$, using the change of variable
$\tilde{\zeta}=\zeta^{\frac{1}{2}}$ and then the Cauchy-Schwarz inequality,

$a=3\pi\int_{0}^{1}(1-\tilde{\zeta}^{2})^{\frac{1}{2}}(1+\tilde{\zeta}^{2})^{-\frac{1}{2}}(1+3\tilde{\zeta}^{2})d\tilde{\zeta}\\\
\leq
3\pi\sqrt{\int_{0}^{1}\frac{d\tilde{\zeta}}{1+\tilde{\zeta}^{2}}}\sqrt{\int_{0}^{1}(1-\tilde{\zeta}^{2})(1+3\tilde{\zeta}^{2})^{2}d\tilde{\zeta}}=3\pi\sqrt{\frac{\pi}{4}\left(1+\frac{5}{3}+\frac{3}{5}-\frac{9}{7}\right)}\leq
12.$

We proceed similarly for $b$ and $c$, which yields

$\displaystyle b$
$\displaystyle=5\pi\int_{0}^{1}(1-\tilde{\zeta}^{2})^{\frac{3}{2}}(1+\tilde{\zeta}^{2})^{-\frac{1}{2}}d\tilde{\zeta}$
$\displaystyle\leq
5\pi\sqrt{\int_{0}^{1}\frac{d\tilde{\zeta}}{1+\tilde{\zeta}^{2}}}\sqrt{\int_{0}^{1}(1-\tilde{\zeta}^{2})^{3}d\tilde{\zeta}}=5\pi\sqrt{\frac{\pi}{7}}\leq
11,$ $\displaystyle c$
$\displaystyle=2\pi\int_{0}^{1}(1-\tilde{\zeta}^{2})^{\frac{5}{2}}(1+\tilde{\zeta}^{2})^{-\frac{3}{2}}d\tilde{\zeta}\leq
2\pi.$

For the sake of simplicity, we bound $b\leq 12$ and $c\leq 12$, so that we
have proved

$|H(r)|\leq 12\left(r^{-1}+r^{-2}+r^{-3}\right).$

Using the expression of $I$ and the estimate on $H$, it holds

$\displaystyle\int_{1}^{\infty}H^{2}(r)\log rdr$ $\displaystyle\leq
144\int_{1}^{\infty}(r^{-1}+r^{-2}+r^{-3})^{2}\log rdr$ $\displaystyle\leq
144\int_{1}^{\infty}\left(r^{-2}+2r^{-3}+3r^{-4}+2r^{-5}+r^{-6}\right)\log
rdr.$

Using $\int_{1}^{\infty}r^{-k}\log rdr=(k-1)^{-2}$, we obtain

$I\leq\frac{144}{32\pi^{4}}\left(1\times
1+2\times\frac{1}{4}+3\times\frac{1}{9}+2\times\frac{1}{16}+1\times\frac{1}{25}\right)=\frac{3597}{400}\frac{1}{\pi^{4}}\leq\frac{1}{10}$

where the last estimate, checked numerically, is given for the sake of
simplicity. ∎

We continue the proof of Lemma 3.3. By Lemma 3.5 and then the Hardy inequality
(see _e.g._ [73, Appendix A.4])

$\int_{0}^{\infty}v_{0}^{2}rdr\leq\int_{0}^{\infty}(v_{0}^{\prime})^{2}r^{3}dr,$

we obtain

$\int_{0}^{1}(G^{\prime})^{2}(\eta)d\eta\leq\frac{1}{10}\int_{1}^{\infty}v_{0}^{2}(r)rdr=\frac{1}{10}\int_{0}^{\infty}v_{0}^{2}(r)rdr\leq\frac{1}{10}\int_{0}^{\infty}(v_{0}^{\prime})^{2}(r)r^{3}dr.$

For the term $V_{7}$, we conclude

$\displaystyle\int_{t}^{t+1}(\partial_{s}V_{7})^{2}(t,s)s^{3}ds$
$\displaystyle=t^{-3}\int_{t}^{t+1}(G^{\prime})^{2}(s-t)s^{3}ds$
$\displaystyle=t^{-3}\int_{0}^{1}(G^{\prime})^{2}(\eta)(t+\eta)^{3}d\eta\leq\left(1+\frac{1}{t}\right)^{3}\int_{0}^{1}(G^{\prime})^{2}(\eta)d\eta,$

and so

(3.24) $\limsup_{t\to\infty}\int_{t\leq|x|\leq t+1}|\nabla
V_{7}|^{2}(t,x)dx\leq\frac{1}{10}\int|\nabla v_{0}(x)|^{2}dx.$

Now, we claim that for $k=1,2,4,6$, for $t$ large and for any $\eta\in[0,1]$,
it holds

(3.25) $|\partial_{s}V_{k}(t,t+\eta)|\lesssim t^{-\frac{5}{2}}.$

Observe that setting $s=t+\eta$, estimate (3.25) implies that

$\int_{t}^{t+1}(\partial_{s}V_{k})^{2}(t,s)s^{3}ds=\int_{0}^{1}(\partial_{s}V_{k})^{2}(t,t+\eta)(t+\eta)^{3}d\eta\\\
\leq(t+1)^{3}\int_{0}^{1}(\partial_{s}V_{k})^{2}(t,t+\eta)\,d\eta\lesssim
t^{-2}.$

Thus, for $k=1,2,4,6$,

(3.26) $\lim_{t\to\infty}\int_{t}^{t+1}(\partial_{s}V_{k})^{2}(t,s)s^{3}ds=0.$

Since estimates (3.24) and (3.26) imply Lemma 3.3, we only have to prove
(3.25). We start with the following technical result.

######  Lemma 3.6.

Let $A>0$ and let $f:\mathbb{R}^{4}\to\mathbb{R}$ be a smooth function such
that $\operatorname{supp}f\subset B_{A}(0)$. Define

$\Omega[f](t)=\int_{|x-t{\mathbf{e}_{1}}|<t}\frac{f(x)}{\left(t^{2}-|x-t{\mathbf{e}_{1}}|^{2}\right)^{\frac{1}{2}}}dx,$

For $t$ large enough,

(3.27)
$\left|(2t)^{\frac{1}{2}}\Omega[f](t)-\int_{x_{1}>0}\frac{f(x)}{x_{1}^{\frac{1}{2}}}dx\right|\leq\frac{C}{t}$

and

(3.28) $\left|\Omega[f](t)\right|\leq Ct^{-\frac{1}{2}}$

where the constant $C$ depends on $A$, $\|f\|_{L^{\infty}}$ and $\|\nabla
f\|_{L^{\infty}}$.

######  Remark 3.7.

We do not assume that $f$ has radial symmetry.

###### Proof.

Observe that (3.28) is a direct consequence of (3.27). We prove (3.27). Since
$t^{2}-|x-t{\mathbf{e}_{1}}|^{2}=2tx_{1}-|x|^{2}$, we rewrite

$(2t)^{\frac{1}{2}}\Omega[f](t)=\int_{x_{1}>\frac{|x|^{2}}{2t}}\frac{f(x)}{\left(x_{1}-\frac{|x|^{2}}{2t}\right)^{\frac{1}{2}}}dx.$

We change variables, setting

$y=\Phi(x)=\left(x_{1}-\frac{|x|^{2}}{2t},x_{2},x_{3},x_{4}\right).$

For $t$ large enough (depending on $A$), the map $\Phi$ is a diffeomorphism
from $B_{2A}(0)$ to its image $\Phi(B_{2A}(0))$, and
$B_{A}(0)\subset\Phi(B_{2A}(0))\subset B_{4A}(0)$. We compute $\Phi^{-1}$.
Using the notation $\bar{x}=(x_{2},x_{3},x_{4})$,
$\bar{y}=(y_{2},y_{3},y_{4})$, we have for $y=\Phi(x)$,

$|\bar{y}|^{2}=|\bar{x}|^{2},\quad x_{1}^{2}-2tx_{1}+2ty_{1}+|\bar{y}|^{2}=0,$

and thus

$\Phi^{-1}(y)=\left(t-\sqrt{t^{2}-2ty_{1}-|\bar{y}|^{2}},y_{2},y_{3},y_{4}\right).$

In particular, on $\Phi(B_{2A}(0))$, it holds

$|\Phi^{-1}(y)-y|\lesssim\frac{1}{t}.$

Let $J_{\Phi}$ denote the Jacobian matrix of $\Phi$. Since $\det
J_{\Phi}(x)=1-\frac{x_{1}}{t}$, we find

$\det J_{\Phi}(\Phi^{-1}(y))=\sqrt{1-2t^{-1}y_{1}-t^{-2}|\bar{y}|^{2}},$

and thus on $\Phi(B_{2A}(0))$, it holds

$|\det J_{\Phi}(\Phi^{-1}(y))-1|\lesssim\frac{1}{t}.$

Let

$\displaystyle X_{A}$ $\displaystyle=\left\\{x\in\mathbb{R}^{4}:x\in
B_{2A}(0)\mbox{ and }x_{1}-\frac{|x|^{2}}{2t}>0\right\\},$ $\displaystyle
Y_{A}$ $\displaystyle=\\{y\in\mathbb{R}^{4}:y\in B_{4A}(0)\mbox{ and
}y_{1}>0\\}.$

Since $\operatorname{supp}f\subset B_{A}(0)$ and $\Phi(X_{A})\subset Y_{A}$,

$\displaystyle\int_{x_{1}-\frac{|x|^{2}}{2t}>0}\frac{f(x)}{\left(x_{1}-\frac{|x|^{2}}{2t}\right)^{\frac{1}{2}}}dx$
$\displaystyle=\int_{X_{A}}\frac{f(x)}{\left(x_{1}-\frac{|x|^{2}}{2t}\right)^{\frac{1}{2}}}dx$
$\displaystyle=\int_{\Phi(X_{A})}\frac{f(\Phi^{-1}(y))}{y_{1}^{\frac{1}{2}}}\frac{dy}{|\det
J_{\Phi}(\Phi^{-1}(y))|}$
$\displaystyle\leq\int_{Y_{A}}\frac{f(\Phi^{-1}(y))}{y_{1}^{\frac{1}{2}}}\frac{dy}{|\det
J_{\Phi}(\Phi^{-1}(y))|}.$

