# Stability of the Double-Cusp Spacetimes and long-time geometrizations

###### Abstract.

Since the early years of General Relativity, understanding the long-time
behavior of the cosmological solutions of Einstein’s vacuum equations has been
a fundamental yet challenging task. Solutions with global symmetries, or
perturbations thereof, have been extensively studied and are reasonably
understood. On the other hand, thanks to the work of Fischer-Moncrief and M.
Anderson, it is known that there is a tight relation between the future
evolution of solutions and the Thurston decomposition of the spatial
3-manifold. Consequently, cosmological spacetimes developing a future
asymptotic symmetry should represent only a negligible part of a much larger
yet unexplored solution landscape. In this work, we revisit a program
initiated by the second named author, aimed at constructing a new type of
cosmological solution first posed by M. Anderson, where (at the right scale)
two hyperbolic manifolds with a cusp separate from each other through a thin
torus neck. Specifically, we prove that the so-called double-cusp solution,
which models the torus neck, is stable under
${\mathbb{S}}^{1}\times{\mathbb{S}}^{1}$ \- symmetry-preserving perturbations.
The proof, which has interest on its own, reduces to proving the stability of
a geodesic segment as a wave map into the hyperbolic plane and partially
relates to the work of Sideris on wave maps and the work of Ringström on the
future asymptotics of Gowdy spacetimes.

Alejandro Bellati

<EMAIL_ADDRESS>
Martín Reiris

<EMAIL_ADDRESS>
Centro de Matemática, Universidad de la República,

Montevideo, Uruguay.

##  1\. Introduction

Since the early years of General Relativity, understanding the long-time
behavior of the cosmological solutions of Einstein’s equations has been a
fundamental yet quite challenging task. Solutions with spatial symmetries,
like the spatially homogeneous Bianchi models or the Gowdy
$\mathbb{T}^{2}$-symmetric spacetimes, have been extensively studied over the
decades and are reasonably well understood [8], [5], [9]. All these models are
very valuable and provide explicit examples of future dynamics but fall short
when the goal is to describe the full set of possible future behaviors. In
this work, we revisit a program initiated by the second named author, aimed at
constructing a new type of cosmological solution first posed by M. Anderson
with a qualitative behavior that is pretty different from any other model
known. As we will explain below, such a solution would provide strong support
to some ideas developed by Fischer-Moncrief and Anderson relating
fundamentally the topology of the Cauchy 3-hypersurfaces to the dynamics of
the cosmological solutions [3], [4], [1], [2].

Motivated by certain considerations on the Thurston geometrization conjecture,
Anderson posed in [2] (see paper’s bottom) the problem of finding a
cosmological solution where coarsely speaking, two hyperbolic 3-manifolds with
a cusp111A cusp is a 3-manifold $(-\infty,0]\times\mathbb{T}^{2}$ with a
hyperbolic metric of the form $g_{H}=dx^{2}+e^{2x}g_{T}$, with $g_{T}$ flat
$x-$independent on $\mathbb{T}^{2}$. Cusps are discussed later in the article.
separate from each other through a thin torus neck. In Anderson’s picture,
hypersurfaces $\Sigma_{k}$ of mean curvature $k\in(-\infty,0)$ evolve in the
expanding direction $k\uparrow 0$, but the geometry at each time $k$ is scaled
so that the mean curvature of $\Sigma_{k}$ is $-3$. Under this scaling, the
two hyperbolic pieces with their corresponding cusp should emerge over time,
separating from each other along an increasingly thin torus neck that develops
asymptotically a $\mathbb{T}^{2}$-symmetry. Figure 1 schematizes that
behavior. This spacetime would comprise a new and non-trivial example of a
cosmological solution of the vacuum Einstein equations whose spatial geometry
(at the mentioned scale) evolves towards the Thurston decomposition of its
Cauchy hypersurface. Furthermore, for this solution, the Fischer-Moncrief’s
reduced volume would decay towards its topological lower bound given by
$(-\sigma/6)^{3/2}$, where $\sigma$ is the Yamabe invariant of the $\Sigma$’s.

The double-cusps solutions were introduced in [6] and are explicit
$\mathbb{T}^{2}={\mathbb{S}}^{1}\times{\mathbb{S}}^{1}$-symmetric solutions on
$\mathbb{R}\times\mathbb{R}\times\mathbb{T}^{2}$ tailored to model the
evolution of the torus neck. As we will see, they enjoy all the required
global and asymptotic properties and are therefore a crucial piece. But before
attempting to study the combined evolution of the torus neck and the two
hyperbolic manifolds with a cusp, it becomes necessary to prove that the
double cusps are future-stable and provide sharp decaying estimates. In this
article, we do that but for perturbations preserving the
$\mathbb{T}^{2}$-symmetry. Quite interestingly, this problem reduces to
proving the stability of a parametrized geodesic segment (that models the
double-cusp) as a wave map from a (flat) 3-dimensional spacetime into the
hyperbolic plane. The stability problem of geodesics as wave maps was studied
by Sideris in [10], but the problem considered in that work is different from
ours. Also, the future evolution of $\mathbb{T}^{3}$-Gowdy spacetimes was
studied by Ringström in [7], through a wave map that, of course, has the same
origin as ours. In that case, any solution defines a loop in hyperbolic space,
while here, it defines a curve with the same ends as the geodesic segment.

Torus neck
---
Hyperbolic piece
---
Hyperbolic piece
---
Figure 1. Expected behavior of the type of solution posed in [2].

In section 2 we introduce the double cusp spacetimes and describe their global
properties. This analysis is not relevant for the technical part of the paper,
sections 4 to 6, but helps to understand the geometric motivation of the
article and to point out certain subtleties that appear when we discuss, in
section 3, the statements of the main results and the stability of the double
cusp as spacetimes.

##  2\. Double-cusps and their global properties

The double cusp spacetimes have metrics of the form [6],

$g=e^{2a}(-dt^{2}+dx^{2})+R(e^{2W}+q^{2}e^{-2W})d\theta_{1}^{2}-2Rqe^{-2W}d\theta_{1}d\theta_{2}+Re^{-2W}d\theta_{2}^{2},$
(1)

over the manifold
$\mathbb{R}_{t}\times\mathbb{R}_{x}\times{\mathbb{S}}^{1}_{\theta_{1}}\times{\mathbb{S}}^{1}_{\theta_{2}}$,
where $a$, $R$, and $q$ depend only on $t$ and $x$. Cauchy hypersurfaces, for
instance, those with $t$ constant, are diffeomorphic to
$\mathbb{R}\times\mathbb{T}^{2}$ and are thus “torus necks”. Metrics of this
form are similar to the $\mathbb{T}^{3}$-Gowdy’s metric, [5], but differ from
them in that $R$ is not taken as a coordinate and that $x$ is not periodic.
The double-cusps are non-stationary spacetimes. We will explicitly present the
forms of $a,R$, and $q$ later below. We will analyze their global geometry,
explaining how their geometry behaves along the CMC foliation (hypersurfaces
with $t$ constant are not CMC). However, before doing that and to motivate how
these solutions arise, we first present the equations for $R$, $W$, $q$, and
$a$. These equations, derived from the Einstein equations, are,

$\displaystyle R_{xx}-R_{tt}=0,$ (2) $\displaystyle
W_{tt}-W_{xx}+\frac{R_{t}}{R}W_{t}-\frac{R_{x}}{R}W_{x}+\frac{(q_{t}^{2}-q_{x}^{2})}{2}e^{-4W}=0,$
(3) $\displaystyle
q_{tt}-q_{xx}+\frac{R_{t}}{R}q_{t}-\frac{R_{x}}{R}q_{x}-4q_{t}W_{t}+4q_{x}W_{x}=0,$
(4) $\displaystyle
a_{tt}-a_{xx}+\tfrac{R_{x}^{2}-R_{t}^{2}}{4R^{2}}+W_{t}^{2}-W_{x}^{2}+\tfrac{1}{4}(q_{t}^{2}-q_{x}^{2})e^{-4W}=0,$
(5)

and,

$\displaystyle
a_{t}\frac{R_{t}}{R}+a_{x}\frac{R_{x}}{R}+\frac{1}{4}\left(\frac{R_{x}^{2}}{R^{2}}+\frac{R_{t}^{2}}{R^{2}}\right)-\frac{R_{xx}}{R}-(W_{x}^{2}+W_{t}^{2})-\frac{1}{4}e^{-4W}(q_{x}^{2}+q_{t}^{2})=0,$
(6) $\displaystyle
a_{x}\frac{R_{t}}{R}+a_{t}\frac{R_{x}}{R}-\frac{R_{tx}}{R}+\frac{R_{x}R_{t}}{2R^{2}}+2W_{t}W_{x}-\frac{1}{2}e^{-4W}q_{x}q_{t}=0.$
(7)

The equations (2), (3), (4) and (5) are the dynamical equations for $R$, $W$,
$q$ and $a$, and (6) and (7) are the constraint equations. The dynamical
equation for $R$ decouples from all the others, and the dynamical equations
for $W$ and $q$ decouple from that of $a$. In certain cases, one can solve
globally for $a_{x}$ and $a_{t}$ from (6)-(7) and then simply perform line
integrals to find $a$. In this case, $a$ is determined from $R,W$, and $q$ up
to an integration constant.

It is crucial but also well known that the equations (3) and (4) are wave map
equations into the hyperbolic plane. This is seen as follows. Think of the
hyperbolic plane $\mathbb{H}$ as
$\mathbb{R}^{2}=\mathbb{R}_{x}\times\mathbb{R}_{y}$ endowed with the metric
$h=4dy^{2}+e^{4y}dx^{2}$. On the other hand consider the manifold
$\mathbb{R}_{t}\times\mathbb{R}_{x}\times{\mathbb{S}}^{1}_{\phi}$ endowed with
the metric $k=4e^{4t}(-dt^{2}+dx^{2})+R^{2}(t,x)d\phi^{2}$ and denote this
Riemannian manifold as $\mathbb{K}$. Then $W$ and $q$ satisfy equations (3)
and (4) if and only if the map $\chi:\mathbb{K}\longrightarrow\mathbb{H}$
given by,

$\chi(t,x,\phi)=(q(t,x),-W(t,x)),$ (8)

is a wave map between the two manifolds. Another way of expressing this is
that equations (3) and (4) are the Euler-Lagrange equations of the action,

$\displaystyle
S=\int\partial_{l}\chi^{i}\partial_{m}\chi^{j}h_{ij}k^{lm}\,dV_{k}=2\pi\int
R(4(W_{x}^{2}+W_{t}^{2})+(q_{x}^{2}+q_{t}^{2})e^{-4W})dtdx.$ (9)

Let us now see the explicit form of the double cusps. First, for all the
double-cusp solutions, one takes $R(t,x)=R_{0}e^{2t}\cosh 2x$ with $R_{0}$ a
constant, which of course solves (2). Second, one requires $W$ and $q$ to be
$t$-independent, i.e. $W=W(x)$ and $q=q(x)$. The Euler-Lagrange equations for
such particular data are equivalent to the Euler-Lagrange equations of the
action,

$F=\int_{\mathbb{R}}|\gamma^{\prime}|^{2}\cosh(2x)dx$ (10)

where $\gamma(x)=\chi(x)$, and whose solutions are well known to be
parametrized geodesic segments of the hyperbolic plane. When the geodesic
segment is vertical and thus has $q$ constant, we say that the double cusp is
polarized. Their explicit form is,

$\displaystyle R=R_{0}e^{2t}\cosh(2x),$ (11) $\displaystyle
W=W_{1}+W_{0}\arctan(e^{2x}),$ (12) $\displaystyle q=q_{0},$ (13)
$\displaystyle
a=a_{0}-\left(\frac{1}{2}+\frac{W_{0}^{2}}{2}\right)\frac{1}{2}\ln(\cosh(2x))+\left(\frac{3}{2}+\frac{W_{0}^{2}}{2}\right)t,$
(14)

with $R_{0}>0,W_{0}\neq 0,W_{1}$, $q_{0}$ and $a_{0}$ constants. The non-
polarized double-cusps are created by transforming polarized ones by an
isometry of the hyperbolic plane (see Figure 2), and the explicit expression
won’t be particularly relevant. From now on, double cusp solutions will be
denoted by $R_{b},W_{b},q_{b}$ and $a_{b}$, where ‘b’ stands for ‘background’.

Formally speaking, the stability problem that we face amounts to the stability
of $R_{b}$, $W_{b}$, $q_{b}$ and $a_{b}$ as particular solutions of a system
of partial differential equations, and in fact this is pretty much the
viewpoint that we take. We will study first the wave equation for $R$, then
the wave map equation for $(W,q)$, and finally, we will study $a$, which will
be determined entirely from them. As it turns out, to control the
perturbations of $R$ and of $(W,q)$, we will use some natural norms that may
seem very suitable and standard from a PDE point of view but that may not
guarantee the stability of $a$, and thus of the spacetime, even for
perturbations of $R$, $W$ and $q$ arbitrarily small with respect to them. In
the next section, we will see that this apparently conflicting point can be
solved using finer norms to measure the smallness of the initial data for
$R,W$, and $q$ over the Cauchy surface $t=0$. The double cusp spacetimes are
stable for small perturbations in the sense of these finer norms, even though
the evolution of $R,W$, and $q$ is controlled with more coarse ones. All these
subtleties have their origin in the very nature of the coordinates $(t,x)$ and
the nature of the Cauchy hypersurface $\\{t=0\\}$, (where we are perturbing
the initial data). In the rest of this section, we review the global
properties of the double cusps. As said, this information will not play a role
when studying the stability of $R$ and of $(W,q)$, which will be treated as a
standard PDE problem, but it will help to understand the discussion about the
stability of $a$ and therefore of the double cusp as a spacetime.

$q$
---
$e^{2W}$
---
Polarized
---
Double Cusp
Non-polarized
---
Double Cusp
$\varphi\in\text{Isom}(\mathbb{H}^{2})$
---
Perturbation
---
Perturbation
---
Figure 2. In blue, two double cusps are represented. A perturbation is
represented in red, as the curve $x\mapsto\chi(t,x)$ for a fixed $t$. As time
evolves, this curve will move. The figure also illustrates how an arbitrary
double cusp can be seen as a polarized double cusp.

A main property of the double cusps is that at each of their two ends, one can
define spacetime coordinates $(t^{\prime},x^{\prime})$ and
$(t^{\prime\prime},x^{\prime\prime})$ where one can observe the spatial scaled
metrics converge towards hyperbolic cusps. This is one of the main properties
making double cusps adequate to model the necks of the solutions posed in [2].
To explain all that, we begin recalling certain notions on hyperbolic
manifolds and flat cone spacetimes. If $(\mathbb{M},g_{H})$ is a hyperbolic
manifold, then $\mathbb{R}_{\tau}\times\mathbb{M}$ endowed with the metric
$g=-d\tau^{2}+\tau^{2}g_{H}$ is a flat spacetime (hence a solution of the
Einstein equations) called a flat cone. The mean curvature of the hypersurface
$\tau=\tau_{0}$ is $k_{0}=-3/\tau_{0}$. Therefore, when the spacetime metric
$g$ is scaled as
$(k_{0}/3)^{2}g=\tau_{0}^{-2}g=d(\tau/\tau_{0})^{2}+(\tau/\tau_{0})^{2}g_{H}$,
then the mean curvature of the hypersurface $\tau=\tau_{0}$ becomes $-3$ and
the induced 3-metric $g_{H}$. This is called CMC scaling and can be made at
any CMC hypersurface $\Sigma_{k}$ of mean curvature $k$ inside a spacetime.
Hyperbolic manifolds of finite volume can be non-compact. When this is so, the
manifold has a finite number of truncated “cusps” of the form
$\mathbb{C}=(-\infty,x_{0}]_{x}\times\mathbb{T}^{2}$ with
$g_{H}=dx^{2}+e^{2x}g_{T}$, where $g_{T}$ is an $x-$independent flat metric on
$\mathbb{T}^{2}$. A cusp spacetime is a flat cone with
$\mathbb{M}=\mathbb{R}_{x}\times\mathbb{T}^{2}$ and
$g_{H}=dx^{2}+e^{2x}g_{T}$, with $g_{T}$ $x-$independent and flat. As
mentioned a few lines above, the two ends of double cusps are asymptotic to
cusp spacetimes as $t\rightarrow\infty$. This behavior is not observed in the
coordinates $t,x$ but rather in new coordinates linearly related to them. On a
double cusp solution, consider the new coordinates,

$\displaystyle
t^{\prime}=-\bigg{(}\frac{1}{2}+\frac{W_{0}^{2}}{2}\bigg{)}x+\bigg{(}\frac{3}{2}+\frac{W_{0}^{2}}{2}\bigg{)}t,$
(15) $\displaystyle
x^{\prime}=-\bigg{(}\frac{1}{2}+\frac{W_{0}^{2}}{2}\bigg{)}t+\bigg{(}\frac{3}{2}+\frac{W_{0}^{2}}{2}\bigg{)}x.$
(16)

These coordinates are plotted in Figure 3. When we fix $x^{\prime}$ and
increase $t^{\prime}$, or when we fix $t^{\prime}$ and increase $x^{\prime}$,
both $x$ and $t$ increase. In this sense, these new coordinates are adapted to
the ‘right’ end. It is on these coordinates that the double-cusp metric
approaches a cusp spacetime metric ($\tau=e^{t^{\prime}}$). This is easy to
show and has been done in [6] in detail. On the ‘left’ end, one can also
define coordinates $x^{\prime\prime},t^{\prime\prime}$ where the evolution
displays a similar behavior. The whole picture is represented in Figure 3.
This phenomenon is best observed globally along the CMC foliation. Indeed,
double cusps admit a global CMC foliation of Cauchy hypersurfaces $\Sigma_{k}$
covering the whole spacetime, where the mean curvature $k$ ranges on
$(-\infty,0)$, [6]. More specifically, there is a Cauchy hypersurface
$\Sigma_{-3}$ of mean curvature $-3$ defined by a graph $t=s(x)$ and any other
leaf of the CMC foliation is obtained translating in $t$ the graph of $s(x)$.
Furthermore, the graph of $s(x)$ approaches a $t^{\prime}={\rm const}$ and
$t^{\prime\prime}={\rm const}$ line as $x$ goes to $\infty$ and $-\infty$
respectively. Hence, one can simultaneously observe the convergence to the
flat cones on the right and left ends by following the CMC foliation
$\Sigma_{k}$, $k\uparrow 0$. If one performs CMC scalings so that the mean
curvature of each leaf $\Sigma_{k}$ becomes $-3$, then a convergence-collapse
picture emerges. Roughly speaking, the scaled metric over the $\Sigma_{k}$
converges to a hyperbolic cusp metric on each of the two ends (one must follow
the $x^{\prime}=$ const and $x^{\prime\prime}=$ const directions), while the
central part collapses its volume while keeping its curvature bounded so that
the narrow necks appear to look like thin and long lines. Figure 3 depicts
this phenomenon.

$t^{\prime}$
---
$t^{\prime\prime}$
---
$t=s(x)$
---

cusp
---

cusp
---

$x$
---
$t$
---
$x^{\prime}$
---
$x^{\prime\prime}$
---
Figure 3. Double cusp’s behavior over the CMC foliation after CMC scalings.

There is a relevant but standard change of variables $(t,x)\rightarrow(R,V)$
given by,

$R=R_{0}e^{2t}\cosh(2x),\quad V=R_{0}e^{2t}\sinh(2x),$ (17)

where one can better observe certain important global facts. The spatial
coordinate $V$ is the conjugate of $R$ and satisfies the wave equation too.
With this change, we have,

$-dt^{2}+dx^{2}=4R^{-2}(-dR^{2}+dV^{2})$ (18)

so light rays in the plane $(t,x)$ are mapped into light rays in the $(R,V)$
plane. The spacetime region is the region $R>|V|$ and the slice $t=0$ maps
into the hyperbola $R^{2}-V^{2}=R_{0}^{2}$, $R>0$. This is displayed in Figure
4. The whole picture proves that $t=0$ is a Cauchy surface for the double cusp
spacetime. It can be seen that the spacetime cannot be smoothly extended to
the past boundary $R=|V|$, which is singular. The CMC curve $t=s(x)$ is mapped
into a curve that is also asymptotic to the lines $R=V,V>0$ and $R=-V,V<0$ but
is not a hyperbola, of course. Though these two Cauchy hypersurfaces look
similar, there is a clear distinction between them: the former approaches the
past boundary faster than the latter and in such a way that the Lorentzian
distance to the past boundary tends to zero for the former but remains bounded
away from zero for the latter. The significant consequence of this is that
when considering the Cauchy problem for the Einstein equations (2)-(7) over
the hypersurface $t=0$, perturbations of the initial data that do not fall to
zero sufficiently fast as $x\rightarrow\infty$ may cause large distortions of
the spacetime. This point will reappear when discussing the double cusp’s
stability statement as a spacetime.

Based on the discussion above, it may seem that making the analysis on the
coordinates $(t,x)$ is inconvenient. However, the great advantage of it is
that it displays $(W_{b},q_{b})$ as a time-independent parametrized geodesic
segment in the hyperbolic plane. This proves to be very useful. Given the
motivation of the stability problem we have discussed, the natural gauge to
work would be the CMC gauge. This choice, however, entangles a number of
difficulties, for instance, finding and dealing with the right shift, which
makes it more complex. We plan to do that analysis elsewhere.

$R$
---
$V$
---
$CMC$
---
$t=const$
---
Figure 4. Different Cauchy surfaces seen in the $R-V$ coordinates.

##  3\. Statements of the main results

To state the main results, we must first introduce a few norms and spaces.
Let,

$\displaystyle m_{0}(t):=\lVert R-R_{b}\rVert_{C^{0}}+\lVert
R_{t}-{R_{b}}_{t}\rVert_{C^{0}},$ (19) $\displaystyle m_{k}(t):=\lVert
R-R_{b}\rVert_{C^{k}}+\lVert R_{t}-{R_{b}}_{t}\rVert_{C^{k-1}},\ {\rm for}\
k\geq 1,$ (20)

where $R,R_{b},R_{t}$ and $R_{bt}$ are considered at time $t$. The quantity
$m_{0}(t)$ measures the $C^{0}$ norm between $R$ and $R_{b}$ and between
$R_{t}$ and $R_{bt}$. The quantity $m_{k}(t)$ instead measures, at time $t$,
the $C^{k}$ norm between $R$ and $R_{b}$ and the $C^{k-1}$ norm between
$R_{t}$ and $R_{bt}$. We also define for $k\geq 1$,

$\displaystyle\widetilde{\mathcal{M}}_{k}(t):=\lVert
W-W_{b}\rVert_{\widetilde{H}_{k}}+\lVert\partial_{t}(W-W_{b})\rVert_{\widetilde{H}_{k-1}}+\lVert
q\rVert_{\widetilde{H}_{k}}+\lVert\partial_{t}q\rVert_{\widetilde{H}_{k-1}},$
(21)

where again, the functions inside the norms are considered at time $t$. Here
$\widetilde{H}_{k}$ is the completion of the space of smooth and compactly
supported functions, $C^{\infty}_{c}(\mathbb{R})$, with respect to the norm,

$\lVert
f\rVert_{\widetilde{H}_{k}}^{2}:=\sum_{i=0}^{k}\int_{\mathbb{R}}(f^{(k)}(x))^{2}\cosh(2x)\
dx.$ (22)

This is a weighted Sobolev space with weight $\cosh(2x)$. Lastly, let
$C_{0}^{k}(\mathbb{R})$ be the space of $C^{k}$ functions $f$ such that for
every $i\leq k$, $f^{(i)}(x)\to 0$ when $\left\lvert x\right\rvert\to+\infty$.

We will first discuss some basic statements about the perturbations of $R$.
Regarding $(W,q)$, we will state only the stability results for the polarized
double cusp $(q_{b}=0)$ as an isometry of the hyperbolic plane relates
polarized and non-polarized double cusps. We will present two separate
stability statements, one for polarized perturbations (i.e. with $q=0$) and
one for general non-polarized perturbations. The former is stronger than the
latter. After stating these results, we will discuss what they imply for the
stability of $a$ and double cusp spacetimes.

######  Lemma 3.1.

For every solution $R$ to equation (2), if $m_{0}(0)<2R_{0}/3$, then
$R(t,x)>0$ when $t\geq 0$. Moreover,

$\lVert R-R_{b}\rVert_{\infty}(t)\leq(t+1)m_{0}(0)\ \ \text{for }t\geq 0,$

and for all multi-index $\alpha\neq(0,0)$, there is a constant $C$ such that,
for $t\geq 0$,

$\lVert\partial^{\alpha}(R-R_{b})\rVert_{\infty}(t)\leq m_{\alpha}(0),$

and

$\bigg{\lVert}\partial^{\alpha}\left(\frac{R_{t}}{R}-\frac{{R_{b}}_{t}}{{R_{b}}}\right)\bigg{\rVert}_{\infty},\bigg{\lVert}\partial^{\alpha}\left(\frac{R_{x}}{R}-\frac{{R_{b}}_{x}}{R_{b}}\right)\bigg{\rVert}_{\infty}(t)\leq
C\frac{t+1}{e^{2t}\cosh(2x)}m_{\left\lvert\alpha\right\rvert+1}(0).$

The first statement below is for polarized perturbations, i.e., $q=0$. It
shows the exponential decay of $\tilde{\mathcal{M}}_{k}$.

######  Theorem 6.3.

Let $k\geq 3$. Let $(R,W)$ be a $C^{2}$ solution of the system eqs. 2 and 3
with $q=0$. Suppose also that
$(R-R_{b}(0,\cdot),\partial_{t}(R-R_{b})(0,\cdot))\in C_{0}^{k}\times
C_{0}^{k-1}$, $m_{0}(0)<2R_{0}/3$ and
$((W-W_{b})(0,\cdot),\partial_{t}(W-W_{b})(0,\cdot))\in\widetilde{H}^{k}\times\widetilde{H}^{k-1}$.
Then, the solution is defined for every $t\geq 0$ and

$\widetilde{\mathcal{M}}_{k}(t)\leq
Ce^{-t}(t+1)\left(\widetilde{\mathcal{M}}_{k}(0)+m_{k}(0)\right).$ (23)

Moreover, the constant $C$ depends only on an upper bound on $m_{k}(0)$ and
$k$.

Observe that except for the condition $m_{0}(0)\leq 2R_{0}/3$, which is
somehow unavoidable, no particular smallness condition is required. In this
sense, this proves that the double cusp is somehow a global attractor among
polarized data.

For non-polarized perturbations ($q\neq 0$) a similar estimate is obtained but
for $\widetilde{\mathcal{M}}_{3}(t)$. A smallness hypothesis must also be
provided.

######  Theorem 6.4.

There is a number $\delta>0$ such that, any $C^{2}$ solution of the system
eqs. 2, 3 and 4, $(R,W,q)$, with initial data satisfying
$(R-R_{b}(0,\cdot),\partial_{t}(R-R_{b})(0,\cdot))\in C_{0}^{k}\times
C_{0}^{k-1}$, $m_{0}(0)<2R_{0}/3$,
$((W-W_{b})(0,\cdot),\partial_{t}(W-W_{b})(0,\cdot))\in\widetilde{H}^{k}\times\widetilde{H}^{k-1}$,
$(q(0,\cdot),\partial_{t}q(0,\cdot))\in\widetilde{H}^{k}\times\widetilde{H}^{k-1}$,
and $m_{3}(0)<\delta$, $\widetilde{\mathcal{M}}_{3}(0)<\delta$, is defined for
all $t\geq 0$ and

$\widetilde{\mathcal{M}}_{3}(t)\leq
Ce^{-t}(t+1)(\widetilde{\mathcal{M}}_{3}(0)+m_{3}(0)).$ (24)

Moreover, the constant $C$ depends only on an upper bound on $m_{3}(0)$ and on
$\widetilde{\mathcal{M}}_{3}(0)$.

This last theorem implies that the red curve in Figure 2 approaches the blue
one exponentially fast as $t\to+\infty$.

We now discuss the consequences of the last two results on $a$ and the
stability of the double cusp spacetimes.

A basic consequence of the previous results is that if the initial data for
$(R,W,q)$ satisfies the hypotheses of either Theorem 6.3 or Theorem 6.4, and
$(R,W,q,a)$ satisfies eqs. 2, 3, 4 and 5, then $a$ is defined for all $t\geq
0$. This happens because the equation (5) is a wave equation with a source
defined for all $t\geq 0$. Furthermore, if the constraint equations are
satisfied at $t=0$, then, by standard arguments, they are satisfied for every
$t\geq 0$. This, in turn, proves that the system, eqs. 2, 3, 4, 5, 6 and 7,
gives a Cauchy development of the perturbed initial data on
$[0,+\infty)\times\mathbb{R}\times\mathbb{T}^{2}$. The argument is standard as
the system (2)-(7) is equivalent to the Einstein equations, and hence we omit
it.