On $Y_{A}$, one has

$|f(\Phi^{-1}(y))-f(y)|\lesssim\frac{1}{t},\quad\left|\frac{1}{\det
J_{\Phi}(\Phi^{-1}(y))}-1\right|\lesssim\frac{1}{t}.$

Thus,

$\left|\int_{Y_{A}}\frac{f(\Phi^{-1}(y))}{y_{1}^{\frac{1}{2}}}\frac{dy}{|\det
J_{\Phi}(\Phi^{-1}(y))|}-\int_{y_{1}>0}\frac{f(y)}{y_{1}^{\frac{1}{2}}}dy\right|\lesssim\frac{1}{t},$

which completes the proof of the lemma. ∎

_Estimate for $V_{1}$._ We observe by the change of variable
$x=x^{\prime}-\eta{\mathbf{e}_{1}}$ that

$V_{1}(t,t+\eta)=3V(t,t+\eta)=\frac{3}{4\pi^{2}t^{3}}\int_{|x-t{\mathbf{e}_{1}}|<t}\frac{v_{0}(x+\eta{\mathbf{e}_{1}})}{(t^{2}-|x-t{\mathbf{e}_{1}}|^{2})^{\frac{1}{2}}}dx$

and so

$\partial_{s}V_{1}(t,t+\eta)=\frac{3}{4\pi^{2}t^{3}}\Omega[\partial_{x_{1}}v_{0}(\cdot+\eta\mathbf{e}_{1})](t)$

For $\eta\in[0,1]$, the support of the function
$x\mapsto\partial_{x_{1}}v_{0}(x+\eta{\mathbf{e}_{1}})$ is included in
$B_{A+1}(0)$ and thus by (3.28), we have
$|\partial_{s}V_{1}(t,t+\eta)|\lesssim t^{-\frac{7}{2}}$ for $t$ large.

_Estimate for $V_{2}$._ By direct computation using the expression of $V$ in
(3.17), we have

$V_{2}(t,s)=5t\partial_{t}V(t,s)=\frac{5t}{4\pi^{2}}\int_{|y|<1}\frac{\sum_{j=1}^{4}y_{j}\partial_{x_{j}}v_{0}(s{\mathbf{e}_{1}}+ty)}{(1-|y|^{2})^{\frac{1}{2}}}dy.$

By the change of variable $y=\frac{x}{t}-{\mathbf{e}_{1}}$, we rewrite

$\displaystyle V_{2}(t,t+\eta)$
$\displaystyle=\frac{5}{4\pi^{2}t^{2}}\int_{|x-t{\mathbf{e}_{1}}|<t}\frac{\left(\frac{x_{1}}{t}-1\right)\partial_{x_{1}}v_{0}(x+\eta{\mathbf{e}_{1}})}{(t^{2}-|x-t{\mathbf{e}_{1}}|^{2})^{\frac{1}{2}}}dx$
$\displaystyle\quad+\frac{5}{4\pi^{2}t^{3}}\int_{|x-t{\mathbf{e}_{1}}|<t}\frac{\sum_{j=2}^{4}x_{j}\partial_{x_{j}}v_{0}(x+\eta{\mathbf{e}_{1}})}{(t^{2}-|x-t{\mathbf{e}_{1}}|^{2})^{\frac{1}{2}}}dx.$

Arguing as for $V_{1}$, using (3.28), for any $\eta\in[0,1]$, we find
$|\partial_{s}V_{2}(t,t+\eta)|\lesssim t^{-\frac{5}{2}}$.

_Estimate for $V_{4}$._ The first term in the expression of $V_{4}$ in (3.18)
is rewritten as

$V_{4,1}(t,t+\eta)=-\frac{1}{2\pi^{2}t^{2}}\int_{|x-t{\mathbf{e}_{1}}|<t}\frac{x_{1}\partial_{x_{1}}^{2}v_{0}(x+\eta{\mathbf{e}_{1}})}{(t^{2}-|x-t{\mathbf{e}_{1}}|^{2})^{\frac{1}{2}}}dx.$

Arguing as for $V_{1}$, using (3.28), for $\eta\in[0,1]$, we find
$|\partial_{s}V_{4,1}(t,t+\eta)|\lesssim t^{-\frac{5}{2}}$. The other terms in
the expression of $V_{4}$ are estimated similarly.

_Estimate for $V_{6}$._ Recall that

$\displaystyle V_{6}(t,t+\eta)$
$\displaystyle=\frac{t^{-\frac{3}{2}}}{4\sqrt{2}\pi^{2}}\left((2t)^{\frac{1}{2}}
\Omega[\partial_{x_{1}}^{2}v_{0}(\cdot+\eta\mathbf{e}_{1})](t)-\int_{x_{1}>0}\frac{\partial_{x_{1}}^{2}v_{0}(x+\eta\mathbf{e}_{1})}{x_{1}^{\frac{1}{2}}}dx\right).$

Thus, by (3.27) of Lemma 3.6, applied to the function
$x\mapsto\partial_{x_{1}}^{3}v_{0}(x+\eta{\mathbf{e}_{1}})$, it follows that
for any $\eta\in(0,1)$, $|\partial_{s}V_{6}(t,t+\eta)|\lesssim
t^{-\frac{5}{2}}$ and the proof of (3.25) is complete. ∎

###  3.3. Exterior energy in space dimension $6$

In this subsection, we consider the linear wave equation in space dimension
$6$:

(3.29) $\partial_{t}^{2}u_{L}-\Delta
u_{L}=0,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{6},\quad|x|>R+|t|.$

With initial data of the form:

(3.30) $\vec{u}_{\restriction t=0}=(0,u_{1}),\quad u_{1}\in
L^{2}(\\{x\in\mathbb{R}^{6},\;|x|>R\\}).$

We note that $t/r^{4}$ is a radial solution of (3.29),(3.30), with initial
data $(0,1/r^{4})$. We will denote by $\Pi_{R}$ the orthogonal projection, on
$\mathrm{span}(1/r^{4})$ in the space of radial functions in
$L^{2}(\\{x\in\mathbb{R}^{6},\;|x|>R\\})$, and by
$\Pi_{R}^{\bot}=\mathrm{Id}-\Pi_{R}$. We will prove:

######  Proposition 3.8.

Let $u$ be a radial solution of (3.29) with initial data of the form (3.30).
Then

(3.31) $\lim_{t\to\infty}\int_{t+R}^{\infty}|\nabla
u_{L}(t,r)|^{2}r^{5}dr=\lim_{t\to\infty}\int_{t+R}^{\infty}|\partial_{t}u_{L}(t,r)|^{2}r^{5}dr\\\
\geq\frac{3}{20}\int_{R}^{\infty}\left|\Pi_{R}^{\perp}u_{1}(r)\right|^{2}r^{5}dr.$

###### Proof.

This is a simple consequence of Proposition 3.2. By using the radiation
profile (see Proposition (A.1)), it suffices to prove the inequality in
(3.31). By scaling, we can assume $R=1$. Consider

$v(t,r)=\int_{r}^{\infty}\rho\partial_{t}u_{L}(t,\rho)d\rho.$

A calculation shows that $v(t,r)$ solves the linear wave equation in
$\mathbb{R}\times\mathbb{R}^{4}$ for $|x|>1+|t|$. Moreover,
$\partial_{r}v(0,r)\in L^{2}(\\{x\in\mathbb{R}^{4},\;|x|>1\\})$ and
$\partial_{t}v(0,r)=0$ since $u_{L}(0,r)=0$. By Proposition 3.2, we have that

$\lim_{t\to\infty}\int_{1+t}^{\infty}(\partial_{t}u_{L}(t,r))^{2}r^{5}dr=\lim_{t\to\infty}\int_{1+t}^{\infty}(\partial_{r}v(t,r))^{2}r^{3}dr\geq\frac{3}{20}\int_{1+t}^{\infty}\left|\nabla\pi_{1}^{\bot}v(0)\right|^{2}r^{3}dr.$

Now $\partial_{r}v(0,r)=-ru_{1}(r)$, and

$\partial_{r}(\pi^{\bot}_{1}v(0,r))=\frac{\partial}{\partial
r}\left(v(0,r)-\frac{v(0,1)}{r^{2}}\right)\\\
=-ru_{1}(r)+\frac{2}{r^{3}}\int_{1}^{\infty}\rho
u_{1}(\rho)d\rho=-r\left(u_{1}(r)-\frac{2}{r^{4}}\int_{1}^{\infty}\rho
u_{1}(\rho)d\rho\right),$

which by a simple calculation equals $-r\Pi^{\bot}_{1}(u_{1})$, and the
proposition follows. ∎

###  3.4. Exterior energy for the linear inhomogeneous wave equation

We deduce from Propositions 3.1 and 3.2 similar lower bounds on the exterior
energy for the inhomogeneous problem (3.2).