######  Corollary 3.2.

Consider initial data to the Einstein’s equations on
$\mathbb{R}\times\mathbb{T}^{2}$, induced by eq. 1 on $t=0$. Suppose that
$R,R_{t},W,W_{t},q_{t}$ and $q_{t}$, at $t=0$, satisfy the hypotheses of
either Theorem 6.3 or Theorem 6.4. Then, the solutions of the system given by
eqs. 2, 3, 4 and 5 are defined for all $t\geq 0$, and the metric, eq. 1, given
by these functions on $[0,+\infty)\times\mathbb{R}\times\mathbb{T}^{2}$ gives
a Cauchy development of the data.

Regarding the stability of $a$, note that the equation (5) implies

$(a-a_{b})_{tt}-(a-a_{b})_{xx}=F(R,W,q)-F(R_{b},W_{b},q_{b}).$ (25)

where the source $F(R,W,q)-F(R_{b},W_{b},q_{b})$ is controlled by eq. 24.
Using this and D’Alembert, we deduce that the contribution of the source to
$\|a(t,\cdot)-a_{b}(t,\cdot)\|_{C^{0}}$ is controlled by
$\tilde{\mathcal{M}}_{2}(0)+m_{2}(0)$. The contribution of the homogeneous
solution is controlled by the initial data norm
$\|a(0,\cdot)-a_{b}(0,\cdot)\|_{C^{0}}+\|a_{t}(0,\cdot)-a_{b}(0,\cdot)\|_{L^{1}}$.
This immediately leads to the following result.

######  Theorem 3.3.

For any $\epsilon>0$ there exist $\delta>0$ such that if the initial data for
$(R,W,q,a)$ satisfies $\widetilde{\mathcal{M}}_{3}(0)+m_{3}(0)\leq\delta$ and
$\|a(0,\cdot)-a_{b}(0,\cdot)\|_{C^{0}}+\|a_{t}(0,\cdot)-{a_{b}}_{t}(0,\cdot)\|_{L^{1}}<\delta$,
then $\|a(t,\cdot)-a_{b}(t,\cdot)\|_{C^{0}}\leq\epsilon\ \forall t\geq 0$.

The spacetime perturbations of the previous theorem are future geodesically
complete. Once again, the functional space for the perturbations of $a$ is
chosen from the point of view of the PDEs. The natural question is if there is
a non-trivial perturbation on these spaces subjected to the constraint
equations. Indeed, this is the case, for instance, if we require stricter
norms for $(R,W,q)$ at $t=0$. By doing this, $a-a_{b}$ naturally belongs to
the above spaces for each $t\geq 0$. An example of these norms could be
$\lVert
f\rVert_{\widetilde{H}_{p,k}}^{2}:=\sum_{i=0}^{k}\int_{\mathbb{R}}(f^{(k)}(x))^{2}\cosh^{p}(2x)\
dx$ for $W$ an $q$, and ${m_{l,k}}(f):=m_{k}(\cosh^{l}(2x)f(x))$ for $R$. With
these norms, using $p=2$ instead of $p=1$, and $l=1$ instead of $l=0$, one can
see that small perturbations in this new sense imply that $a_{t}$ and $a_{x}$
can be solved out in eqs. 6 and 7. Furthermore,
$\|a(t,\cdot)-a_{b}(t,\cdot)\|_{C^{0}}+\|a_{t}(t,\cdot)-{a_{b}}_{t}(t,\cdot)\|_{L^{1}}$
is finite for each $t\geq 0$, and arbitrarily small by reducing the values of
$m_{l,1}(R-R_{b})(0)$ and $\widetilde{\mathcal{M}}_{2,3}(0)$, where

$\displaystyle\widetilde{\mathcal{M}}_{p,k}(t):=\lVert
W-W_{b}\rVert_{\widetilde{H}_{p,k}}+\lVert\partial_{t}(W-W_{b})\rVert_{\widetilde{H}_{p,k-1}}+\lVert
q\rVert_{\widetilde{H}_{p,k}}+\lVert\partial_{t}q\rVert_{\widetilde{H}_{p,k-1}}.$

The proof of the Theorems 6.3, 6.4, are done first for compactly supported
perturbations, i.e., solutions with initial data differing only on a compact
set from that of the background. This is done in section 4 and 5. The latter
section is the central part of the paper. Finally, in section 6, we give a
rather general argument to extend this simplified versions to larger
functional spaces, proving theorem 6.3 and 6.4.

##  4\. Compactly supported polarized perturbations

In this section, we shall address compactly supported perturbations with
$q=0$, namely, a solution $(R,W)$ of the system given by eqs. 2 and 3 whose
initial data differs from that of the background only on a compact set.
Although we will not use the results found in this section explicitly in the
non-polarized case, the computations used in the non-polarized case rely on
the ones developed here. Furthermore, here the computations are more clear and
yield stronger results.

At first, the solutions are not defined for all $t\geq 0$. All the estimates
concerning will be for $t\geq 0$ in the interval of existence. Furthermore, we
also assume $m_{0}(0)<2R_{0}/3$ to ensure $R>0$ for $t\geq 0$. Throughout the
work, depending on the kind of computations, we will use $\partial_{x}f$ or
$f_{x}$. Also $\partial^{\alpha}f$ with $\alpha$ a multi-index. We say that a
function $f(t,x)$ is of locally x-compact support if for any interval
$[T_{1},T_{2}]$ there is a compact subset $K\subset\mathbb{R}$ such that
$f(t,x)=0$ if $t\in[T_{1},T_{2}]$ and $x\notin K$. Recall the notation
$m_{k}$. We summarize some elementary properties in the following lemma.

######  Lemma 4.1.

If $(R,W)$ is a solution such that $R-R_{b}$ and $W-W_{b}$ at $t=0$ are
compactly supported and $m_{0}(0)<\frac{2}{3}R_{0}$ then, for $t\geq 0$, and
where defined, $R>0$ and $(R-R_{b},W-W_{b})$ is of locally x-compact support.

###### Proof.

The use of D’Alembert’s formula gives $R\geq R_{b}-m_{0}(0)(t+1),t\geq 0$.
With this, it can be seen that both claims about $R$ are true. For $W$, use a
finite speed propagation argument. ∎

###  4.1. The basic energy inequality

In order to obtain the asymptotic stability of our solution, we need some
useful energy. Consider the change of variable $z=R^{1/2}(W-W_{b})$. In this
new variable, the equation for $W$ becomes

$z_{tt}-z_{xx}+zG=g\ \ \ \textrm{with
}G=\frac{R_{t}^{2}-R_{x}^{2}}{4R^{2}}\textrm{ and
}g=R^{1/2}\left(\frac{R_{x}}{R}-\frac{{R_{b}}_{x}}{R_{b}}\right){W_{b}}_{x}.$
(26)

Now $z$ is of locally x-compact support, then it makes sense to define the
energy

$E:=\frac{1}{2}\int_{\mathbb{R}}z_{t}^{2}+z_{x}^{2}+z^{2}G_{b}\ dx,$

where $G_{b}=\tfrac{1}{\cosh^{2}(2x)}$. We need some estimates involving
$R-R_{b}$ so that we can control $E$. These properties are given in the
following lemma.

######  Lemma 4.2 (Coefficients estimates).

For all $t\geq 0$, we have the following estimates:

$\displaystyle\lVert R-R_{b}\rVert_{\infty}(t)\leq(t+1)m_{0}(0),$ (27)
$\displaystyle\lVert R_{x}-{R_{b}}_{x}\rVert_{\infty}(t),\lVert
R_{t}-{R_{b}}_{t}\rVert_{\infty}(t)\leq m_{1}(0).$ (28)

Furthermore, as we are assuming $m_{0}(0)\leq\frac{2}{3}R_{0}$ we also have
that $R\sim R_{b}$, i.e, there is a constant $d>0$ such that

$\frac{1}{d}\leq\bigg{\lVert}\frac{{R_{b}}}{R}\bigg{\rVert}_{\infty}(t)\leq
d,$ (29)

and this, in turn, implies the existence of a constant $C>0$ such that

$\displaystyle\bigg{\lVert}\frac{R_{t}}{R}-\frac{{R_{b}}_{t}}{R}\bigg{\rVert}_{\infty}(t),\bigg{\lVert}\frac{R_{x}}{R}-\frac{{R_{b}}_{x}}{R_{b}}\bigg{\rVert}_{\infty}(t)\leq\frac{C(t+1)}{e^{2t}\cosh(2x)}m_{1}(0),$
(30) $\displaystyle\lVert
G_{b}-G\rVert_{\infty}(t)\leq\frac{C(t+1)}{e^{2t}\cosh(2x)}m_{1}(0),$ (31)
$\displaystyle\lVert
g\rVert_{\infty}(t)\leq\frac{C(t+1)}{e^{t}\cosh^{3/2}(2x)}m_{1}(0).$ (32)

###### Proof.

Use D’Alembert to derive the first estimates. The other follows from direct
computation using these. ∎

######  Proposition 4.3 (The basic energy inequality).

Let $(R,z)$ be a solution such that $R-R_{b}$, $z$ and $z_{t}$ are compactly
supported at $t=0$, with $m_{0}(0)<2R_{0}/3$. Then there is a constant $C$
such that, where defined,

$E^{1/2}(t)\leq C(E^{1/2}(0)+m_{1}(0)),\ \text{with }\ t\geq 0.$ (33)

Furthermore, the constant $C$ depends only on a bound on $m_{1}(0)$.

###### Proof.

The function $z$ is of locally x-compact support as $W-W_{b}$ is. For this
reason, we can derive under the integral and use integration by parts. Then,
we use the equation to obtain

$\displaystyle\dot{E}$
$\displaystyle=\int_{\mathbb{R}}z_{t}z_{tt}+z_{x}z_{xt}+zz_{t}G_{b}\
dx=\int_{\mathbb{R}}z_{t}(z_{tt}-z_{xx}+zG_{b})\ dx$
$\displaystyle=\int_{\mathbb{R}}z_{t}z(G_{b}-G)\ dx+\int_{\mathbb{R}}z_{t}g\
dx$ $\displaystyle\leq\underbrace{\sqrt{\int_{\mathbb{R}}z_{t}^{2}\
dx}\sqrt{\int_{\mathbb{R}}z^{2}(G_{b}-G)^{2}\ dx}}_{\text{First
term}}+\underbrace{\sqrt{\int_{\mathbb{R}}z_{t}^{2}\
dx}\sqrt{\int_{\mathbb{R}}g^{2}\ dx}}_{\text{Second term}}.$

In the first term, the first integral is bounded by $E^{1/2}$. For the second
integral, use the estimate (31) to find out that this term is less or equal to

$C(t+1)m_{1}(0)e^{-2t}E^{1/2}\sqrt{\int_{\mathbb{R}}\frac{z^{2}}{\cosh^{2}(2x)}\
dx},$

but the integrand in the last integral is just $4z^{2}G_{b}$, so in the end,
our first term is less or equal to

$C(t+1)e^{-2t}m_{1}(0)E,$

where we have adjusted the constant $C$. Let us control the second term. This
term is a product of two integrals. The first one is less or equal to
$E^{1/2}$. For the second integral, the use of the estimate (32) yields

$\int_{\mathbb{R}}g^{2}\ dx\leq C(t+1)e^{-t}m_{1}(0).$

Using all these observations, we get

$\displaystyle\dot{E}$ $\displaystyle\leq
C(t+1)e^{-2t}m_{1}(0)E+C(t+1)e^{-t}m_{1}(0)E^{1/2}$ (34) $\displaystyle\leq
C(t+1)e^{-2t}m_{1}(0)E+C(t+1)e^{-t}E+C(t+1)e^{-t}m_{1}^{2}(0)$
$\displaystyle\leq C(t+1)e^{-t}E+C(t+1)e^{-t}m_{1}^{2}(0),$

where in the last inequality, the constant depends on $m_{1}(0)$. This
inequality implies the thesis. ∎

Note that since $R\sim R_{b}=R_{0}e^{2t}\cosh(2x)$, this implies exponential
decay in our original variables plus decay as $x$ goes to infinity. However,
we do not have future existence for all $t\geq 0$. This matter is the
objective of the following subsection.

###  4.2. Higher order energies

Let $\alpha$ be a multi-index $\alpha=(m,n)$. The first letter will refer to
time derivatives, and the second will refer to spatial derivatives. Let us
denote

$E^{\alpha}(t):=\frac{1}{2}\int_{\mathbb{R}}(\partial^{\alpha}z)_{t}^{2}+(\partial^{\alpha}z)_{x}^{2}+(\partial^{\alpha}z)^{2}G_{b}\
dx.$

The most important of these energies are the ones with $\alpha=(0,m)$ because
they are related to $H^{k}$ norms of $z$ and $\partial_{t}z$. This fact is
important since, at $t=0$, they involve only the initial data. One could be
tempted to do an argument similar to the one made in the derivation of eq. 33,
but using $E^{(0,m)}$. If we do this, we will discover that the growth of
$E^{(0,m)}$ is bounded by a polynomial of degree $m$. This result is not bad,
but we found another, longer way to obtain better estimates. First, we derive
estimates for $E^{(n,0)}$, and then we pass our estimates to $E^{(0,m)}$ using
the equations satisfied by $z$ and its derivatives. Now, more coefficient
estimates are needed.

######  Lemma 4.4.

Suppose $m_{0}(0)<2R_{0}/3$, then:

  1. a)

For all multi-index $\alpha\neq(0,0)$ we have

$\lVert\partial^{\alpha}(R-R_{b})\rVert_{\infty}(t)\leq m_{\alpha}(0)\ \ t\geq
0.$ (35)

  2. b)

For every multi-index $\alpha$, $\partial^{\alpha}R/R$ and
$\partial^{\alpha}R^{1/2}/R^{1/2}$ are bounded to the future. Moreover, the
bound depends only on a bound on $m_{\left\lvert\alpha\right\rvert}(0)$.

  3. c)

For every multi-index $\alpha$, $\partial^{\alpha}R_{t}/R$,
$\partial^{\alpha}R_{x}/R$, $\partial^{\alpha}(R_{t}/R)$ and
$\partial^{\alpha}(R_{x}/R)$ are bounded to the future. Moreover, the bound
depends only on a bound on $m_{\left\lvert\alpha\right\rvert+1}(0)$.

  4. d)

For all multi-index $\alpha$ there is a constant $C$ such that

$\bigg{\lVert}\partial^{\alpha}\left(\frac{R_{t}}{R}-\frac{{R_{b}}_{t}}{{R_{b}}}\right)\bigg{\rVert}_{\infty},\bigg{\lVert}\partial^{\alpha}\left(\frac{R_{x}}{R}-\frac{{R_{b}}_{x}}{R_{b}}\right)\bigg{\rVert}_{\infty}(t)\leq
C\frac{t+1}{e^{2t}\cosh(2x)}m_{\left\lvert\alpha\right\rvert+1}(0).$ (36)

  5. e)

Estimates for $G_{b}-G$: for every multi-index $\alpha$ there is a constant
$C$ such that

$\lVert\partial^{\alpha}(G-G_{b})\rVert_{\infty}(t)\leq
C\frac{(t+1)}{e^{2t}\cosh(2x)}m_{\left\lvert\alpha\right\rvert+1}(0).$ (37)

Moreover, the constant depends only on a bound on
$m_{\left\lvert\alpha\right\rvert+1}(0)$.

  6. f)

For every multi-index $\alpha$ there is a constant $C$ such that

$\left\lvert\partial^{\alpha}G\right\rvert\leq\frac{C}{\cosh{2x}}=2C\sqrt{G_{b}}.$
(38)

Furthermore, the constant $C$ depends on $\alpha$ and on a bound on
$m_{\left\lvert\alpha\right\rvert+1}(0)$.

  7. g)

For every multi-index $\alpha$ there is a constant $C$ such that
$\left\lvert\partial^{\alpha}{W_{b}}_{x}\right\rvert\leq C/\cosh(2x)$. In
addition, if $\alpha$ is not purely spatial then
$\partial^{\alpha}{W_{b}}_{x}=0$.

  8. h)

Estimates for $g$: For every multi-index $\alpha$ there is a constant $C$ such
that

$\lVert\partial^{\alpha}g\rVert_{\infty}\leq
C\frac{t+1}{e^{2t}\cosh^{3/2}(2x)}m_{\left\lvert\alpha\right\rvert+1}(0).$
(39)

Additionally, the constant $C$ just depends on a bound on
$m_{\left\lvert\alpha\right\rvert+1}(0)$.

###### Proof.

Item a) is a direct consequence of D’Alembert’s Formula. Item b) and c) are
just computations using a) and the fact that $R\sim R_{b}$. For these
computations, it is often helpful to use recursion formulas, such as

$\partial^{\alpha}\left(\frac{R_{t}}{R}\right)=\frac{\partial^{\alpha}R_{t}}{R}-\sum_{0\leq\beta<\alpha}\binom{\alpha}{\beta}\frac{\partial^{\alpha-\beta}R\partial^{\beta}(\tfrac{R_{t}}{R})}{R},$

or

$\left\lvert\frac{\partial^{\alpha}(R^{1/2})}{R^{1/2}}\right\rvert\leq\left\lvert\frac{\partial^{\alpha}R}{2R}\right\rvert+\sum_{0<\alpha<\beta}\binom{\alpha}{\beta}\left\lvert\frac{\partial^{\alpha-\beta}R^{1/2}\partial^{\beta}(R^{1/2})}{2R}\right\rvert.$

Item d) is proved similarly, and item e) is a consequence of a)-d). Item f)
follows from item e). Item g) follows from direct computation, and finally, g)
is a consequence of d), g), and b). ∎

######  Proposition 4.5 (Energy estimates for time derivatives).

Let $(R,z)$ be a solution such that $R-R_{b}$, $z$ and $z_{t}$ are compactly
supported at $t=0$, with $m_{0}(0)<2R_{0}/3$, and let $\alpha=(m,0)$. Then,
there is a constant $C$ such that, where defined,

$\sqrt{E^{(m,0)}}(t)\leq
C(\sqrt{E^{(m,0)}}(0)+\ldots+\sqrt{E^{(1,0)}}(0)+\sqrt{E}(0)+m_{\left\lvert\alpha\right\rvert+1}(0)),\text{
for }t\geq 0.$

Furthermore, the constant $C$ just depends on $m$ and on a bound on
$m_{\left\lvert\alpha\right\rvert+1}(0)$.

###### Proof.

We have already proved the case $\alpha=0$ in the Proposition 4.3. Let us
proceed by induction. Deriving the equation with respect to $m>0$ yields:

$(\partial^{m}_{t}z)_{tt}-(\partial^{m}_{t}z)_{xx}+(\partial^{m}_{t}z)G+\sum_{i=0}^{m-1}\binom{\alpha}{\beta}\underbrace{\partial_{t}^{i}z}_{\begin{subarray}{c}\text{in
}E^{(i,0)}\\\ \text{already
controlled}\end{subarray}}\underbrace{\partial_{t}^{m-i}(G-G_{b})}_{\lx@cref{creftype~refnum}{eq:Galpha_est}}=\underbrace{\partial^{m}_{t}g}_{\textit{\lx@cref{creftype~refnum}{eq:galpha_est}}}.$

Here, we have used that $\partial_{t}^{i}G_{b}=0$ in the last term before the
equal sign. Now, as we did in the proof of Proposition 4.3), we can derive
$E^{(m,0)}$ with respect to time, integrate by parts, use the equation and
control each of the terms appearing. The new terms are the ones that have a
curly bracket in the above equation. Below these brackets, it is specified how
to control these terms. ∎

The following lemma goes in the direction of proving the desired estimates for
$E^{(0,n)}$. Here we use the notation $\left\lvert\alpha\right\rvert=m+n$ for
$\alpha=(m,n)$.

######  Lemma 4.6.

In the above assumptions, suppose $n\geq 1$ and $m\geq 0$, then there is a
constant $C$ such that, where defined,

$\sqrt{E^{(m,n)}}(t)\leq
C\sqrt{E^{(m+1,n-1)}}(t)+C\sum_{0\leq\beta\leq(m,n-1)}\sqrt{E^{\beta}}(t)+Cm_{\left\lvert\alpha\right\rvert}(0)\
\text{for }t\geq 0,$

where $\alpha=(m,n)$. Now if $m\geq 1$ and $n\geq 0$ then there is a constant
$C$ such that, where defined,

$\sqrt{E^{(m,n)}}(t)\leq
C\sqrt{E^{(m-1,n+1)}}(t)+C\sum_{0\leq\beta\leq(m-1,n)}\sqrt{E^{\beta}}(t)+Cm_{\left\lvert\alpha\right\rvert}(0)\
\text{for }t\geq 0.$

The constants just depend on a bound on
$m_{\left\lvert\alpha\right\rvert+1}(0)$, and on $m$ and $n$.

###### Proof.

$\displaystyle E^{(m,n)}(t)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}\underbrace{[(\partial_{t}^{m}\partial_{x}^{n}z)_{t}]^{2}}_{\text{It
is in
}E^{(m+1,n-1)}}+[(\partial_{t}^{m}\partial_{x}^{n}z)_{x}]^{2}+\underbrace{(\partial_{t}^{m}\partial_{x}^{n}z)^{2}}_{\text{It
is in }E^{(m,n-1)}}\underbrace{G_{b}}_{\leq 1}\ dx$ $\displaystyle\leq
E^{(m+1,n-1)}(t)+E^{(m,n-1)}(t)+\frac{1}{2}\int_{\mathbb{R}}[(\partial_{t}^{m}\partial_{n}^{x}z)_{x}]^{2}\
dx.$

To bound the last term, derive the eq. 26 $m-$times with respect to time and
$n-1-$times with respect to $x$. The derived equation gives us the following
estimate,

$\int_{\mathbb{R}}(\partial^{m}_{t}\partial_{x}^{n+1}z)^{2}\leq
C\underbrace{\int_{\mathbb{R}}(\partial_{t}^{m+2}\partial_{x}^{n-1}z)^{2}}_{\text{It
is in
}E^{(m+1,n-1)}}+C\sum_{\scriptscriptstyle\beta=(0,0)}^{\scriptscriptstyle(m,n-1)}\underbrace{\int_{\mathbb{R}}(\partial^{\beta}z\partial^{\alpha-\beta}G)^{2}}_{\leq
CE^{\beta}}+\underbrace{\int_{\mathbb{R}}(\partial_{t}^{m}\partial_{x}^{n-1}g)^{2}}_{\leq
Cm_{\left\lvert\alpha\right\rvert}^{2}(0)}.$

Using this and putting a square root, we arrive at the first claim stated in
the lemma. Keeping track of the constant, we see that the assertion about its
dependence is true. For the second inequality stated in the lemma, the same
reasoning works. ∎

######  Corollary 4.7.

In the above assumptions, given $n$, there is a constant $C$ such that, where
defined,

$\sqrt{E^{(0,n)}}(t)\leq
C(\sqrt{E^{(n,0)}}(t)+\ldots+\sqrt{E^{(1,0)}}(t)+\sqrt{E}(t))+Cm_{n}(0),$ (40)

and

$\sqrt{E^{(n,0)}}(t)\leq
C(\sqrt{E^{(0,n)}}(t)+\ldots+\sqrt{E^{(0,1)}}(t)+\sqrt{E}(t))+Cm_{n}(0).$ (41)

The constant depends only on a bound on $m_{n+1}(0)$ and on $n,m$.

###### Proof.

Use the previous lemma and induction. ∎

######  Proposition 4.8 (Energy estimates for spatial derivatives).

Let $(R,z)$ be a solution such that $R-R_{b}$, $z$ and $z_{t}$ are compactly
supported at $t=0$, with $m_{0}(0)<2R_{0}/3$. Then there is a constant $C$
such that, where defined,

$\sqrt{E^{(0,n)}}(t)\leq
C(\sqrt{E^{(0,n)}}(0)+\ldots+\sqrt{E^{(0,1)}}(0)+\sqrt{E}(0))+Cm_{n+1}(0).$
(42)

The constant $C$ just depends on $n$ and on a bound on $m_{n+1}(0)$.

###### Proof.

First, note that we have already proved the corresponding theorem for
$E^{(n,0)}$ in the Proposition 4.5. We first use the eq. 40, then the
Proposition 4.5, and finally the eq. 41, this time evaluated at $t=0$. ∎

###  4.3. Relation with $H^{k}$-norms and returning to our original variables.

Let us define

$\mathcal{M}_{k}[z](t):=\lVert
z\rVert_{H^{k}}(t)+\lVert\partial_{t}z\rVert_{H^{k-1}}(t),\ \ \
\mathcal{A}(t):=\frac{1}{2}\int_{\mathbb{R}}z^{2}\ dx.$

Note that $\dot{\mathcal{A}}\leq E^{1/2}\mathcal{A}^{1/2}\leq
C(E^{1/2}(0)+m_{1}(0))\mathcal{A}^{1/2}$. Therefore

$\mathcal{A}^{1/2}(t)\leq\mathcal{A}^{1/2}(0)+C(E^{1/2}(0)+m_{1}(0))t.$

######  Theorem 4.9 ($H^{k}$-evolution).

Let $(R,z)$ be a solution such that $R-R_{b}$, $z$ and $z_{t}$ are compactly
supported at $t=0$, with $m_{0}(0)<2R_{0}/3$, and let $k>0$. Then there is a
constant $C$ such that

$\mathcal{M}_{k}(t)\leq C\left(\mathcal{M}_{k}(0)+m_{k}(0)\right)(t+1),\
\text{for }t\geq 0.$ (43)

The constant $C$ here depends on a bound on $m_{k}(0)$ and on $k$.
Furthermore, the solution $(R,z)$ is defined for all $t\geq 0$.

###### Proof.

Just note that $\mathcal{M}_{k}$ involves $(\partial^{i}_{x}z)^{2}$ with
$i=0,...,k$ and $(\partial^{j}_{x}\partial_{t}z)^{2}$ with $j=0,...,k-1$, and
that all of these terms appear in the t-derivative or the x-derivative term of
one of the followings quantities:
$\mathcal{A},E,E^{(0,1)},\ldots,E^{(0,k-1)}$. Using this, with the above
computation for $\mathcal{A}$ and eq. 42 yields eq. 43. Lastly, if we call
$T_{+}$ the supreme of times $T$ such that the solution is defined on $[0,T]$,
then $\mathcal{M}_{k}(t)$ is bounded on $[0,T_{+}]$. By a continuity lemma, it
follows that $T_{+}=+\infty$. ∎

This result allows us to prove the existence of solutions with certain decay
in our original variables.

######  Lemma 4.10 (Passage lemma).

Given $R$ a solution of eq. 2 with $m_{0}(0)<2R_{0}/3$, and let $k>0$. Then
for every solution $z$ to eq. 26, of locally x-compact support, we have that
$W-W_{b}$ is of locally x-compact support and

$\frac{1}{C}e^{-t}\lVert z\rVert_{H^{k}}(t)\leq\lVert
W-W_{b}\rVert_{\widetilde{H}_{k}}(t)\leq Ce^{-t}\lVert z\rVert_{H^{k}}(t)$

where $\widetilde{H}^{k}$ is the weighted Sobolev space, with $\cosh(2x)\ dx$
as the weight. Here $C$ is a constant that depends on a bound on $m_{k}(0)$
and $k$.

###### Proof.

The proof is just computations and the use of the estimates derived in Lemma
4.4. ∎

In the following corollary, we summarize what we have obtained so far. Recall
the notation

$\widetilde{\mathcal{M}}_{k}(t):=\lVert
W-W_{b}\rVert_{\widetilde{H}_{k}}(t)+\lVert\partial_{t}(W-W_{b})\rVert_{\widetilde{H}_{k-1}}(t).$
(44)

######  Corollary 4.11.

Given $(R,W)$ a smooth solution of the system eqs. 2 and 3 with $q=0$. Suppose
also that the initial data $R,R_{t},W,W_{t}$ differs from that of the
background in a compact set and that $m_{0}(0)<2R_{0}/3$. Then, the solution
is defined for every $t\geq 0$ and

$\widetilde{\mathcal{M}}_{k}(t)\leq
Ce^{-t}(t+1)\left(\widetilde{\mathcal{M}}_{k}(0)+m_{k}(0)\right).$

Moreover, the constant $C$ depends only on a bound on $m_{k}(0)$ and $k$.

##  5\. Compactly supported non-polarized perturbations

The objective of this section is to study non-polarized perturbations, i.e.,
we allow our perturbations to have $q\neq 0$. As before, we first study
compactly supported perturbations.