######  Lemma 3.9.

Let $u_{0}\in\dot{H}^{1}(\mathbb{R}^{4})$ and $f\in
L^{1}((0,\infty),L^{2}(\mathbb{R}^{4}))$ have radial symmetry. Let $u$ be the
solution of (3.2). Then

(3.32) $\|u_{0}\|_{\dot{H}^{1}}\leq 2\lim_{t\to+\infty}\|\nabla
u(t)\|_{L^{2}(|x|>t)}+2\left\|1\\!\\!1_{\\{|x|>|t|\\}}f\right\|_{L^{1}((0,\infty),L^{2})},$

and for $R>0$,

(3.33)
$\|\pi_{R}^{\bot}u_{0}\|_{\dot{H}^{1}(R)}\leq\sqrt{\frac{20}{3}}\left(\lim_{t\to+\infty}\|\nabla
u(t)\|_{L^{2}(|x|>t+R)}+\left\|1\\!\\!1_{\\{|x|>|t|+R\\}}f\right\|_{L^{1}((0,\infty),L^{2})}\right).$

###### Proof.

Let $v$ be the solution of

$\left\\{\begin{aligned} \partial_{t}^{2}v-\Delta
v&=1\\!\\!1_{\\{|x|>|t|\\}}f\\\ \vec{v}_{\restriction
t=0}&=(0,0).\end{aligned}\right.$

On the one hand, by standard energy computations and the Cauchy-Schwarz
inequality, one sees that

$\left|\frac{d}{dt}\|\nabla_{t,x}v(t)\|_{L^{2}}\right|\leq\|1\\!\\!1_{\\{|x|>|t|\\}}f(t)\|_{L^{2}}$

and so by integration on $(0,\infty)$,

$\|\nabla_{t,x}v(t)\|_{L^{2}}\leq\|1\\!\\!1_{\\{|x|>|t|\\}}f\|_{L^{1}L^{2}},$

where in this proof, we denote
$\|\cdot\|_{L^{1}L^{2}}=\|\cdot\|_{L^{1}((0,\infty),L^{2}(\mathbb{R}^{4}))}$.

On the other hand, we observe that the function $u-v$ satisfies

$\left\\{\begin{aligned}
\partial_{t}^{2}(u-v)-\Delta(u-v)&=1\\!\\!1_{\\{|x|\leq|t|\\}}f\\\
(\vec{u}-\vec{v})_{\restriction t=0}&=(u_{0},0)\end{aligned}\right.$

and so $u-v$ coincides in the region {$|x|>|t|$} with the solution $u_{L}$ of
the free wave equation (3.1) with initial data $(u_{0},0)$. As a consequence,
by Proposition 3.1,

$\displaystyle\|u_{0}\|_{\dot{H}^{1}}$ $\displaystyle\leq
2\lim_{t\to+\infty}\|\nabla u_{L}\|_{L^{2}(|x|>t)}$ $\displaystyle\leq
2\lim_{t\to+\infty}\|\nabla u\|_{L^{2}(|x|>t)}+2\lim_{t\to+\infty}\|\nabla
v\|_{L^{2}(|x|>t)}$ $\displaystyle\leq 2\lim_{t\to+\infty}\|\nabla
u\|_{L^{2}(|x|>t)}+2\,\|1\\!\\!1_{\\{|x|>|t|\\}}f\|_{L^{1}L^{2}},$

which completes the proof of (3.32).

The proof of (3.33) is similar, using the solution $v$ of

$\left\\{\begin{aligned} \partial_{t}^{2}v-\Delta
v&=1\\!\\!1_{\\{|x|>|t|+R\\}}f\\\ \vec{v}_{\restriction
t=0}&=(0,0).\end{aligned}\right.$

and Proposition 3.2 instead of Proposition 3.1. ∎

##  4\. Rigidity theorem part I: compactly supported pertubations and constant
sign solutions

In this section, we will start the proof of Theorems 2 and 1. We will prove
the two theorems at the same time, reducing the co-rotational wave maps
equation to a critical wave equation in four space dimension (see §4.1). In
§4.2, we consider solutions with initial data that are equal to a stationary
solution for large $r$. In §4.3 we will treat the case of solutions that have
a constant sign outside a wave cone. Section 5 concerns solutions that are odd
in time.

###  4.1. From co-rotational wave maps to four dimensional waves

Let $R>0$, $(\psi_{0},\psi_{1})\in\mathbf{H}(R)$. Let $\psi$ be the solution
of (1.1) on $\\{|x|>R+|t|\\}$. Let

$u(t,r)=\sqrt{\frac{2}{3}}\frac{\psi}{r},\;r>R+|t|\quad\text{and}\quad(u_{0},u_{1})(r)=\vec{u}_{\restriction
t=0}(r),\;r>R.$

Then $(u_{0},u_{1})\in\mathcal{H}(R)$ and

$\partial_{t}^{2}u-\partial_{r}^{2}u-\frac{3}{r}\partial_{r}u-\frac{1}{r^{2}}u+\frac{\sin(\sqrt{6}ru)}{\sqrt{6}ru}=0,\quad
r>R+|t|.$

We write this equation as

(4.1)
$\partial_{t}^{2}u-\partial_{r}^{2}u-\frac{3}{r}\partial_{r}u=\frac{1}{r^{3}}\Lambda(ru),$

where

(4.2) $\Lambda(U)=U-\frac{\sin(\sqrt{6}U)}{\sqrt{6}}.$

We note that $\Lambda$ is odd, smooth, and that it satisfies:

(4.3) $\forall
k\in\\{0,1,2\\},\quad\left|\left(\frac{d}{dU}\right)^{k}\left(\Lambda(U)-U^{3}\right)\right|\lesssim\min\left(U^{5-k},U^{3-k}\right).$

As a consequence, we have the following Lipschitz bound:

(4.4) $\left|\Lambda(U)-\Lambda(V)\right|\lesssim|U-V|(U^{2}+V^{2}).$

Note that (4.1) with $\Lambda(U)=U^{3}$ is exactly the $4D$ radial wave
equation (and that in this case, (4.3) is trivially satisfied). We have the
following small data statement:

######  Proposition 4.1.

Let $\Lambda\in C^{\infty}(\mathbb{R})$, odd, such that (4.3) holds. Then
there exists $\varepsilon>0$ with the following property. For all
$(u_{0},u_{1})\in\mathcal{H}(R)$ such that
$\|u_{L}\|_{(L^{3}L^{6})(R)}\leq\varepsilon$, the solution $u$ of (4.1) with
initial data $(u_{0},u_{1})$ is defined for $r>R+|t|$ and

$\sup_{t\in\mathbb{R}}\|\vec{u}(t)-\vec{u}_{L}(t)\|_{\mathcal{H}(R+|t|)}+\|u-u_{L}\|_{(L^{3}L^{6})(R)}\lesssim\|u_{L}\|^{3}_{(L^{3}L^{6})(R)},$

where $u_{L}$ is the solution of the linear wave equation (3.1) with initial
data $(u_{0},u_{1})$.

###### Sketch of proof.

Using equation (4.1), the Strichartz estimate outside the cone (2.6) and a
straightforward bootstrap argument, one first obtains:

$\|u\|_{(L^{3}L^{6})(R)}\lesssim\|u_{L}\|_{(L^{3}L^{6})(R)}.$

The statement then follows from (4.1) and (2.6). ∎

Recall that $Q(r)=2\arctan r$ is a stationary solution of (1.1). By the above
reduction, we obtain a solution
$\widetilde{W}=\sqrt{\frac{2}{3}}\frac{1}{r}(\pi-Q)$ of (4.1) (with $\Lambda$
is defined by (4.2)), such that for large $r$,

$\left|\widetilde{W}(r)-\frac{2\sqrt{2}}{\sqrt{3}r^{2}}\right|\lesssim\frac{1}{r^{4}},\quad
r\geq 1.$

Rescaling, we obtain a solution $W(r)=\lambda\widetilde{W}(\lambda r)$ of
(4.1) such that

(4.5) $\left|W(r)-\frac{8}{r^{2}}\right|\leq\frac{C}{r^{4}},$

that is, with exactly the same asymptotic behaviour as the stationary solution
$W$ of (1.7) defined in the introduction. We denote by $\mathcal{H}(0^{+})$
the set of functions $(u_{0},u_{1})$ defined on $(0,\infty)$ such that

$\forall R>0,\quad(u_{0},u_{1})_{\restriction(R,\infty)}\in\mathcal{H}(R),$

and define similarly $\dot{H}^{1}(0^{+})$. Theorems 2 and 1 reduce to the
following rigidity theorem:

######  Theorem 4.2.

Let $\Lambda\in C^{\infty}(\mathbb{R})$ odd, such that (4.3) holds. Assume
that the wave equation (4.1) admits a stationary solution
$W\in\dot{H}^{1}(0^{+})$ such that (4.5) holds. Let
$(u_{0},u_{1})\in\mathcal{H}(0^{+})$, and assume that
$\forall\lambda\in\mathbb{R}$, $(u_{0},u_{1})$ is not equal to $(\lambda
W(\lambda\cdot),0)$. Then there exists $R>0$ such that the solution $u$ with
initial data $(u_{0},u_{1})$ is defined on $\\{|x|>R+|t|\\}$ and

(4.6)
$\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R+|t|}|\nabla_{t,x}u(t,x)|^{2}dx>0.$

######  Claim 4.3.

Theorem 4.2 implies Theorems 2 and 1.

###### Proof.

The wave equation (1.7) is (4.1) with $\Lambda(U)=U^{3}$. The ground state
$W(x)=(1+|x|^{2}/8)^{-1}$ is a stationary solution of (1.7) that satisfies
(4.5). Thus Theorem 4.2 implies that for any solution $u$ of (1.7) that is not
a stationary solution, there exists $R>0$ such that $u$ is defined for
$\\{|x|>R+|t|\\}$ and

(4.7)
$\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R+|t|}(\partial_{t}u(t,x))^{2}+|\nabla_{x}u(t,x)|^{2}dx>0.$

Next, observe that from finite speed of propagation and small data theory, if

$\sum_{\pm}\inf_{t\to\pm\infty}\int_{|x|>R+|t|}(\partial_{t}u(t,x))^{2}+|\nabla_{x}u(t,x)|^{2}dx=0,$

then (4.7) cannot hold, concluding the proof of Theorem 2.