###  5.1. Bounded distance from the background

Again, let us consider a solution to the system eqs. 2, 3 and 4. We ask this
solution to be smooth and to differ initially from the background only on a
compact set. Due to a finite speed propagation argument, $R-R_{b}$, $W-W_{b}$,
and $q-q_{b}$, where defined, are of locally x-compact support. These
properties allow us to integrate by parts and derive under the integral.

In what follows, we will use a special connection. First of all, recall
$\chi:\mathbb{K}\to\mathbb{H}$. Note that its differential, $D\chi$, can be
regarded as a section of the fiber-bundle $\mathbb{K}\to
T^{*}\mathbb{K}\otimes\chi^{-1}T\mathbb{H}$, where $\chi^{-1}T\mathbb{H}$ is
the pullback bundle over $\mathbb{K}$. In $\chi^{-1}T\mathbb{H}$, we have the
pullback connection and the pullback metric, and in the bundle
$T^{*}\mathbb{K}$ we have the metric $k$ and its connection. The product
connection on $T^{*}\mathbb{K}\otimes\chi^{-1}T\mathbb{H}$ preserves the
product metric. Using this connection the equations eqs. 3 and 4 are just
$\nabla^{a}\partial_{a}\chi^{i}=0$, i.e, $tr(\nabla D\chi)=0$. Consider the
energy-momentum tensor given by
$T_{ab}:=\partial_{a}\chi^{i}\partial_{b}\chi^{j}h_{ij}-\frac{1}{2}g_{ab}g^{\alpha\beta}h_{ij}\partial_{\alpha}\chi^{i}\partial_{\beta}\chi^{j}$.
A direct computation shows that
$\nabla^{a}T_{ab}=\partial_{b}\chi^{i}h_{ij}\nabla^{a}\partial_{a}\chi^{j}$.
The last term, with $j=1,2$, are exactly the equations eq. 3 and eq. 4.
Accordingly, the solutions to these equations have divergence null energy-
momentum tensor. Recall the metric
$k=4e^{4t}(-dt^{2}+dx^{2})+R^{2}(t,x)d\psi^{2}$, and consider the the vector
field $N=\partial_{t}/2e^{2t}$, the slice $S_{t}=\\{t\\}\times\mathbb{R}\times
S^{1}$ and the vector field given by $Y^{a}=T^{a}_{\ b}X^{b}$. This vector
field is the dual vector to the one form $T(\cdot,X)$. Using Stokes222As the
solutions we are treating now differ from the background only on a compact
set, we only have these two terms contributing to the flux. we have:

$\int_{[0,t_{0}]\times\mathbb{R}\times S^{1}}\nabla^{a}(T_{ab}X^{b})\
dVol=-\int_{S_{0}}X^{a}T_{ab}X^{b}\ dS_{0}+\int_{S_{t}}X^{a}T_{ab}X^{b}\
dS_{t}$

Now $\nabla^{a}(T_{ab}X^{b})=T_{ab}{\nabla^{a}X^{b}}$. A computation shows
that

$\displaystyle\nabla^{a}X^{b}$
$\displaystyle=\frac{1}{4e^{6t}}(\partial_{x})^{a}(\partial_{x})^{b}+R^{-2}\frac{R_{t}}{R}\frac{1}{2e^{2t}}(\partial_{\psi})^{a}(\partial_{\psi})^{b},$
$\displaystyle T_{00}$
$\displaystyle=T_{11}=\frac{1}{2}(\lVert\partial_{t}\chi\rVert_{h}^{2}+\lVert\partial_{x}\chi\rVert_{h}^{2}),$
$\displaystyle T_{22}$
$\displaystyle=\frac{R^{2}}{2}\frac{\lVert\partial_{t}\chi\rVert_{h}^{2}-\lVert\partial_{x}\chi\rVert_{h}^{2}}{4e^{4t}}.$

Using this

$\displaystyle\partial_{t}\int_{\mathbb{R}}\frac{1}{2}(\lVert\partial_{t}\chi\rVert_{h}^{2}+\lVert\partial_{x}\chi\rVert_{h}^{2})\frac{R}{e^{2t}}dx$
$\displaystyle=-2\int_{\mathbb{R}}\frac{\lVert\partial_{t}\chi\rVert_{h}^{2}}{2}\left(1+\dfrac{R_{t}}{2R}\right)\frac{R}{e^{2t}}\
dx$
$\displaystyle-2\int_{\mathbb{R}}\frac{\lVert\partial_{x}\chi\rVert_{h}^{2}}{2}\left(1-\dfrac{R_{t}}{2R}\right)\frac{R}{e^{2t}}\
dx.$

Let us momentarily call the left integral $\mathcal{E}$ (without
$\partial_{t}$). Since $m_{0}(0)<\frac{2}{3}R_{0}$, eq. 36 holds. Using this
estimate yields $\partial_{t}\mathcal{E}\leq
C(t+1)(m_{1}(0)+2)e^{-2t}\mathcal{E}$. Integrating we obtain
$\mathcal{E}(t)\leq\exp(C(m_{1}(0)+2))\mathcal{E}(0)$ for the future. Now

$\displaystyle\left\lvert
W-W_{b}\right\rvert\leq\int_{-\infty}^{x}\left\lvert(W-W_{b})_{x}\right\rvert\
dx$ $\displaystyle\leq C\sqrt{\int_{\mathbb{R}}(W-W_{b})_{x}^{2}\cosh(2x)\
dx}$ $\displaystyle\leq C\mathcal{E}^{1/2}(t)\leq C\
exp(C(m_{1}(0)+2))\mathcal{E}^{1/2}(0),$

and analogously for $q-q_{b}=q$. This result allows us to conclude that the
distance between the background solution and a solution $(W,q)$, whose initial
data is a compactly supported perturbation of that of the background, is
bounded. Geometrically, this means that if we start with a solution that is a
compactly supported perturbation of a geodesic, then this perturbation evolves
at a bounded distance. Moreover, this distance depends on a bound on
$m_{1}(0)$ and on $\widetilde{\mathcal{M}}_{1}(0)$, where now

$\widetilde{\mathcal{M}}_{1}[W-W_{b},q](t)=(\lVert
W-W_{b}\rVert_{\widetilde{H}^{1}}+\lVert(W-W_{b})_{t}\rVert_{\widetilde{H}^{0}}+\lVert
q\rVert_{\widetilde{H}^{1}}+\lVert q_{t}\rVert_{\widetilde{H}^{0}})(t).$

We define $\widetilde{\mathcal{M}}_{k}[\cdot,\cdot]$ in the same manner but
using $k$ and $k-1$ norms. Again, whenever we say
$\widetilde{\mathcal{M}}_{k}$ without explicit mention of the functions, the
reader should interpret $\widetilde{\mathcal{M}}_{k}[W-W_{b},q]$.

###  5.2. The basic energy inequality

In this section, we derive the basic energy inequality from which exponential
decay will follow. We will consider a smooth solution to the system formed by
eqs. 2, 3 and 4, whose initial data differs only on a compact set from the
data of the background, eqs. 11, 12, 13 and 14, and such that
$m_{0}(0)<2R_{0}/3$. These requirements are imposed throughout all the section
5, and, for the sake of concreteness, will be referred to as the assumptions.
Consider the change of variable $z=R^{1/2}(W-W_{b})$ and $v=\tilde{R}^{1/2}q$,
where $\tilde{R}=Re^{-4W}$. This change yields the following system of PDEs
for $z$ and $v$:

$\displaystyle z_{tt}-z_{xx}+zG+B=g,$ (45) $\displaystyle
v_{tt}-v_{xx}+v(G+4{W_{b}}_{x}^{2})+D=0,$ (46)

where

$\left\\{\begin{array}[]{ll}\begin{aligned}
D&=4v(\tilde{W}_{x}^{2}-\tilde{W}_{t}^{2})+8v\tilde{W}_{x}{W_{b}}_{x},\\\
&+2v(q_{x}^{2}-q_{t}^{2})e^{-4W}\end{aligned}&B=R^{1/2}\left(\frac{q_{t}^{2}-q_{x}^{2}}{2}\right)e^{-4W},\\\\[14.22636pt]
q_{t}=\tilde{R}^{-1/2}\left(v_{t}-\dfrac{R_{t}}{2R}v+2R^{-1/2}(z_{t}-\dfrac{R_{t}}{2R}v)\right),&G=\frac{R_{t}^{2}-R_{x}^{2}}{4R},\\\
q_{x}=\tilde{R}^{-1/2}\left(v_{x}-\dfrac{R_{t}}{2R}v+2R^{-1/2}(z_{x}-\dfrac{R_{x}}{2R}z)+2{W_{b}}_{x}v\right),&{W_{b}}_{x}=\frac{W_{0}}{\cosh(2x)},\\\
\tilde{W}=W-W_{b},&\tilde{W}_{t}=R^{-1/2}\left(z_{t}-\dfrac{R_{t}}{2R}z\right),\\\
\tilde{W}_{x}=R^{-1/2}\left(z_{x}-\dfrac{R_{x}}{2R}z\right),&g=R^{1/2}\left(\frac{R_{x}}{R}-\frac{{R_{b}}_{x}}{R_{b}}\right){W_{b}}_{x}.\\\
\end{array}\right.$

The system displayed is exactly the system of the polarized case plus powers
of first and zeroth derivatives. Our objective of finding exponential decay is
translated, under this change of variable, to prove that $z$ and $v$ grow at
most polynomially. In order to achieve this, we define the following energies:

$\mathcal{A}:=\frac{1}{2}\int_{\mathbb{R}}z^{2}+v^{2}\ dx,$ (47)
$\mathcal{E}_{0}:=E[z,v]:=\frac{1}{2}\int_{\mathbb{R}}z_{x}^{2}+z_{t}^{2}+z^{2}G_{b}\
dx+\frac{1}{2}\int_{\mathbb{R}}v_{x}^{2}+v_{t}^{2}+v^{2}(G_{b}+4{W_{b}}_{x}^{2})\
dx,$ (48) $E_{1}:=E[z_{t},v_{t}]+E[z,v],\ \
\mathcal{E}_{1}:=E[z_{x},v_{x}]+E[z,v].$ (49)

Recall that $G_{b}=\frac{1}{\cosh^{2}(2x)}$ and
${W_{b}}_{x}^{2}=\frac{W_{0}^{2}}{\cosh^{2}(2x)}$, so all the energies defined
above are positive definite. Our first objective is to control
$\mathcal{E}_{1}$. This control will be achieved by a series of lemmas (Lemma
5.1, Lemma 5.2 and Lemma 5.3), which culminate in the Theorem 5.4. The outline
here is similar to that of the polarized case. As before, we need to use the
estimates for time derivatives to get control of $\mathcal{E}_{1}$. The reader
should think of Lemma 5.1 as Proposition 4.5, the difference being that, due
to the factor $\mathcal{E}_{1}$ in the right-hand of Equation 51, Lemma 5.1
almost controls $E_{1}$. In the same way, the reader should think of Lemmas
5.2 and 5.3 as Lemma 4.6, serving as a passage between time and spatial
energies, this time being more subtle than before.

######  Lemma 5.1.

There are numbers $n_{i},m_{i}$ such that, for every solution to the system
given by eqs. 2, 45 and 46, satisfying the assumptions, there is a constant
$C>0$, depending only on a bound on $m_{2}(0)$ and on
$\mathcal{E}_{0}(0)+\mathcal{A}(0)$, such that

$\displaystyle\partial_{t}\mathcal{A}\leq\mathcal{A}^{1/2}\mathcal{E}_{0}^{1/2},$
(50) $\displaystyle\begin{aligned} \dot{E}_{1}&\leq\
C(t+1)e^{-2t}E_{1}+C(t+1)e^{-t}m_{2}(0)E_{1}^{1/2}\\\
&+\sum_{i}Ce^{-t}E_{1}\mathcal{E}_{1}^{n_{i}/2}\mathcal{A}^{m_{i}/2}.\end{aligned}$
(51)

###### Proof.

Since we are working with compactly supported solutions, we can derive inside
the integral. Doing this with $\mathcal{A}$ and using Cauchy-Schwarz yields

$\partial_{t}\mathcal{A}=\int_{\mathbb{R}}zz_{t}\
dx\leq\mathcal{A}^{1/2}\mathcal{E}^{1/2}_{0}.$

Now let’s compute $\partial_{t}\mathcal{E}_{0}$. Deriving under the integral
and using parts

$\displaystyle\dot{\mathcal{E}}_{0}$
$\displaystyle=\int_{\mathbb{R}}z_{t}(z_{tt}-z_{xx}+zG_{b})+v_{t}(v_{tt}-v_{xx}+v(G_{b}+4{W_{b}}_{x}^{2}))\
dx$
$\displaystyle=\int_{\mathbb{R}}\underbrace{z_{t}[z(G_{b}-G)+g]}_{A}-z_{t}B+\underbrace{v_{t}[v(G_{b}-G)]}_{C}-v_{t}D\
dx.$

The terms $A$ and $C$ are controlled as in the polarized case:

$\int_{\mathbb{R}}A+C\ dx\leq
C(t+1)e^{-2t}\mathcal{E}_{0}+C(t+1)e^{-t}m_{1}(0)\mathcal{E}_{0}^{1/2}.$

The new thing here is to control the terms involving $B$ and $D$. A careful
inspection of $B$ and $D$, using the equations below the eqs. 45 and 46, shows
that they can be written as a linear combination of the terms displayed below

$B\ \ \left\\{\begin{aligned} &R^{-1/2}e^{-4\tilde{W}}\alpha_{1},\text{where
}\begin{aligned} \alpha_{1}=&\
v_{t}^{2},\dfrac{R_{t}}{R}vv_{t},\dfrac{R_{x}}{R}vv_{x},v^{2}G,{W_{b}}_{x}^{2}v^{2},\\\
&\dfrac{R_{x}}{R}{W_{b}}_{x}v^{2},{W_{b}}_{x}v_{x}v,v_{x}^{2}\end{aligned}\\\
&\begin{aligned} R^{-1}e^{-4\tilde{W}}\alpha_{2},\text{where }\alpha_{2}=&\
v_{t}vz_{t},\dfrac{R_{t}}{R}zv_{t}v,\dfrac{R_{t}}{R}v^{2}z_{t},Gv^{2}z,v_{x}vz_{x},\\\
&\dfrac{R_{x}}{R}zv_{x}v,\dfrac{R_{x}}{R}v^{2}z_{x},{W_{b}}_{x}v^{2}z_{x},{W_{b}}_{x}v^{2}z\dfrac{R_{x}}{2R}\end{aligned}\\\
&R^{-3/2}e^{-4\tilde{W}}\alpha_{3},\text{where
}\alpha_{3}=v^{2}z_{t}^{2},v^{2}z^{2}G,\dfrac{R_{t}}{R}z_{t}zv^{2},v^{2}z_{x}^{2},\dfrac{R_{x}}{R}z_{x}zv^{2}\end{aligned}\right.$
(52)

and

$D\ \ \left\\{\begin{aligned}
&R^{-1/2-i/2}e^{-4\widetilde{W}}\alpha_{4},\text{where
}\alpha_{4}=v\alpha_{i},\ \ \text{for }i=1,2,3.\\\
&R^{-1}\alpha_{5},\text{where
}\alpha_{5}=vz_{x}^{2},Gz^{2}v,\dfrac{R_{x}}{2R}z_{x}zv,vz_{t}^{2},\dfrac{R}{2R_{t}}z_{t}zv,\dfrac{R_{x}}{R}{W_{b}}_{x}vz\\\
&R^{-1/2}\alpha_{6},\text{where
}\alpha_{6}={W_{b}}_{x}vz_{x}\end{aligned}\right.$ (53)

Because of this, to bound $\int_{\mathbb{R}}z_{t}B+v_{t}D\ dx$, it suffices to
deal with a sum of terms of the form

$\int_{\mathbb{R}}R^{-m/2}{e^{-4\tilde{W}j}}z_{t}\alpha\ dx\ \ \text{or }\ \
\int_{\mathbb{R}}R^{-m/2}e^{-4\tilde{W}j}v_{t}\alpha\ dx\ \ \text{where
}\begin{aligned} m&\geq 1,j=0,1\\\ \alpha&=\alpha_{i}\ \text{for
}i=1,..,6\end{aligned}.$

Let us call the term $z_{t}$ (or $v_{t}$) next to $\alpha$ the main
derivative. The strategy is as follows:

  1. 1.

If $\alpha$ has at least one derivative, apply Cauchy-Schwarz to this
derivative and the main derivative and use $\lVert\cdot\rVert_{\infty}$ for
the remaining terms inside $\alpha$. The use of Cauchy-Schwarz gives a bound
$\mathcal{E}_{0}$. For $R^{-m}e^{-4\tilde{W}}$ just use that $\tilde{W}$ is
bounded and that $R^{-m}\leq CR_{b}^{-m}\leq Ce^{-t}$. Regarding the use of
$\lVert\cdot\rVert_{\infty}$, note that by Sobolev embedding

$\lVert v\rVert_{\infty},\lVert z\rVert_{\infty}\leq
C(\mathcal{A}^{1/2}+\mathcal{E}_{0}^{1/2}),\ \ \ \ \ \lVert
v_{t}\rVert_{\infty},\lVert v_{x}\rVert_{\infty},\lVert
z_{t}\rVert_{\infty},\lVert z_{x}\rVert_{\infty}\leq C\mathcal{E}_{1}^{1/2}.$

Proceeding in this way, we produce bounds of the form:

$Ce^{-t}\mathcal{E}_{0}\mathcal{E}_{1}^{n/2}\mathcal{A}^{m/2}\ \ \ \ \
\text{for some }n,m\geq 0.$ (54)

where we have used that $\mathcal{E}_{0}\leq\mathcal{E}_{1}$.

  2. 2.

If $\alpha$ comprises only zero derivative terms, then looking at the terms
that constitute $D$ and $B$, and recalling that $G\leq\tfrac{C}{\cosh(2x)}$,
we see that all these terms are multiplied by $\tfrac{1}{\cosh(2x)}$. Apply
Cauchy-Schwarz between the main derivative and $\tfrac{v}{\cosh(2x)}$ (or
$\tfrac{w}{\cosh(2x)}$), and $\lVert\cdot\rVert_{\infty}$ and Sobolev
embedding for the other terms. The use of Cauchy-Schwarz gives the bound
$\mathcal{E}_{0}$. Finally, again we bound $R^{-m}e^{-4\tilde{W}}\leq
Ce^{-t}$. This procedure yields bounds of the form

$Ce^{-t}\mathcal{E}_{0}\mathcal{E}_{0}^{n/2}\mathcal{A}^{m/2}\ \ \ \ \
\text{for some }n,m\geq 0.$ (55)

Using the bounds (54),(55) with the fact that $\mathcal{E}_{0}\leq E_{1}$, and
the control for $A$ and $C$ we obtain

$\displaystyle\dot{\mathcal{E}_{0}}$ $\displaystyle\leq\
C(t+1)e^{-2t}m_{1}(0)E_{1}+C(t+1)e^{-t}m_{1}(0)E_{1}^{1/2}$ (56)
$\displaystyle+\sum_{i}Ce^{-t}E_{1}\mathcal{E}_{1}^{n_{i}/2}\mathcal{A}^{m_{i}/2},\text{
where }n_{i},m_{i}\geq 0.$

Now in order to obtain an estimate for $\dot{E}_{1}$ we will bound
$\dot{E}[z_{t},q_{t}]$. Deriving under the integral and using parts

$\displaystyle\dot{E}[z_{t},q_{t}]$
$\displaystyle=\int_{\mathbb{R}}z_{tt}((z_{t})_{tt}-(z_{t})_{xx}+(z_{t})G_{b})+v_{tt}((v_{t})_{tt}-(v_{t})_{xx}+v_{t}(G_{b}+4{W_{b}}_{x}^{2}))\
dx$

Now deriving the system, eqs. 45 and 46, respect to $t$ we find that

$\displaystyle(z_{t})_{tt}-(z_{t})_{xx}+z_{t}G+G_{t}z+B^{\prime}=g_{t},$
$\displaystyle\begin{aligned}
&(v_{t})_{tt}-(v_{t})_{xx}+v_{t}(G+4{W_{b}}_{x}^{2})+vG_{t}+D^{\prime}=0.\end{aligned}$

Therefore

$\displaystyle\dot{E}[z_{t},q_{t}]$
$\displaystyle=\int_{\mathbb{R}}\underbrace{z_{tt}[z_{t}(G_{b}-G)+G_{t}z+g_{t}]}_{F}-z_{tt}B^{\prime}+\underbrace{v_{tt}[v_{t}(G_{b}-G)+G_{t}v]}_{G}-v_{tt}D^{\prime}\
dx.$

Again, the terms $F$ and $G$ are controlled as in the polarized case:

$\int_{\mathbb{R}}F+G\ dx\leq
C(t+1)e^{-2t}E_{1}+C(t+1)e^{-t}m_{2}(0)E_{1}^{1/2}.$

Now when we derive $B$ and $D$, we find (see eqs. 52 and 53)

$\displaystyle(R^{-m/2})_{t}e^{-4\tilde{W}}\alpha+R^{-m/2}(-4\tilde{W}_{t})e^{-4\tilde{W}}\alpha+R^{-m/2}e^{-4\tilde{W}}\alpha_{t}$
$\displaystyle=(R^{-m/2})_{t}e^{-4\tilde{W}}\beta+R^{-m/2}\beta,$

where $\beta$ could be $\alpha_{t}$, $\alpha$ or $-4\tilde{W}_{t}\alpha$, with
$\alpha=\alpha_{i},\ i=1,..,6$. Here
$\left\lvert(R^{-m/2})_{t}\right\rvert=\left\lvert-m\tfrac{R_{t}}{R}R^{-m/2}\right\rvert\leq
CR^{-m/2}$, so it suffices to bound terms of the form

$\int_{\mathbb{R}}R^{-m/2}e^{-4\tilde{W}}z_{tt}\beta\ dx\ \ \ \text{or}\ \ \
\int_{\mathbb{R}}R^{-m/2}e^{-4\tilde{W}}v_{tt}\beta\ dx\ \ \
\beta=\alpha,\alpha_{t},-4\widetilde{W}_{t}\alpha.$

The fundamental fact about the products inside $\beta$ is that they have, at
most, a second derivative since $\alpha$ is a product of one or zero
derivatives of $v$ and $q$. In other words, if a product inside $\beta$ has a
second derivative, this derivative is raised to the power of 1. In addition,
this second derivative could be $z_{tt}$, $v_{tt}$, $z_{xt}$ or $v_{xt}$. Let
us call the term $z_{tt}$ (or $v_{tt}$) next to $\beta$ the main derivative.
The strategy is similar to the previous one:

  1. 1.

If $\beta$ has a second derivative, apply Cauchy-Schwarz to this derivative
and the main derivative and use $\lVert\cdot\rVert_{\infty}$ for the remaining
terms of $\beta$. If it does not have a second derivative but has at least one
derivative, apply Cauchy-Schwarz to this derivative and the main derivative.
The use of Cauchy-Schwarz gives a term $E_{1}$. None of the remaining terms of
$\beta$ will be a second derivative so, using Sobolev embedding, these
$\lVert\cdot\rVert_{\infty}$ will be bounded precisely as before:

$\lVert v\rVert_{\infty},\lVert z\rVert_{\infty}\leq
C(\mathcal{A}^{1/2}+\mathcal{E}_{0}^{1/2}),\ \ \ \ \ \lVert
v_{t}\rVert_{\infty},\lVert v_{x}\rVert_{\infty},\lVert
z_{t}\rVert_{\infty},\lVert z_{x}\rVert_{\infty}\leq C\mathcal{E}_{1}^{1/2}.$

Once more, $R^{-m/2}e^{-4\tilde{W}}\leq Ce^{-t}$. With this procedure, we
obtain bounds of the form

$\leq Ce^{-t}E_{1}\mathcal{E}_{1}^{n/2}\mathcal{A}^{m/2}\ \ \ \ \ \text{for
some }n,m\geq 0.$ (57)

  2. 2.

If $\beta$ comprises only zero-derivative terms then, looking at the terms
that constitute $D^{\prime}$ and $B^{\prime}$, and recalling that
$G\leq\tfrac{C}{\cosh(2x)}$ and $\partial_{t}G\leq\tfrac{C}{\cosh(2x)}$, we
see that all these terms are multiplied by $\tfrac{1}{\cosh(2x)}$. Then apply
Cauchy-Schwarz between the main derivative and $\tfrac{v}{\cosh(2x)}$ (or
$\tfrac{w}{\cosh(2x)}$). For the remaining terms inside $\beta$, use
$\lVert\cdot\rVert_{\infty}$ and Sobolev embedding as above. The use of Cauchy
Schwarz will give a term bounded by $CE_{1}$. Finally, again, we bound
$R^{-m}e^{-4\tilde{W}}\leq Ce^{-et}$. This procedure yields bounds of the form

$\leq Ce^{-t}E_{1}\mathcal{E}_{0}^{n/2}\mathcal{A}^{m/2}\ \ \ \ \ \text{for
some }n,m\geq 0.$ (58)

Now $\dot{E}_{1}=\dot{\mathcal{E}}_{0}+\dot{E}[z_{t},q_{t}]$. Using eqs. 57
and 58, the control of $F+G$ and eq. 56, we obtain:

$\displaystyle\dot{E}_{1}$ $\displaystyle\leq\
C(t+1)e^{-2t}m_{2}(0)E_{1}+C(t+1)e^{-t}m_{2}(0)E_{1}^{1/2}$
$\displaystyle+\sum_{i}Ce^{-t}E_{1}\mathcal{E}_{1}^{n_{i}/2}\mathcal{A}^{m_{i}/2},\text{
where }n_{i},m_{i}\geq 0.$

∎

######  Lemma 5.2.

There are some numbers $n_{i},m_{i}$ such that, if during an interval of time
$[0,T]$ we have a solution satisfying the assumptions with $\left\lvert
z_{x}\right\rvert,\left\lvert v_{x}\right\rvert<1$, then, there is a constant
$C>0$, depending only on a bound on $m_{1}(0)$ and on $\mathcal{E}_{0}(0)$,
such that

$\mathcal{E}_{1}\leq
CE_{1}+C(t+1)^{2}e^{-2t}m_{1}(0)^{2}+\sum_{i}Ce^{-t}E_{1}^{1+n_{i}/2}\mathcal{A}^{m_{i}/2}\
\text{ for }t\in[0,T].$ (59)

###### Proof.

We know that $\mathcal{E}_{1}=E[z,v]+E[z_{x},v_{x}]\leq E_{1}+E[z_{x},v_{x}]$
so in order to bound $\mathcal{E}_{1}$ with $E_{1}$ we need to control
$E[z_{x},v_{x}]$ by $E_{1}$. Note that

$\displaystyle E[z_{x},v_{x}]$ $\displaystyle=\int
z_{xx}^{2}+z_{xt}^{2}+z_{x}^{2}G_{b}+v_{xx}^{2}+v_{xt}^{2}+v_{x}^{2}(G_{b}+4{W_{b}}_{x}^{2})\
dx$ $\displaystyle\leq\int_{\mathbb{R}}z_{xx}^{2}+v_{xx}^{2}+CE_{1}.$

Now, using the equation,

$\displaystyle z_{xx}^{2}\leq C(z_{tt}^{2}+z^{2}G^{2}+g^{2}+B^{2}),$
$\displaystyle w_{xx}^{2}\leq C(w_{tt}^{2}+w^{2}G^{2}+D^{2}),$

so

$\displaystyle\int_{\mathbb{R}}z_{xx}^{2}+w_{xx}^{2}$ $\displaystyle\leq
CE_{1}+\int_{\mathbb{R}}g^{2}\ dx+\int_{\mathbb{R}}B^{2}+D^{2}\ dx$
$\displaystyle\leq$ $\displaystyle\
CE_{1}+C(t+1)^{2}e^{-2t}m_{1}(0)^{2}+\int_{\mathbb{R}}B^{2}+D^{2}\ dx.$

So now we need to control the last integral by $E_{1}$. In order to do this,
we will use the hypothesis that $\left\lvert z_{x}\right\rvert,\left\lvert
v_{x}\right\rvert<1$. Now, remember the general form of $B$ and $D$ (eqs. 52
and 53). It follows that we need to bound

$\int R^{-k}\alpha^{2}e^{-8\tilde{W}j}\ dx\ \text{where }k\geq 1,j=0,1\
\text{and }\alpha=\alpha_{i}\ \text{for }i=1,...,6.$

Here, we have used that $2ab\leq a^{2}+b^{2}$ many times. The strategy is as
follows:

  1. 1.