Let $\psi$ as in Theorem 4.2. Thus
$(\psi_{0},\psi_{1})\in\mathbf{H}_{\ell,m}$, which implies $(\psi_{0}-\pi
m,\psi_{1})\in\mathbf{H}_{\ell-m,0}$. Let

$u(t,r)=\sqrt{\frac{2}{3}}\frac{\psi(t,r)-\pi
m}{r},\quad(u_{0},u_{1})(r)=\vec{u}(0,r).$

Since $(\psi_{0}-\pi m,\psi_{1})\in\mathbf{H}(R)$ for all $R$, we have
$(u_{0},u_{1})\in\mathcal{H}(0^{+})$. By the assumption on
$(\psi_{0},\psi_{1})$, we have that $(u_{0},u_{1})$ is not identically $0$,
and that it is not equal to
$\pm\left(\frac{\sqrt{2}}{\sqrt{3}r}(\pi-Q(\lambda\cdot)),0\right)$ for all
$\lambda>0$. By the definition of $W$, we deduce that $(u_{0},u_{1})$
satisfies the assumptions of Theorem 4.2. Furthermore,

$\int_{R+|t|}^{+\infty}\left((\partial_{t}u)^{2}+(\partial_{r}u)^{2}\right)r^{3}dr=\frac{2}{3}\int_{R+|t|}^{+\infty}\left((\partial_{t}\psi)^{2}+\left(\partial_{r}\psi-r^{-1}(\psi-\pi
m)\right)^{2}\right)rdr\\\
\lesssim\int_{R+|t|}\left(\partial_{t}\psi)^{2}+(\partial_{r}\psi)^{2}+\frac{1}{r^{2}}(\psi-\pi
m)^{2}\right)rdr,$

which shows that the conclusion (4.6) of Theorem 4.2 implies

(4.8)
$\sum_{\pm}\lim_{t\to\pm\infty}\int_{R+|t|}^{\infty}\left((\partial_{t}\psi)^{2}+(\partial_{r}\psi(t,r))^{2}+\frac{1}{r^{2}}(\psi(t,r)-\pi
m)^{2}\right)rdr>0.$

It remains to prove that (4.8) implies the conclusion (1.13) of Theorem 1.
This is elementary, but requires some preliminaries on equation (1.1) and we
postpone the proof to Section 7, after the proof of Claim 7.1. ∎

The remainder of this section and the next section are dedicated to the proof
of Theorem 4.2. Until the end of Section 5, $\Lambda$ is an odd, smooth
function on $\mathbb{R}$ satisfying (4.3) and such that there exists a
stationary solution $W\in\dot{H}^{1}$ of (4.1) satisfying (4.5).

###  4.2. Compactly supported perturbation of a stationary solution

We first prove:

######  Proposition 4.4.

Let $(u_{0},u_{1})\in\mathcal{H}(0^{+})$, radial. Assume that there exists
$R_{0}>0$ and $\lambda\in\mathbb{R}$ such that

$\forall r>R_{0},\quad(u_{0},u_{1})(r)=\left(\lambda W(\lambda r),0\right),$

and that the preceding equality does not hold for all $r>0$. Then there exists
$R>0$ such that the solution $u$ of (4.1) with initial data $(u_{0},u_{1})$ is
defined for $r>R+|t|$ and

$\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R+|t|}|\nabla_{t,x}u(t,x)|^{2}dx>0.$

###### Proof.

Rescaling, we may assume that $\lambda=1$. Let

$h=u-W,\quad(h_{0},h_{1})=(u_{0}-W,u_{1})=\vec{h}_{\restriction t=0}.$

Then the equation (4.1) satisfied by $u$ is equivalent to

(4.9) $\partial_{t}^{2}h-\Delta
h=\frac{1}{r^{3}}\left(\Lambda(r(W+h))-\Lambda(rW)\right).$

Furthermore by (4.3)

(4.10)
$\frac{1}{r^{3}}\left|\Lambda(r(W+h))-\Lambda(rW)\right|\lesssim|h|(h^{2}+W^{2}).$

We define

$\rho=\inf\left\\{\sigma>0,\;\int_{|x|>\sigma}|\nabla
h_{0}|^{2}+h_{1}^{2}=0\right\\}$

By the assumptions of the proposition, $0<\rho<\infty$. We fix a small
$\varepsilon>0$ such that

(4.11) $0<\|(h_{0},h_{1})\|_{\mathcal{H}(\rho-\varepsilon)}\ll 1.$

Using the Strichartz estimates (2.6), and standard small data theory, we
obtain that (4.9) has a solution $h$ with initial data $(h_{0},h_{1})$ defined
for $\\{|x|>\rho-\varepsilon+|t|\\}$, and that

$\sup_{t\in\mathbb{R}}\|\vec{h}\|_{\mathcal{H}(\rho-\varepsilon+|t|)}\lesssim\varepsilon_{0}.$

By the radial Sobolev embedding and the property (4.5) of $W$, we obtain that
there exists a constant $C>0$ such that

$\forall t\in\mathbb{R},\;\forall r>\rho-\varepsilon+|t|,\quad
h^{2}+W^{2}\leq\frac{C}{r^{2}}.$

Combined with (4.10), and taking a smaller $\varepsilon$ if necessary, we see
that Proposition 4.7 in [23] implies that the following holds for all $t>0$ or
for all $t<0$:

(4.12)
$\int_{|x|\geq\rho-\varepsilon+|t|}|\nabla_{t,x}h|^{2}dx\geq\frac{1}{8}\int_{|x|\geq\rho-\varepsilon}\left(|\nabla
h_{0}|^{2}+h_{1}^{2}\right)dx>0,$

which concludes the proof, since $W\in\dot{H}^{1}(0^{+})$. ∎

Proposition 4.4 reduces the proof of Theorem 4.2 to

######  Proposition 4.5.

Let $R_{0}>0$ and $u$ be a radial solution of (4.1) for $|x|>|t|+R_{0}$, with
initial data in $\mathcal{H}(R_{0})$. Assume

$\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R_{0}+|t|}|\nabla_{t,x}u(t,x)|^{2}dx=0.$

Then there exists $R\geq R_{0}$ such that one of the following holds:

(4.13) $\forall r>R,\quad(u_{0},u_{1})(r)=\left(0,0\right),$

or

(4.14) $\exists\lambda>0,\;\iota\in\\{\pm 1\\},\;\forall
r>R,\quad(u_{0},u_{1})(r)=\iota\left(\lambda W(\lambda r),0\right).$

The proof of Proposition 4.5 is divided into §4.3, and Section 5.

###  4.3. Constant sign solutions

In this section, we prove

######  Proposition 4.6.

Let $R_{0}>0$ and $u$ be a radial solution of (4.1) defined for
$\\{|x|>|t|+R_{0}\\}$ with

$\vec{u}_{\restriction t=0}=(u_{0},u_{1})\in\mathcal{H}(R_{0}).$

Assume

(4.15)
$\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>|t|+R_{0}}|\nabla_{t,x}u(t,x)|^{2}dx=0.$

Then one of the following holds:

(4.16) $\displaystyle\exists t_{0}\in\mathbb{R},\;\exists R>R_{0},\quad\forall
r>|t_{0}|+R,\quad u(t_{0},r)=0$ (4.17)
$\displaystyle\exists\mu\in\mathbb{R}\setminus\\{0\\},\;\exists
R>R_{0},\;\forall r>R,\quad(u_{0},u_{1})(r)=\left(\mu W(\mu r),0\right).$

Proposition 4.6 reduces the proof of Proposition 4.5 to the classification of
solutions satisfying (4.15) and (4.16), which we will carry out in the next
section. Note that (4.16) is not sufficient to conclude immediately that $u$
is identically $0$ outside a wave cone, since $\partial_{t}u(t_{0},r)$ does
not have to vanish for $r>|t_{0}|+R$.

We split the proof of Proposition 4.6 in a few lemmas.

######  Lemma 4.7.

The exists a small $\varepsilon_{0}>0$ with the following property. Let $u$ be
a radial solution of (4.1) on $\\{|x|>R+|t|\\}$, where $R\geq 0$. Assume

(4.18)
$\displaystyle\int_{|x|>R}|\nabla_{t,x}u(0,x)|^{2}dx\leq\varepsilon_{0}^{2}$
(4.19)
$\displaystyle\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R+|t|}|\nabla_{t,x}u(t,x)|^{2}dx=0.$

Then

(4.20) $\displaystyle\forall\rho\geq
R,\quad|u_{0}(\rho)|\approx\frac{1}{\rho}\|u\|_{L^{4}(\rho)}\approx\frac{1}{\rho}\|u_{0}\|_{\dot{H}^{1}(\rho)}$
(4.21) $\displaystyle\forall r\geq\rho\geq R,\quad|u_{0}(r)|\leq
2\left(\frac{\rho}{r}\right)^{\frac{11}{6}}|u_{0}(\rho)|.$

###### Proof.

Let

$u_{+}(t)=\frac{u(t)+u(-t)}{2},\quad u_{-}(t)=\frac{u(t)-u(-t)}{2}.$

Then

(4.22) $\left\\{\begin{aligned} \left|\partial_{t}^{2}u_{+}-\Delta
u_{+}\right|&\lesssim u_{+}(u_{+}^{2}+u_{-}^{2})\\\ \vec{u}_{+\restriction
t=0}&=(u_{0},0),\end{aligned}\right.$

where we have used that, $\Lambda$ being odd,

$\left|\Lambda(ru(t))+\Lambda(ru(-t))\right|=\left|\Lambda(ru(t))-\Lambda(-ru(-t))\right|\\\
\lesssim r^{3}|u(t)+u(-t)|\left(u^{2}(t)+u^{2}(-t)\right),$

by (4.4). By the small data theory for (4.1) (see Proposition 4.1), the
condition (4.18) implies $\|u\|_{(L^{3}L^{6})(R)}\lesssim\varepsilon_{0}^{2}$.
Combining with the equation (4.22) and Strichartz estimates, we deduce that
for all $\rho\geq R$,

(4.23)
$\|u_{+}\|_{(L^{3}L^{6})(\rho)}+\sup_{t\in\mathbb{R}}\left\|\vec{u}_{+}(t)\right\|_{\mathcal{H}(\rho+|t|)}\lesssim\|u_{0}\|_{\dot{H}^{1}(\rho)}.$