If $\alpha$ is composed only by zeroth derivative terms, then $\alpha^{2}$
always has $\frac{1}{\cosh^{2}(2x)}$ as one of its factors and always has at
least four terms (powers of $z$ and $v$). Choose two of these, for example,
$v$ and $w$, and use $\lVert\cdot\rVert_{\infty}$ with the remaining terms and
Cauchy-Schwarz with these two terms together with $\frac{1}{\cosh^{2}(2x)}$.
Using Cauchy-Schwarz gives us a term $\mathcal{E}_{0}$. Again, use Sobolev
embedding to bound the infinite norms of the remaining terms and
$R^{-k}e^{-8\tilde{W}}\leq Ce^{-2t}$. This procedure yields bounds of the form

$Ce^{-t}\mathcal{E}_{0}^{1+n/2}\mathcal{A}^{m/2}\ \ \ (\Rightarrow\leq
Ce^{-t}E_{1}^{1+n/2}\mathcal{A}^{m/2})\ \ \ \ \ \text{for some }n,m\geq 0.$

  2. 2.

If $\alpha^{2}$ has at least one derivative then $\alpha$ has at least one
derivative then $\alpha^{2}$ has at least two derivatives. Bound the remaining
terms by the infinite norms and apply Cauchy-Schwarz to these two derivatives.
The use of Cauchy-Schwarz will give a term bounded by
$C\mathcal{E}_{0}<E_{1}$. For the infinite norms, if we have $\lVert
z\rVert_{\infty}$, $\lVert z_{t}\rVert_{\infty}$, $\lVert v\rVert_{\infty}$
and $\lVert v_{t}\rVert_{\infty}$ then use Sobolev embedding to bound these
terms by

$C(\mathcal{A}^{1/2}+\mathcal{E}_{0}^{1/2})\ \ \text{or }\ E_{1}^{1/2}.$

Regarding terms like $\lVert z_{x}\rVert_{\infty}$ and $\lVert
v_{x}\rVert_{\infty}$ just use the hypothesis that they are less than $1$.
Lastly, $R^{-k}e^{-8\tilde{W}}\leq Ce^{-t}$. This procedure gives a bound of
the form

$Ce^{-t}E_{1}^{1+n/2}\mathcal{A}^{m/2}\ \ \ \ \text{for some }n,m\geq 0.$

Putting it all together, we find

$E[z_{x},v_{x}]\leq
CE_{1}+C(t+1)^{2}e^{-2t}m_{1}(0)^{2}+\sum_{i}Ce^{-t}E_{1}^{1+n_{i}/2}\mathcal{A}^{m_{i}/2},$

which is the desired conclusion. ∎

######  Lemma 5.3.

There are some numbers $n_{i},m_{i}$ such that, for every solution satisfying
the assumptions, there is a constant $C>0$, depending only on a bound on
$m_{1}(0)$ and $\mathcal{E}_{0}(0)$, such that

$E_{1}\leq
C\mathcal{E}_{1}+C(t+1)^{2}e^{-2t}m_{1}(0)^{2}+\sum_{i}Ce^{-t}\mathcal{E}_{1}^{1+n_{i}/2}\mathcal{A}^{m_{i}/2},$
(60)

in particular, evaluating at $t=0$,

$E_{1}(0)\leq C(\mathcal{E}_{1}(0)+m_{1}^{2}(0)),$ (61)

and now the constant also depends on a bound on
$\mathcal{A}(0)+\mathcal{E}_{1}(0)$.

###### Proof.

Notice that this is the inequality of the previous lemma but with $E_{1}$ and
$\mathcal{E}_{1}$ reversed. Following the same argument leads to the need to
control $\int B^{2}+D^{2}\ dx$ by $\mathcal{E}_{1}$. This control is more
straightforward than before. The strategy used in the previous lemma works
with minor modifications. The first step already led to bounds of the form
$Ce^{-t}\mathcal{E}_{0}^{1+n/2}\mathcal{A}^{m/2}$, which is less than
$Ce^{-t}\mathcal{E}_{1}^{1+n/2}\mathcal{A}^{m/2}$. In the second step, the use
of Cauchy-Schwarz gave $\mathcal{E}_{0}$, which is fine. Regarding the
infinite norms:

$\lVert z\rVert_{\infty},\lVert v\rVert_{\infty},\lVert
z_{t}\rVert_{\infty},\lVert z_{x}\rVert_{\infty},\lVert
v_{t}\rVert_{\infty},\lVert
v_{x}\rVert_{\infty}\leq\mathcal{A}^{1/2}+\mathcal{E}_{1}^{1/2},$

which is also fine. ∎

######  Theorem 5.4.

There is some $\delta>0$ such that, for every solution to the system
satisfying the assumptions and $\mathcal{E}_{1}^{1/2}(0),m_{2}(0)<\delta$,
there is a constant $C$, such that

$E_{1}^{1/2}<C(E_{1}^{1/2}(0)+m_{2}(0))\ \ \forall t\in[0,T),$ (62)

and

$\mathcal{E}_{1}^{1/2}<C(\mathcal{E}_{1}^{1/2}(0)+m_{2}(0))\ \ \forall
t\in[0,T),\\\ $ (63)

where $T$ is the supremum of times $T^{\prime}$ such that the solution is
defined on $[0,T^{\prime}]$. Furthermore, the constant just depends on a bound
on $m_{2}(0),\mathcal{E}_{1}(0)$ and on $\mathcal{A}^{1/2}(0)$.

###### Proof.

Let $\delta^{\prime}>0$ be such that $\delta^{\prime}<1$ and such that if
$\mathcal{E}_{1}^{1/2}<\delta^{\prime}$ then $\left\lvert
z_{x}\right\rvert,\left\lvert v_{x}\right\rvert<1$. The existence of
$\delta^{\prime}$ is justified by Sobolev embedding. Now suppose that
$\mathcal{E}_{1}^{1/2}(0),m_{2}(0)<\delta<<\delta^{\prime}$. The value of
$\delta$ will be specified in a moment. The only property that we will use now
is that since $\delta<\delta^{\prime}$ and $\mathcal{E}_{1}^{1/2}(0)<\delta$,
then $\mathcal{E}_{1}^{1/2}(t)<\delta^{\prime}$ for at least an interval of
time. Consider

$\widetilde{T}=\sup\\{s:s\leq T\text{ and
}\mathcal{E}_{1}^{1/2}(t)<\delta^{\prime}\text{ for
}t\in[0,\widetilde{T})\\}.$

For $t\in[0,\tilde{T})$, we have $\left\lvert z_{x}\right\rvert,\left\lvert
v_{x}\right\rvert<1$, and therefore we are allowed to apply Lemma 5.2. This
lemma asserts that

$\mathcal{E}_{1}\leq
CE_{1}+C(t+1)^{2}e^{-2t}m_{1}(0)^{2}+\sum_{i}Ce^{-t}E_{1}^{1+n_{i}/2}\mathcal{A}^{m_{i}/2}\
\forall t\in[0,\widetilde{T}).$

Now by the evolution equation for $\mathcal{A}$, eq. 50, we know that

$\mathcal{A}^{1/2}\leq\mathcal{A}^{1/2}(0)+\int_{0}^{t}\mathcal{E}_{1}^{1/2}(s)\
ds\leq\mathcal{A}^{1/2}(0)+t\ \ \ \ \ \forall t\in[0,\widetilde{T}),$

so $e^{-t}\mathcal{A}^{m_{i}}$ is bounded, in $[0,\widetilde{T})$, by a
constant that just depends on a bound on $\mathcal{A}^{1/2}(0)$. Hence
$\left\lvert\mathcal{E}_{1},m_{1}\right\rvert<1$ and
$e^{-t}\mathcal{A}^{m_{i}}$ is bounded for $t\in[0,\widetilde{T})$. Using
these bounds and the previous lemma, we see that $E_{1}$ is bounded in
$[0,\widetilde{T})$, and the bound depends on a bound on $m_{1}(0)$,
$\mathcal{E}_{1}^{1/2}(0)$ and on $\mathcal{A}^{1/2}(0)$. As a consequence,
$E_{1}^{1+n_{i}/2}<CE_{1}\ \forall t\in[0,\widetilde{T})$, where again, $C$
depends on a bound on $m_{1}(0)$, $\mathcal{E}_{1}^{1/2}(0)$ and on
$\mathcal{A}^{1/2}(0)$. Using this fact, we specialize the conclusion of the
Lemma 5.2, obtaining

$\mathcal{E}_{1}\leq CE_{1}+C(t+1)^{2}e^{-2t}m_{1}^{2}(0)\ \forall
t\in[0,\widetilde{T}).$ (64)

Here the constant $C$ depends on a bound on $m_{1}(0)$,
$\mathcal{E}_{1}^{1/2}(0)$ and on $\mathcal{A}^{1/2}(0)$. Now, we proceed to
control our energies. First, by the Equation 51 and the fact that
$\mathcal{E}_{1}<1$ in $[0,\widetilde{T})$ we know that

$\displaystyle\dot{E}_{1}$ $\displaystyle\leq\
C(t+1)e^{-2t}E_{1}+C(t+1)e^{-t}m_{2}(0)E_{1}^{1/2}$
$\displaystyle+CE_{1}\sum_{i}e^{-t}\mathcal{A}^{m_{i}/2}\ \ \ \forall
t\in[0,\widetilde{T}).$

Now, use that $(t+1)e^{-t},(t+1)e^{-2t}<Ce^{-t/2}$ and
$e^{-t}\mathcal{A}^{m_{i}}<Ce^{-t/2}\ \forall i,\forall\
t\in[0,\widetilde{T})$. This yields

$\dot{E}_{1}\leq Ce^{-t/2}(E_{1}+m_{2}(0)E_{1}^{1/2}).$

Using Gronwall

$E_{1}^{1/2}\leq C(E_{1}^{1/2}(0)+m_{2}(0))\ \forall t\in[0,\widetilde{T}).$
(65)

The constant here depends on a bound on $m_{2}(0),\mathcal{E}_{1}^{1/2}(0)$
and on $\mathcal{A}^{1/2}(0)$. Now, using eq. 64 and eq. 65 we get

$\displaystyle\mathcal{E}_{1}^{1/2}$ $\displaystyle\leq
CE_{1}^{1/2}+Cm_{1}(0)\leq C(E_{1}^{1/2}(0)+m_{2}(0))$ $\displaystyle\leq
C(\mathcal{E}_{1}^{1/2}(0)+m_{2}(0))\ \forall t\in[0,\widetilde{T}),$

where in the last inequality, we have used eq. 61. Summarizing

$\mathcal{E}_{1}^{1/2}\leq C(\mathcal{E}_{1}^{1/2}(0)+m_{2}(0))\ \forall
t\in[0,\widetilde{T}).$ (66)

As we are asking $\mathcal{E}_{1}^{1/2}(0),m_{2}(0)$ to be less than $\delta$,
we have $\mathcal{E}_{1}^{1/2}\leq C2\delta\ \forall t\in[0,\widetilde{T})$.
Now, if we require $\delta<\frac{\delta^{\prime}}{4C}$ then
$\mathcal{E}_{1}^{1/2}(t)<\delta^{\prime}/2<\delta^{\prime}\ \forall
t\in[0,\widetilde{T})$ and hence $\widetilde{T}=T$. To sum up, if
$m_{2}(0),\mathcal{E}_{1}^{1/2}(0)<\delta$ then

$E_{1}^{1/2}\leq C(E_{1}^{1/2}(0)+m_{2}(0))\ \ \forall t\in[0,T),$

and since we have (66), then we also have

$\mathcal{E}_{1}^{1/2}\leq C(\mathcal{E}_{1}^{1/2}(0)+m_{2}(0))\ \ \forall
t\in[0,T).$

∎

####  5.2.1. Gaining one more derivative

In order to obtain existence for all $t\geq 0$, what we have to do is to
extend Theorem 5.4 to $E_{2}:=E_{1}+E[z_{tt},v_{tt}]$ and
$\mathcal{E}_{2}:=\mathcal{E}_{1}+E[z_{xx},v_{xx}]$. In this way, we control
one more derivative, and then by a continuity lemma, existence for all time to
the future is guaranteed . The arguments here are essentially the same as
above. For this reason, we only state the results.

######  Lemma 5.5.

There are numbers $n_{i},m_{i},r_{i}$ such that, for every solution to the
system given by eqs. 2, 45 and 46, satisfying the assumptions, there is a
constant $C>0$, depending only on a bound on $m_{3}(0)$ and on
$\mathcal{E}_{0}(0)+\mathcal{A}(0)$, such that

$\displaystyle\dot{E}_{2}$ $\displaystyle\leq\
C(t+1)e^{-2t}E_{2}+C(t+1)e^{-t}m_{2}(0)E_{2}^{1/2}$ (67)
$\displaystyle+\sum_{i}Ce^{-t}E_{2}^{1+r_{i}/2}\mathcal{E}_{2}^{n_{i}/2}\mathcal{A}^{m_{i}/2}.$

######  Lemma 5.6.

There are some numbers $n_{i},m_{i}$ such that, if during an interval of time
$[0,T]$ we have a solution satisfying the assumptions with $\left\lvert
z_{x}\right\rvert,\left\lvert v_{x}\right\rvert,\left\lvert
z_{xx}\right\rvert,\left\lvert v_{xx}\right\rvert<1$, then there is a constant
$C>0$, depending only on a bound on $m_{2}(0)$ and on
$\mathcal{E}_{0}(0)+\mathcal{A}(0)$, such that

$\mathcal{E}_{2}\leq
CE_{2}+C(t+1)^{2}e^{-2t}m_{2}(0)^{2}+\sum_{i}Ce^{-t}E_{2}^{1+n_{i}/2}\mathcal{A}^{m_{i}/2}\
\ \text{for }t\in[0,T].$ (68)

######  Lemma 5.7.

There are some numbers $n_{i},m_{i},r_{i}$ such that, for every solution
satisfying the assumptions, there is a constant $C>0$, depending only on a
bound on $m_{2}(0)$ and $\mathcal{E}_{0}(0)$, such that

$E_{2}\leq
C\mathcal{E}_{2}+C(t+1)^{2}e^{-2t}m_{2}(0)^{2}+\sum_{i}Ce^{-t}\mathcal{E}_{2}^{1+n_{i}/2}\mathcal{A}^{m_{i}/2}+\sum_{i}Ce^{-t}\mathcal{A}^{l_{i}/2}\mathcal{E}_{1}^{r_{i}/2}m_{2}^{2}(0),$
(69)

in particular, evaluating at $t=0$

$E_{2}(0)\leq C(\mathcal{E}_{2}(0)+m_{2}^{2}(0)),$ (70)

and now the constant also depends on a bound on
$\mathcal{A}(0)+\mathcal{E}_{1}(0)$.