Using the lower bound for the exterior energy (Lemma 3.9) and equation (4.22)
again, we obtain

(4.24) $\left\|\pi_{\rho}^{\bot}u_{0}\right\|_{\dot{H}^{1}(\rho)}\leq
C\left(\|u_{+}^{3}\|_{(L^{1}L^{2})(\rho)}+\|u_{-}^{2}u_{+}\|_{(L^{1}L^{2})(\rho)}\right)\leq
C\varepsilon_{0}^{2}\|u_{0}\|_{\dot{H}^{1}(\rho)}.$

Since

$\pi_{\rho}^{\bot}u_{0}(r)=u_{0}(r)-\frac{\rho^{2}}{r^{2}}u_{0}(\rho),\quad\left\|\frac{1}{r^{2}}\right\|_{\dot{H}^{1}(\rho)}=\frac{1}{\sqrt{2}\rho},$

we deduce from (4.24), taking $\varepsilon_{0}$ small enough,

(4.25) $\|u_{0}\|_{\dot{H}^{1}(\rho)}\leq C\rho|u_{0}(\rho)|.$

Next, we deduce from (4.24), (4.25) and Sobolev embedding that for $\rho\geq
R$,

(4.26)
$\left\|u_{0}-\frac{\rho^{2}}{r^{2}}u_{0}(\rho)\right\|_{L^{4}(\rho)}\leq
C\varepsilon^{2}_{0}\rho|u_{0}(\rho)|.$

This implies, noting that
$\left\|\frac{1}{r^{2}}\right\|_{L^{4}(\rho)}=\frac{1}{\sqrt{2}\rho}$,

(4.27)
$\left|\|u_{0}\|_{L^{4}(\rho)}-\frac{\rho|u_{0}(\rho)|}{\sqrt{2}}\right|\leq
C\varepsilon^{2}_{0}\rho|u_{0}(\rho)|.$

Combining with (4.25) and Sobolev embedding, we obtain (4.20). We also obtain,
from (4.27),

(4.28)
$\|u_{0}\|^{4}_{L^{4}(\rho)}\leq\frac{\rho^{4}}{4^{-}}|u_{0}(\rho)|^{4},$

where $4^{-}$ is a positive constant, smaller than $4$, that can be chosen
arbitrarily close to $4$ provided $\varepsilon_{0}$ is small enough. Letting

$f(\rho)=\int_{\rho}^{+\infty}|u_{0}(r)|^{4}r^{3}dr=\|u_{0}\|^{4}_{L^{4}(\rho)},$

we can write (4.28) as

(4.29) $f(\rho)\leq-\frac{\rho}{4^{-}}f^{\prime}(\rho),\quad\rho\geq R,$

that is

(4.30) $\frac{f^{\prime}(\rho)}{f(\rho)}\leq-\frac{4^{-}}{\rho},\quad\rho\geq
R.$

Integrating, we deduce that for $r\geq\rho\geq R$,

(4.31) $\frac{f(r)}{f(\rho)}\leq\frac{\rho^{4^{-}}}{r^{4^{-}}},$

that is

$\|u_{0}\|^{4}_{L^{4}(r)}\leq\left(\frac{\rho}{r}\right)^{4^{-}}\|u_{0}\|^{4}_{L^{4}(\rho)}.$

Combining with (4.27), we deduce (4.21). ∎

######  Lemma 4.8.

Let $u$, $R$ and $\varepsilon_{0}$ be as in Lemma 4.7 and assume that for all
$t$, $u(t)$ is not identically $0$ on $[R+|t|,+\infty)$. Then $u$ never
vanishes and has a constant sign on $\\{|x|>R+|t|\\}$. Furthermore, for all
$t\in\mathbb{R}$, there exists $\ell(t)\neq 0$ such that

(4.32) $\forall r>R+|t|,\quad\left|r^{2}u(t,r)-\ell(t)\right|\leq
Cr^{4}|u(t,r)|^{3}.$

######  Remark 4.9.

Recall that by the radial Sobolev inequality, for all $t$,
$\lim_{r\to\infty}ru(t,r)=0$. Thus (4.32), dividing by $r^{2}u(t,r)$, implies

(4.33) $\lim_{r\to\infty}r^{2}u(t,r)=\ell(t).$

###### Proof.

Taking a smaller $\varepsilon_{0}>0$ if necessary, we see by finite speed of
propagation and the small data theory that for every time $t_{0}$, the
function $(t,r)\mapsto u(t+t_{0},r)$ satisfies the assumptions of Lemma 4.7
with $R$ replaced by $R+|t_{0}|$. By the conclusion of this lemma and the
assumption that for all $t$, $u(t,r)$ is not identically $0$ for $r>R+|t|$, we
obtain that $u$ does not vanish for $r>R+|t|$. Since $u$ is a continuous
function, we deduce that it has a constant sign for $r>R+|t|$. We will assume
to fix ideas $u(t,r)>0$ for all $r>|t|+R$. By (4.21),

(4.34) $\forall r>\rho\geq R,\quad 0<u_{0}(r)\leq
2\left(\frac{\rho}{r}\right)^{\frac{11}{6}}u_{0}(\rho).$

Next, we observe that for all $(t,r)$ with $r>t+R$,

$|u_{-}(t,r)|=\frac{1}{2}\left|u(t)-u(-t)\right|\leq\frac{u(t)+u(-t)}{2}=u_{+}(t),$

where we have used that $u$ is positive. The first inequality in (4.24),
together with (4.23) implies, for $\rho\geq R$,

(4.35) $\left\|\pi_{\rho}^{\bot}u_{0}\right\|_{\dot{H}^{1}(\rho)}\leq
C\|u_{+}^{3}\|_{(L^{1}L^{2})(\rho)}\leq C\|u_{0}\|^{3}_{\dot{H}^{1}(\rho)}.$

Using (4.20) and Sobolev inequalities as in the proof of Lemma 4.7, we deduce

(4.36)
$\left|\|u_{0}\|_{L^{4}(\rho)}-\frac{\rho}{\sqrt{2}}u_{0}(\rho)\right|\leq
C\rho^{3}u_{0}(\rho)^{3},\quad\rho\geq R.$

Letting as before $f(\rho)=\|u_{0}\|^{4}_{L^{4}(\rho)}$, we deduce, for
$r\geq\rho$,

$\left|f(r)+\frac{r}{4}f^{\prime}(r)\right|\leq
Cr^{6}|u_{0}(r)|^{6}\leq\frac{C}{r^{5}}\rho^{11}u_{0}(\rho)^{6},$

that is

$\left|\frac{d}{dr}\left(r^{4}f(r)\right)\right|\leq\frac{C}{r^{2}}\rho^{11}u_{0}(\rho)^{6}.$

As a consequence, $r^{4}f(r)$ has a limit $\ell^{\prime}\geq 0$ as
$r\to\infty$ and

$\left|\ell^{\prime}-\rho^{4}f(\rho)\right|\leq
C\rho^{10}u_{0}(\rho)^{6},\quad\rho\geq R.$

Recall that by (4.20), $\rho u_{0}(\rho)\approx f(\rho)^{1/4}$ goes to $0$ as
$\rho\to\infty$. By (4.36),

$\left|f(\rho)-\frac{\rho^{4}}{4}u_{0}(\rho)^{4}\right|\leq
C\rho^{6}u_{0}(\rho)^{6},$

and thus

$\left|\frac{4\ell^{\prime}}{\rho^{8}}-u_{0}(\rho)^{4}\right|\leq
C\rho^{2}|u_{0}(\rho)|^{6}.$

This implies $\ell^{\prime}>0$ since $u_{0}(\rho)\neq 0$ and $\rho
u_{0}(\rho)$ is small for $\rho$ large. Using the bound
$|A-B|\leq\frac{|A^{4}-B^{4}|}{A^{3}}$, $A,B>0$, we also deduce

$\left|\frac{\sqrt{2}{\ell^{\prime}}^{\frac{1}{4}}}{\rho^{2}}-u_{0}(\rho)\right|\leq
C\rho^{2}|u_{0}(\rho)|^{3},\quad\rho\geq R.$

This yields (4.32) for $t=0$. Applying this to the solution $(t,r)\mapsto
u(t+t_{0},r)$ which satisfies the same assumptions as $u$ with $R$ replaced by
$R+|t|$, we deduce (4.32) for all $t\in\mathbb{R}$. ∎

######  Lemma 4.10.

Let $u$, $R$, $\varepsilon_{0}$ be as in Lemmas 4.7 and 4.8. Then $\ell$ is
independent of $t$ and

(4.37) $\forall t\in\mathbb{R},\;\forall r\geq
R+|t|,\quad|u(t,r)|+\frac{1}{r}\|u(t)\|_{\mathcal{H}(R)}\lesssim\frac{|\ell|}{r^{2}}.$

###### Proof.

We prove that $\ell$ is constant by contradiction. Let
$(\tau,t)\in\mathbb{R}^{2}$, and assume $\ell(t)>\ell(\tau)$. Since $\ell$
does not vanish, and $u$ has a constant sign, we can also assume without loss
of generality (using Remark 4.9), $\ell(\tau)>0$.

Fix a small $\delta>0$, to be specified. By (4.33), we have that for large $r$

$0<(1-\delta)r^{2}u(t,r)\leq\ell(t)\leq(1+\delta)r^{2}u(t,r),$

and similarly

$0<(1-\delta)r^{2}u(\tau,r)\leq\ell(\tau)\leq(1+\delta)r^{2}u(\tau,r).$

As a consequence,

$r^{2}(u(t,r)-u(\tau,r))\geq\frac{1}{1+\delta}\ell(t)-\frac{1}{1-\delta}\ell(\tau)=c>0$

for $\delta>0$ small enough. Thus for large $r$,

(4.38) $u(t,r)-u(\tau,r)\geq\frac{c}{r^{2}}.$

Since $u$ is a finite energy solution of (4.1), on $\\{r>R+|t|\\}$, we have

$u(t)-u(\tau)=\int_{\tau}^{t}\partial_{t}u(s)ds\in L^{2}(\rho)$

for large $\rho$, contradicting (4.38). Thus $\ell$ is independent of $t$.

Going back to (4.32), we obtain

$\forall\rho>R+|t|,\quad|\rho^{2}u(t,\rho)|-C\rho^{4}|u(t,\rho)|^{3}\leq\ell.$

Since by (4.20)

$\rho|u(t,\rho)|\approx\|\vec{u}(t)\|_{\mathcal{H}(\rho)}\lesssim\varepsilon_{0},\quad\rho\geq
R+|t|,$

we deduce (4.37) using that $\varepsilon_{0}$ is small. ∎

###### Proof of Proposition 4.6.

We let $u$ be as in the proposition, and choose $R$ large, so that

(4.39) $\int_{|x|>R}|\nabla u_{0}|^{2}+|u_{1}|^{2}dx\leq\varepsilon_{0},$

where $\varepsilon_{0}$ is small as in the preceding lemmas. We assume that
(4.16) does not hold so that (taking a larger $R$ if necessary), $u$ satisfies
the assumptions of Lemmas 4.8 and 4.10. We assume to fix ideas that $u$ is
positive. Let $\ell$ be as in the two Lemmas, so that by (4.32), (4.37),

(4.40) $\forall t\in\mathbb{R},\;\forall r\geq
R+|t|,\quad\left|u(t,r)-\frac{\ell}{r^{2}}\right|\leq
C\frac{\ell^{3}}{r^{4}},$

for some absolute constant $C>0$. Rescaling, we may assume that $\ell=8$, so
that, by the assumption (4.5) on $W$,

(4.41) $\forall t\in\mathbb{R},\;\forall r\geq
R+|t|,\quad\left|u(t,r)-W(r)\right|\leq\frac{K}{r^{4}},$

for some absolute constant $K>0$. Let, for $\rho\geq R$,

$M(\rho)=\sup_{\begin{subarray}{c}t_{0}\in\mathbb{R}\\\
\sigma\geq\rho+|t_{0}|\end{subarray}}\sigma^{2}|u(t_{0},\sigma)-W(\sigma)|.$

By (4.41),

(4.42) $\forall\rho\geq R,\quad M(\rho)\leq\frac{K}{\rho^{2}}.$

We will show that for $R^{\prime}\geq R$ large enough,

(4.43) $\forall\rho\geq R^{\prime},\quad M(2\rho)\geq\frac{1}{2}M(\rho).$

Note that (4.42) and (4.43) would imply, for $\rho>R^{\prime}$ and
$k\in\mathbb{N}$,

$M(\rho)\leq 2^{k}M(2^{k}\rho)\leq K\frac{1}{2^{k}\rho^{2}}$

and thus $M(\rho)=0$. Thus $u(t,r)=W(r)$ for $r\geq R^{\prime}+|t|$, as
desired. It remains to prove (4.43).

We fix $t_{0}\in\mathbb{R}$ and $\rho\geq R^{\prime}$. We let

$u(t)=W+h(t),\quad h_{\pm}(t)=\frac{h(t)\pm h(2t_{0}-t)}{2}.$

Using the equation $-\Delta W=\frac{1}{r^{3}}\Lambda(rW)$, and the Taylor
expansion

$\Lambda(w+\varepsilon)=\Lambda(w)+\Lambda^{\prime}(w)\varepsilon+\Lambda_{2}(w,\varepsilon)\varepsilon^{2},$

where

$\Lambda_{2}(w,\varepsilon)=\int_{0}^{1}\Lambda^{\prime\prime}(w+\theta\varepsilon)(1-\theta)d\theta,\quad|\Lambda_{2}(w,\varepsilon)|\lesssim|w|+|\varepsilon|,$

by (4.3). As a consequence, denoting $h_{t_{0}}(t)=h(2t_{0}-t)$,

(4.44) $\partial_{t}^{2}h_{+}-\Delta h_{+}\\\
=\frac{1}{2r^{3}}\Big{\\{}\Lambda^{\prime}(rW)(rh_{+})+\Lambda_{2}(rW,rh)(rh)^{2}+\Lambda_{2}(rW,rh_{t_{0}})(rh_{t_{0}})^{2}\Big{\\}},$

with the initial data $\vec{h}_{+\restriction t=0}=(u(t_{0})-W,0)$.

We have

(4.45) $|\Lambda^{\prime}(rW)|\lesssim
r^{2}W^{2},\quad|\Lambda_{2}(rW,rh)|\lesssim r|W|+r|h|.$

Let $\sigma\geq\rho+|t_{0}|$. We have

$r\geq\sigma+|t-t_{0}|\Longrightarrow r\geq\max(\rho+|t|,\rho+|2t_{0}-t|),$

and thus

(4.46)
$r\geq\sigma+|t-t_{0}|\Longrightarrow|h_{+}(t,r)|+|h_{t_{0}}(t,r)|+|h(t,r)|\lesssim\frac{M(\rho)}{r^{2}}.$

Combining (4.44), (4.45), (4.46) and the lower bound for the exterior energy
(Lemma 3.9), we obtain

(4.47)
$\left\|\pi_{\sigma}^{\bot}h_{+}(t_{0})\right\|_{\dot{H}^{1}(\sigma)}\\\
\lesssim\sum_{k=1,2,3}\left\|W\right\|_{(L^{3}L^{6})(\sigma+|t-t_{0}|)}^{3-k}\left(\int_{\mathbb{R}}\left(\int_{\sigma+|t-t_{0}|}^{\infty}\frac{M(\rho)^{6}}{r^{9}}dr\right)^{\frac{1}{2}}dt\right)^{\frac{k}{3}}.$

Let
$\varepsilon(\sigma)=\left\|W\right\|_{(L^{3}L^{6})(\sigma+|t|)}^{2}=\left\|W\right\|_{(L^{3}L^{6})(\sigma+|t-t_{0}|)}^{2}$.
Note that $\varepsilon(\sigma)$ is finite, independent of $t_{0}$, and goes to
$0$ as $\sigma$ goes to infinity. The inequality (4.47) implies

(4.48)
$\left\|h_{+}(t_{0},r)-\frac{\sigma^{2}}{r^{2}}h_{+}(t_{0},\sigma)\right\|_{\dot{H}^{1}_{r}(\sigma)}\leq\frac{C\varepsilon(\sigma)}{\sigma}M(\rho)+\frac{CM(\rho)^{3}}{\sigma^{3}},\quad\sigma\geq\rho+|t_{0}|.$

By the radial Sobolev inequality,

$\left|h_{+}(t_{0},2\sigma)-\frac{\sigma^{2}}{(2\sigma)^{2}}h_{+}(t_{0},\sigma)\right|\leq\frac{C\varepsilon(\sigma)}{\sigma^{2}}M(\rho)+\frac{CM(\rho)^{3}}{\sigma^{4}}.$

As a consequence

$\frac{1}{4}|h_{+}(t_{0},\sigma)|\leq\frac{C\varepsilon(\sigma)}{\sigma^{2}}M(\rho)+\frac{CM(\rho)^{3}}{\sigma^{4}}+|h_{+}(t_{0},2\sigma)|\\\
\leq\frac{C\varepsilon(\sigma)}{\sigma^{2}}M(\rho)+\frac{CM(\rho)^{3}}{\sigma^{4}}+\frac{M(2\sigma)}{4\sigma^{2}},$

where we have used (4.46) and the fact that $2\sigma\geq 2\rho+2|t_{0}|\geq
2\rho+|t_{0}|$, so that
$|h_{+}(t_{0},2\sigma)|\leq\frac{1}{(2\sigma)^{2}}M(2\sigma)$.

Going back to the definition of $h_{+}$, we deduce

$\sigma^{2}|u(t_{0},\sigma)-W(\sigma)|\leq
M(2\rho)+C\tilde{\varepsilon}(\rho)M(\rho),$

where
$\tilde{\varepsilon}(\rho)=\varepsilon(\rho)+\frac{M(\rho)^{2}}{\rho^{2}}$
tends to $0$ as $\rho$ tends to infinity. This holds for all
$t_{0}\in\mathbb{R}$ and $\sigma\geq\rho+|t_{0}|$. Hence

$M(\rho)\leq M(2\rho)+C\tilde{\varepsilon}(\rho)M(\rho).$

Taking $\rho$ large enough, we obtain (4.43), concluding the proof. ∎

##  5\. Rigidity theorem, part II: odd solutions

In this section, we conclude the proof of Proposition 4.5, and thus of the
rigidity theorem (Theorem 4.2) by the following proposition:

######  Proposition 5.1.

Let $R>0$ and $u$ a radial solution of (4.1) on $\\{r\geq R+|t|\\}$, such that

(5.1) $\displaystyle(u,\partial_{t}u)_{\restriction t=0}=(0,u_{1}),\quad
u_{1}\in L^{2}(R)$ (5.2)
$\displaystyle\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R+|t|}|\nabla_{t,x}u(t,x)|^{2}dx=0.$

Then

$\forall t\in\mathbb{R},\;\forall r>R+|t|,\quad u(t,r)=0.$

We first check that Proposition 4.5 (and thus Theorem 4.2) follows from
Propositions 4.6 and 5.1. Let $u$ be as in Proposition 4.5. Then by
Proposition 4.6, one of the properties (4.16) or (4.17) holds. We must prove
that on of the conclusions (4.13) or (4.14) of Proposition 4.5 hold. Property
(4.17) implies immediately (4.14). If (4.16) holds, we can assume, translating
in time and taking a larger $R$, that $u_{0}(0,r)=0$ for all $r>R$. Thus $u$
satisfies (after a time translation), the assumptions of Proposition 5.1, and
the conclusion of this proposition implies (4.13).

We will prove Proposition 5.1 in §5.3 after giving two preliminary results. In
§5.1, we prove a gain of regularity for a solution $u$ of (4.1) satisfying the
assumptions of this proposition. More precisely, we prove that for such a
solution, $(\partial_{t}u,\partial_{t}^{2}u)(t)$ is in
$\mathcal{H}(R^{\prime}+|t|)$ for all $t$ (where $R^{\prime}$ is large) and
that $\partial_{t}u$ is non-radiative, in the sense that

(5.3)
$\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R^{\prime}+|t|}|\nabla_{t,x}\partial_{t}u(t,x)|^{2}dx=0.$

In §5.2, we give an explicit, radial, smooth approximate solution $a$ of (4.1)
on $|x|>|t|$, which satisfies the non-radiative property (5.3) and whose
initial data is of the form$(0,a_{1}(r))$, where $a_{1}\notin L^{2}(R)$ for
$R>0$ (indeed $a_{1}(r)$ equals $\frac{1}{r^{2}(\log r)^{1/2}}$ up to a
multiplicative constant).

The proof of Proposition 5.1 in §5.3 consists in showing that the initial data
$u_{1}(r)$ of a solution as in Proposition 5.1 is of the same order as or
greater than $a_{1}(r)$ for large $r$, contradicting the fact that $u_{1}\in
L^{2}(R)$. This is done by carefully comparing $u_{1}$ with an appropriate
rescaling of $a_{1}$, using again the improved lower energy bound of Lemma
3.9.

###  5.1. Gain of regularity

In this subsection we prove:

######  Lemma 5.2.

Let $u$ be as in Proposition 5.1. Then there exists a large constant
$R^{\prime}\geq R$ such that

(5.4) $\displaystyle ru_{1}\in\dot{H}^{1}(R^{\prime})$ (5.5)
$\displaystyle\forall\rho\geq
R^{\prime},\quad\frac{1}{\rho}\|u_{1}\|_{\dot{H}^{1}(\rho)}\approx|u_{1}(\rho)|$

Furthermore, $(\partial_{t}u,\partial_{t}^{2}u)$ is the restriction to
$\\{r>R^{\prime}+|t|\\}$ of a $C^{0}(\mathbb{R},\dot{H}^{1}\times L^{2})$
function and

(5.6)
$\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R^{\prime}+|t|}|\nabla_{t,x}\partial_{t}u|^{2}dx=0.$

###### Proof of Lemma 5.2.

_Step 1: gain of decay._ We let, for $\rho\geq R+|T|$, $T\in\mathbb{R}$,

(5.7) $\eta(T,\rho)=\max_{0\leq t\leq T}\|\partial_{t}u(t)\|_{L^{2}(\rho)}.$

In this step, we prove that if $R^{\prime}$ is large enough and $\rho\geq
R^{\prime}+|T|$, then $r\nabla u(T)\in L^{2}(2\rho)$ and

(5.8) $\left\|r\nabla
u(T)\right\|_{L^{2}(2\rho)}\lesssim\sqrt{\int_{\rho}^{\infty}|u(T,r)|^{2}r^{3}dr}\lesssim
T\eta(T,\rho).$

Indeed, using that $u(t,r)=\int_{0}^{t}\partial_{t}u(\tau,r)d\tau$, we have,
for all $t\geq 0$ and $\rho\geq R+|t|$,

(5.9)
$\|u(t)\|_{L^{2}(\rho)}\leq\int_{0}^{t}\|\partial_{t}u(\tau)\|_{L^{2}(\rho)}d\tau\leq|t|\eta(t,\rho).$

By Lemma 4.7, for $\rho>R^{\prime}+|T|$, $R^{\prime}$ large,

(5.10) $|u(T,\rho)|\approx\frac{1}{\rho}\|\nabla u(T)\|_{L^{2}(\rho)}.$

Integrating, we obtain, for $\sigma>R^{\prime}+|T|$,

$\int_{\sigma}^{+\infty}|u(T,\rho)|^{2}\rho^{3}d\rho\approx\int_{\sigma}^{+\infty}\int_{\rho}^{+\infty}|\partial_{r}u(T,r)|^{2}r^{3}dr\rho\,d\rho,$

and thus, by Tonelli’s Theorem,

$\int_{\sigma}^{\infty}|u(T,\rho)|^{2}\rho^{3}d\rho\approx\int_{\sigma}^{+\infty}(\partial_{r}u(T,r)|^{2}(r^{5}-\sigma^{2}r^{3})dr.$

Combining with (5.9), we obtain, for $\sigma>R^{\prime}+|t|$,

$\int_{2\sigma}^{\infty}|\partial_{r}u(T,r)|^{2}r^{5}dr\lesssim\int_{\sigma}^{\infty}|u(T,r)|^{2}r^{3}dr\lesssim
T^{2}\eta(T,\sigma)^{2},$

which is exactly (5.8).

_Step 2. Gain of regularity._ In this step, we prove (5.4) and (5.5). By
(5.8), for $T\in[0,1]$, $\rho>R^{\prime}+|T|$,

(5.11)
$\int_{2\rho}^{\infty}r^{5}\left|\frac{1}{T}\partial_{r}u(T,r)\right|^{2}dr\lesssim\eta(T,\rho)^{2}\lesssim\eta(1,R^{\prime})^{2}.$

As a consequence, there exist $v_{1}\in L^{2}([2R^{\prime},\infty),r^{5}dr)$
and a sequence $\\{T_{n}\\}_{n}$ going to $0$ as $n$ goes to infinity such
that

(5.12)
$\frac{1}{T_{n}}\partial_{r}u(T_{n})\xrightharpoonup[n\to\infty]{}v_{1}\text{
weakly in }L^{2}\left([2R^{\prime},\infty),r^{5}dr\right).$

We have

$\frac{1}{T_{n}}u(T_{n},2\rho)=-\frac{1}{T_{n}}\int_{2\rho}^{\infty}\partial_{r}u(T_{n},r)dr,$

and thus, using the weak convergence of $\partial_{r}u(T_{n})$,

(5.13)
$\lim_{n\to\infty}\frac{1}{T_{n}}u(T_{n},2\rho)=-\int_{2\rho}^{\infty}v_{1}(r)dr,$

for all $\rho\geq R^{\prime}$. Next, we observe that by (5.11) and a radial
Sobolev inequality,

$\frac{1}{T_{n}^{2}}|u(T_{n},2\rho)|^{2}\lesssim\frac{1}{\rho^{4}}\eta(1,R^{\prime})^{2}$

for large $n$. This yields by the dominated convergence theorem that the
convergence (5.13) also holds locally in $L^{2}(R^{\prime},\infty)$. Since

$\lim_{t\to 0^{+}}\frac{1}{t}u(t)\to u_{1}\text{ in }L^{2}(2R),$

we deduce, by uniqueness of the limit that for $\rho>R^{\prime}$

(5.14) $u_{1}(2\rho)=-\int_{2\rho}^{\infty}v_{1}(r)dr,$

where $\int_{R^{\prime}}^{\infty}(v_{1}(r))^{2}r^{5}dr<\infty$. Hence (5.4)
(taking a larger $R^{\prime}$).

The estimate (5.5) follows from (5.10), dividing by $T$ and letting $T$ go to
$0$.

_Step 3. Proof that $\partial_{t}u$ is non-radiative._ In this step we prove
(5.6). We fix a small constant $\varepsilon_{0}>0$ to be specified. Taking a
larger $R^{\prime}$ if necessary, we can assume
$\|r\partial_{r}u_{1}\|_{L^{2}}+\|u_{1}\|_{H^{1}(R^{\prime})}\leq\varepsilon_{0}$.
We define $\tilde{u}_{1}$ by

$\tilde{u}_{1}(r)=u_{1}(R^{\prime})\text{ if
}r<R^{\prime},\quad\tilde{u}_{1}(r)=u_{1}(r)\text{ if }r\geq R^{\prime}.$

Using that by the radial Sobolev embedding

$|u_{1}(R^{\prime})|\lesssim\frac{1}{(R^{\prime})^{2}}\|r\partial_{r}u_{1}\|_{L^{2}(R^{\prime})}\lesssim\frac{\varepsilon_{0}}{(R^{\prime})^{2}},$

we obtain

(5.15) $\|\tilde{u}_{1}\|_{H^{1}}\lesssim\varepsilon_{0}.$

Let $\tilde{u}$ be the solution of (4.1) with initial data $(0,\tilde{u}_{1})$
at $t=0$. Since $(0,\tilde{u}_{1})\in(\dot{H}^{2}\cap\dot{H}^{1})\times
H^{1})$, a standard persistence of regularity argument yields

$\overrightarrow{\partial_{t}u}\in
C^{0}\left(\mathbb{R},(\dot{H}^{2}\cap\dot{H}^{1})\times H^{1}\right).$

We observe that $\partial_{t}\tilde{u}$ satisfies the equation

(5.16)
$\partial_{t}^{2}(\partial_{t}\tilde{u})-\Delta\partial_{t}\tilde{u}=\frac{1}{r^{2}}\Lambda^{\prime}(r\tilde{u})\partial_{t}\tilde{u},$

with initial data

(5.17) $\overrightarrow{\partial_{t}\tilde{u}}_{\restriction
t=0}=(\tilde{u}_{1},0)\in\dot{H}^{1}\times L^{2}.$

Noting that

$\frac{1}{r^{2}}\left|\Lambda^{\prime}(r\tilde{u})\right|\lesssim\tilde{u}^{2},\quad\left\|\tilde{u}^{2}\right\|_{(L^{3/2}L^{3})}=\left\|\tilde{u}\right\|_{(L^{3}L^{6})}^{2}\lesssim\varepsilon_{0},$

we deduce by Strichartz estimates, and taking $\varepsilon_{0}$ small enough,

(5.18)
$\|\partial_{t}\tilde{u}\|_{(L^{3}L^{6})}\lesssim\|\tilde{u}_{1}\|_{\dot{H}^{1}}.$

As a consequence,
$\frac{1}{r^{2}}\Lambda^{\prime}(r\tilde{u})\partial_{t}\tilde{u}\in(L^{1}L^{2})$.
By finite speed of propagation,

(5.19) $r>R+|t|\Longrightarrow u(t,r)=\tilde{u}(t,r).$

Thus the assumption (5.2) implies

$\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>R+|t|}|\nabla_{t,x}\tilde{u}(t,x)|^{2}dx=0.$

Combining with Proposition A.1 in the appendix, we obtain (5.6) for
$\tilde{u}$ and thus, by (5.19), for $u$. ∎

###  5.2. Approximate non-radiative solution

######  Lemma 5.3.

There exists a $C^{\infty}$ function $a(t,r)$ defined for $t\in\mathbb{R}$,
$r>2$ such that $\partial_{t}a\in C^{0}(\mathbb{R},\mathcal{H}(1))$ and

(5.20) $\displaystyle\forall t\in\mathbb{R},\;\forall r>2,\quad
a(t,r)=-a(-t,r)$ (5.21) $\displaystyle\forall r>2,\quad
a_{1}(r):=\partial_{t}a(0,r)=\frac{2}{\sqrt{3}r^{2}(\log r)^{1/2}}$ (5.22)
$\displaystyle\forall t\in\mathbb{R},\;\forall
r>\max(|t|,2),\quad|a(t,r)|+|t\partial_{t}a(t,r)|\lesssim\frac{|t|}{r^{2}(\log
r)^{1/2}}$ (5.23)
$\displaystyle\sum_{\pm}\lim_{t\to\pm\infty}\int_{|x|>|t|}|\nabla_{t,x}\partial_{t}a(t,x)|^{2}dx=0$

and $b=\partial_{t}^{2}a-\Delta a-\frac{1}{r^{3}}\Lambda(ra)$ satisfies

(5.24) $\forall t\in\mathbb{R},\;\forall
r>\max(|t|,2),\quad|b(t,r)|+|t\partial_{t}b(t,r)|\lesssim\frac{|t|}{r^{4}(\log
r)^{5/2}}.$

###### Proof.

Let

$a(t,r)=\frac{2t}{\sqrt{3}r^{2}(\log
r)^{1/2}}-\frac{t^{3}}{3\sqrt{3}r^{4}(\log r)^{3/2}}.$

It is easy to see that $a$ satisfies (5.20), (5.21), (5.22) and (5.23). Using
the formula

(5.25) $\Delta\left(\frac{1}{r^{\alpha}(\log
r)^{\beta}}\right)=\frac{1}{r^{\alpha+2}}\left(\frac{\alpha(\alpha-2)}{(\log
r)^{\beta}}+\frac{(2\alpha-2)\beta}{(\log
r)^{\beta+1}}+\frac{\beta(\beta+1)}{(\log r)^{\beta+2}}\right)$

we can prove that $b$ satisfies (5.24). Indeed by (4.3),

$\displaystyle\left|a^{3}-\frac{1}{r^{3}}\Lambda(ra)\right|\lesssim
r^{2}|a|^{5}\lesssim\frac{|t|}{r^{4}(\log r)^{5/2}},\quad r>\max(|t|,2),$
$\displaystyle\left|\partial_{t}\left(a^{3}-\frac{1}{r^{3}}\Lambda(ra)\right)\right|\lesssim
r^{2}a^{4}|\partial_{t}a|\lesssim\frac{1}{r^{4}(\log r)^{5/2}},\quad
r>\max(|t|,2),$

and $\partial_{t}^{2}a-\Delta a-a^{3}$ is a linear combination of:

$\displaystyle\frac{t}{r^{4}(\log r)^{5/2}},\quad\frac{t^{3}}{r^{6}(\log
r)^{5/2}},\quad\frac{t^{3}}{r^{6}(\log r)^{7/2}},\quad\frac{t^{5}}{r^{8}(\log
r)^{5/2}}$ $\displaystyle\frac{t^{7}}{r^{10}(\log
r)^{7/2}},\quad\frac{t^{9}}{r^{12}(\log r)^{9/2}}.$

The key point is the double (approximate) cancellation given by:

$\partial_{t}^{2}\left(\frac{t^{3}}{3\sqrt{3}r^{4}(\log
r)^{3/2}}\right)=-\Delta\left(\frac{2t}{\sqrt{3}r^{2}(\log
r)^{1/2}}\right)+l.o.t$

and

$\Delta\left(\frac{t^{3}}{3\sqrt{3}r^{4}(\log
r)^{3/2}}\right)=\left(\frac{2t}{\sqrt{3}r^{2}(\log
r)^{1/2}}\right)^{3}+l.o.t.,$

where $l.o.t.$ is dominated by $t/r^{4}(\log r)^{5/2}$ for $r\geq|t|$. The
nonlinearity appears in the last line: the approximate solution $a$ is not the
perturbation of a linear solution, but a fully nonlinear object. ∎

######  Remark 5.4.

An explicit computation shows that

$1\\!\\!1_{\\{|x|>1+|t|\\}}a\in L^{3}L^{6}.$

However $\partial_{t}a(0)$ barely fails to be in $L^{2}$. This fact will be
crucial in the proof of Proposition 5.1.

###  5.3. Proof of the rigidity result

We are now in position to prove Proposition 5.1.

_Step 1. Preliminaries._ We argue by contradiction, assuming that $u_{1}$ is
not identically $0$ for $r>R$. By (5.5) in Lemma 5.2, $u_{1}$ does not change
sign, and we can assume that $u_{1}(r)>0$ for large $r$. Furthermore, by (5.4)
in Lemma 5.2, and the radial Sobolev inequality,

(5.26) $\lim_{r\to\infty}r^{2}u_{1}(r)=0.$

We fix $\rho\gg 1$ and choose $\lambda=\lambda(\rho)$ such that
$u_{1}(\rho)=\partial_{t}a_{\lambda}(0,\rho)$, where $a$ is the non-radiative
approximate solution given by Lemma 5.3 and

$a_{\lambda}(t,r)=\lambda a(\lambda t,\lambda r).$

In other words,

(5.27) $u_{1}(\rho)=\frac{2}{\sqrt{3}\rho^{2}\log(\rho\lambda)^{1/2}}.$

Note that the equation (5.27) always has a unique solution
$\lambda(\rho)>1/\rho$, and that by (5.26) and (5.27),

(5.28) $\lim_{\rho\to\infty}\rho\lambda(\rho)=\infty.$

We also note that by (5.22),

(5.29)
$r>\max(|t|,2/\lambda)\Longrightarrow|a_{\lambda}(t,r)|+|t\partial_{t}a_{\lambda}(t,r)|\lesssim\frac{|t|}{r^{2}\log^{1/2}(r\lambda)}.$

We have

(5.30) $\partial_{t}^{2}a_{\lambda}-\Delta
a_{\lambda}-\frac{1}{r^{3}}\Lambda(ra_{\lambda})=b_{[\lambda]},$

where $b_{[\lambda]}(t,r)=\lambda^{3}b(\lambda t,\lambda r)$ satisfies, by
(5.24),

(5.31)
$r>\max(|t|,2/\lambda)\Longrightarrow\left|b_{[\lambda]}(t,r)\right|+\left|t\partial_{t}b_{[\lambda]}(t,r)\right|\lesssim\frac{|t|}{r^{4}\log(\lambda
r)^{5/2}}.$

Let $w=\partial_{t}a_{\lambda}-\partial_{t}u$, so that $w$ satisfies the
equation

(5.32) $\partial_{t}^{2}w-\Delta
w=\frac{1}{r^{2}}\Lambda^{\prime}(ra_{\lambda})\partial_{t}a_{\lambda}-\frac{1}{r^{2}}\Lambda^{\prime}(ru)\partial_{t}u+\partial_{t}b_{[\lambda]}$

with initial data $w(0)=(\partial_{t}a_{\lambda}(0)-u_{1},0)$. By explicit
computation, using (5.31),

(5.33)
$\|\partial_{t}b_{[\lambda]}\|_{(L^{1}L^{2})(\rho)}\lesssim\frac{1}{\rho(\log(\lambda\rho))^{5/2}}.$

_Step 2. Channels of energy._ Fix a small $\varepsilon_{0}>0$, to be specified
later. In this step we prove that if $\rho$ is chosen large enough,

(5.34)
$\left\|u_{1}-\partial_{t}a_{\lambda}(0)\right\|_{\dot{H}^{1}(\rho)}\lesssim\frac{1}{\rho(\log(\lambda\rho))^{5/2}}.$

By the equation (5.32), Strichartz inequalities and finite speed of
propagation,

(5.35)
$\|w\|_{(L^{2}L^{8})(\rho)}+\|w\|_{(L^{3}L^{6})(\rho)}+\sup_{t\in\mathbb{R}}\left\|\vec{w}(t)\right\|_{\mathcal{H}(\rho+|t|)}\\\
\lesssim\|w(0)\|_{\dot{H}^{1}(\rho)}+\left\|\frac{1}{r^{2}}\Lambda^{\prime}(ra_{\lambda})\partial_{t}a_{\lambda}-\frac{1}{r^{2}}\Lambda^{\prime}(ru)\partial_{t}u+\partial_{t}b_{[\lambda]}\right\|_{(L^{1}L^{2})(\rho)}.$

By the exterior energy lower bound of Lemma 3.9 (which also applies to linear
solutions outside a wave cone by finite speed of propagation)

(5.36)
$\|\pi_{\rho}^{\bot}w(0)\|_{\dot{H}^{1}(\rho)}\lesssim\left\|\frac{1}{r^{2}}\Lambda^{\prime}(ra_{\lambda})\partial_{t}a_{\lambda}-\frac{1}{r^{2}}\Lambda^{\prime}(ru)\partial_{t}u+\partial_{t}b_{[\lambda]}\right\|_{(L^{1}L^{2})(\rho)}.$

Since by the choice of $\lambda$, $w(0,\rho)=0$, we obtain