######  Theorem 5.8.

There is some $\delta>0$ such that, for every solution to the system
satisfying the assumptions and $\mathcal{E}_{2}^{1/2}(0),m_{3}(0)<\delta$, the
solution is defined for every $t\geq 0$. Furthermore,

$E_{2}^{1/2}<C(E_{2}^{1/2}(0)+m_{3}(0))\ \ \forall t\geq 0,$ (71)

and

$\mathcal{E}_{2}^{1/2}<C(\mathcal{E}_{2}^{1/2}(0)+m_{3}(0))\ \ \forall t\geq
0,$ (72)

for some constant $C$ that just depends on a bound on
$m_{3}(0),\mathcal{E}_{2}(0)$ and on $\mathcal{A}^{1/2}(0)$.

###### Proof.

Having the three previous lemmas, we can repeat the proof of the Theorem 5.4,
which will give the desired conclusions for $t\in[0,T)$, where $T$ is the
supremum of times $T^{\prime}$ such that the solution is defined on
$[0,T^{\prime}]$. As a consequence,
$\mathcal{A}^{1/2}(t)+\mathcal{E}_{2}^{1/2}(t)+E_{2}^{1/2}(t)$ is bounded in
$[0,T)$. By Sobolev embedding, the $C^{2}-$norm of the solution is bounded in
$[0,T)$. By the continuity lemma $T=+\infty$. ∎

######  Corollary 5.9.

There is a number $\delta>0$, such that for any smooth solution of the system
eqs. 2, 3 and 4, $(R,W,q)$, with initial data that differs from that of the
background only on a compact set, the following holds. If $m_{3}(0)$ and
$\widetilde{\mathcal{M}}_{3}(0)$ are both less than $\delta$, then the
solution is defined for all $t\geq 0$ and

$\widetilde{\mathcal{M}}_{3}(t)\leq
Ce^{-t}(t+1)(\widetilde{\mathcal{M}}_{3}(0)+m_{3}(0)).$

Moreover, the constant $C$ just depends on a bound on $m_{3}(0)$ and on
$\widetilde{\mathcal{M}}_{3}(0)$.

###### Proof.

The previous theorem, together with the estimate for $\mathcal{A}$, gives a
bound for $\mathcal{M}_{2}$. Furthermore, $\mathcal{A}(0)+\mathcal{E}_{2}(0)$
is bounded by $\mathcal{M}_{2}(0)$. Now applying the passage lemma, lemma 4.10
(which also works in this non-polarized case), we have the result for
$W-W_{b},q$. ∎

##  6\. General perturbations

So far, we have proved results concerning compactly supported perturbations.
In this section, we provide an argument to generalize these results for
solutions in a larger functional space. To do this, we approximate the initial
data given by a sequence of initial data which, as before, differ from that of
the background only on a compact set. We will see that the sequence of
solutions converges to the solution with the given initial data in a suitable
sense. Consider $(R_{1},W_{1},q_{1})$ and $(R_{2},W_{2},q_{2})$, two solutions
with initial data that differs from that of the background only on a compact
set. Define $z_{1}=R_{1}^{1/2}(W_{1}-W_{b})$, $v_{1}=R^{1/2}e^{-2W}q$ and
similarly $z_{2}$ and $v_{2}$. Then, for each $i$, we have

$\displaystyle(z_{i})_{tt}-(z_{i})_{xx}+f(G_{i},R_{i}^{-1/2},\tfrac{{R_{i}}_{t}}{R_{i}},\tfrac{{R_{i}}_{x}}{R_{i}},v_{i},{v_{i}}_{x},{v_{i}}_{t},z_{i},{z_{i}}_{t},{z_{i}}_{x},{W_{b}}_{x})=g_{i},$
$\displaystyle(v_{i})_{tt}-(v_{i})_{xx}+h(G_{i},R_{i}^{-1/2},\tfrac{{R_{i}}_{t}}{R_{i}},\tfrac{{R_{i}}_{x}}{R_{i}},v_{i},{v_{i}}_{x},{v_{i}}_{t},z_{i},{z_{i}}_{t},{z_{i}}_{x},{W_{b}}_{x})=0,$

where $f$ and $h$ are polynomials on these variables. For instance, in the
polarized case, $h\equiv 0$ and $f=zG$. For short, let us simply put
$f(A_{i},B_{i},C)$ where
$A_{i}=(G_{i},R_{i}^{-1/2},\tfrac{{R_{i}}_{t}}{R_{i}},\tfrac{{R_{i}}_{x}}{R_{i}})$,
$B_{i}=(v_{i},{v_{i}}_{x},{v_{i}}_{t},z_{i},{z_{i}}_{t},{z_{i}}_{x})$ and
$C={W_{b}}_{x}$. In addition, put $\Delta z=z_{2}-z_{1}$ and analogously, also
define $\Delta v$, $\Delta A$, $\Delta B$, $\Delta f$, $\Delta g$ and $\Delta
R$. Taking the difference of the equations for $i=1$ and $i=2$ yields

$\displaystyle(\Delta z)_{tt}-(\Delta z)_{xx}+\Delta f=\Delta g,$ (73)
$\displaystyle(\Delta v)_{tt}-(\Delta v)_{xx}+\Delta h=0.$ (74)

In order to control the sequence, we need to introduce the following energy

$\mathcal{H}_{n}\\!=\\!\\!\frac{1}{2}\\!\sum_{i=0}^{n}\\!\int_{\mathbb{R}}[(\partial_{x}^{i}\Delta
z)_{x}]^{2}+[(\partial_{x}^{i}\Delta z)_{t}]^{2}+(\partial_{x}^{i}\Delta
z)^{2}+[(\partial_{x}^{i}\Delta v)_{x}]^{2}+[(\partial_{x}^{i}\Delta
v)_{t}]^{2}+(\partial_{x}^{i}\Delta v)^{2}dx.$

Again, some estimates are required to bound $\mathcal{H}_{n}$. For
convenience, we introduce the notation

$m_{k}[f](t):=\lVert f\rVert_{C^{k}}(t)+\lVert\partial_{t}f\rVert_{C^{k}}(t),\
\ \ m_{k}(t):=\sup_{i}m_{k}[R_{i}](t).$

Also note that we now have two definitions for $\mathcal{E}_{n}$, namely, one
for $(z_{1},v_{1})$ and one for $(z_{2},v_{2})$. Now, we will call
$\mathcal{E}_{n}$ to the maximum of these two.

######  Lemma 6.1.

For every $\alpha$ multi-index, there is a constant $C$ such that

$\left\lvert\partial^{\alpha}\Delta g\right\rvert\leq
C\frac{m_{\left\lvert\alpha\right\rvert+1}[\Delta
R](0)(t+1)}{\cosh(2x)^{3/2}e^{t}},$ (75)

and also

$\lVert\partial^{\alpha}\Delta A\rVert\leq
C\frac{m_{\left\lvert\alpha\right\rvert+1}[\Delta
R](0)}{e^{t}\cosh(2x)}(t+1).$ (76)

Moreover, the constant $C$ just depends on $\alpha$, and on a bound on
$m_{\left\lvert\alpha\right\rvert+1}(0)$.

###### Proof.

The same kind of computations shown in the proof of the Lemma 4.4 works. ∎

######  Proposition 6.2 (Control of $\mathcal{H}_{n}$).

Let $n\geq 1$ and suppose that for all $t\geq 0$

$\mathcal{E}_{n}(t)\leq K(\mathcal{E}_{n}(0)+m_{n+1}(0)),$

for some constant $K$. Then, there is a polynomial $P(t)$ and a constant $C$
such that

$\dot{\mathcal{H}}_{n}\leq CP(t)(\mathcal{H}_{n}+m_{n+1}^{2}[\Delta R](0)).$

Moreover, the constant $C$, and $P(t)$, just depends on a bound on $K$,
$\mathcal{A}(0)$, $\mathcal{E}_{n}(0)$ and on $m_{n+1}(0)$.

###### Proof.

Deriving inside the integral, applying parts, using the equation, Cauchy-
Schwarz and the eq. 75, we obtain

$\displaystyle\dot{\mathcal{H}}_{n}$ $\displaystyle\leq
C\mathcal{H}_{n}+C\mathcal{H}_{n}^{1/2}m_{n+1}[\Delta
R](0)+C\sum_{i=0}^{n}\int_{\mathbb{R}}(\partial^{i}_{x}\Delta
z)_{t}\partial^{i}_{x}(\Delta f)+(\partial_{x}^{i}\Delta
v)_{t}\partial_{x}^{i}(\Delta h)\ dx.$

so we need to bound the last two terms. Since these terms are completely
similar, we only show how to control the first term. To do that, let us call
$\zeta(\lambda)=(\Delta A,\Delta B,0)\lambda+(A_{1},B_{1},C)$, and observe
that

$\displaystyle\left\lvert\Delta f\right\rvert$ $\displaystyle=\left\lvert
f(A_{1},B_{1},C)-f(A_{2},B_{2},C)\right\rvert\leq\underset{\lambda\in[0,1]}{\sup}\lVert\nabla
f(\zeta(\lambda))\rVert(\lVert\Delta A\rVert+\lVert\Delta B\rVert).$

Now, $\nabla f$ is a polynomial which is evaluated in a point between
$(A_{1},B_{1},C)$ and $(A_{2},B_{2},C)$. Each of the components in this
vectors are bounded by $C(t+1)$ where $C$ is a constant that depends on a
bound on $\mathcal{E}_{1}(0)$, $\mathcal{A}(0)$, $m_{2}(0)$ and $K$. As a
consequence, the term involving $\nabla f$ grows polynomially. Using this and
the eq. 76 we get

$\left\lvert\Delta f\right\rvert\leq P(t)(C\frac{m_{1}[\Delta
R](0)(t+1)e^{-t}}{\cosh(2x)}+\lVert\Delta B\rVert),$

and hence

$\int_{\mathbb{R}}(\Delta z)_{t}\left\lvert\Delta f\right\rvert\ dx\leq
CP(t)(\mathcal{H}_{n}^{1/2}m_{n+1}[\Delta R](0)+\mathcal{H}_{n}).$

For the other terms, a slightly different argument is needed. Given $i\geq 1$
we are going to control the term involving $(\partial^{i}_{x}\Delta
z)_{t}\partial^{i}_{x}(\Delta f)$, only using that $m_{i+1}$ and
$\mathcal{E}_{i}$ are bounded. In this way, we can apply this up to $i=n$
without needing more hypotheses. Consider

$\displaystyle\partial_{x}^{i}(\Delta f)$
$\displaystyle=\partial_{x}^{i}(f(A_{1},B_{1},C)-f(A_{2},B_{2},C))=\partial_{x}\partial_{x}^{i-1}(f(A_{1},B_{1},C)-f(A_{2},B_{2},C))$
$\displaystyle=\partial_{x}(\widehat{f}(\widehat{A}_{1},\widehat{B}_{1},\widehat{C})-\widehat{f}(\widehat{A}_{2},\widehat{B}_{2},\widehat{C})),$

here $\hat{A}_{1}$ is a vector formed by the elements of $A_{1}$ and their
$i-1$ derivatives with respect to $x$, and similarly for $\widehat{A}_{2}$,
$\widehat{B}_{1},\widehat{B}_{2}$ and $\widehat{C}$. Now $\widehat{f}$ is a
polynomial on these variables. Hence

$\displaystyle\partial_{x}^{i}(\Delta
f)=\nabla\widehat{f}(\widehat{A}_{2},\widehat{B}_{2},\widehat{C})\cdot(\partial_{x}\widehat{A}_{2},\partial_{x}\widehat{B}_{2},\partial_{x}\widehat{C})-\nabla\widehat{f}(\widehat{A}_{1},\widehat{B}_{1},\widehat{C})\cdot(\partial_{x}\widehat{A}_{1},\partial_{x}\widehat{B}_{1},\partial_{x}\widehat{C})$
$\displaystyle=(I)+(II),\ \ \text{where }$ $\displaystyle\ \ \
(I)=\nabla\widehat{f}(\widehat{A}_{2},\widehat{B}_{2},\widehat{C})\cdot(\partial_{x}\Delta\widehat{A},\partial_{x}\Delta\widehat{B},0)$
$\displaystyle\ \ \
(II)=[\nabla\widehat{f}(\widehat{A}_{2},\widehat{B}_{2},\widehat{C})-\nabla\widehat{f}(\widehat{A}_{1},\widehat{B}_{1},\widehat{C})]\cdot(\partial_{x}\widehat{A}_{1},\partial_{x}\widehat{B}_{1},\partial_{x}\widehat{C}).$

For (II), observe that

$(II)\leq
C\underset{i,j,t\in[0,1]}{\sup}\left\lvert\partial^{i}\partial^{j}\widehat{f}(\widehat{\zeta}(t))\right\rvert\lVert(\Delta\widehat{A},\Delta\widehat{B},0)\rVert\lVert(\partial_{x}\widehat{A}_{1},\partial_{x}\widehat{B}_{1},\partial_{x}\widehat{C})\rVert,$

where C is a constant that does not depend on anything. The first term in this
product is a polynomial which is evaluated in a point between
$(\widehat{A}_{1},\widehat{B}_{1},\widehat{C})$ and
$(\widehat{A}_{2},\widehat{B}_{2},\widehat{C})$. The terms in
$\widehat{A}_{j}$, $j=1,2$, are $G_{j}$,$R^{-1/2}_{j}$, ${R_{j}}_{t}/R_{j}$,
${R_{j}}_{x}/R_{j}$, and their $x-$derivatives up to order $i-1$. Since
$m_{i}$ is bounded, all these terms are bounded. The terms in
$\widehat{B}_{j}$ are composed of the $x-$derivatives of
$z_{j},{z_{j}}_{t},{z_{j}}_{x},{v_{j}},{v_{j}}_{t},{v_{j}}_{x}$ up to order
$i-1$. Since $\mathcal{E}_{i}$ is bounded, each term grows, at most, linearly.
Lastly, $\hat{C}$ is bounded. As a result, the first term in this product is
bounded by $CP(t)$, where $P(t)$ is a polynomial and $C$ is a constant that
depends on a bound on $\mathcal{E}_{i}(0),\mathcal{A}(0)$, $m_{i+1}(0)$ and
$K$. For the middle term, we use the eq. 76 for $\Delta\widehat{A}$, and
$\lVert\Delta\widehat{B}\rVert\leq C\mathcal{H}_{n}^{1/2}$, which is true by
Sobolev embedding. This yields

$\left\lvert(II)\right\rvert\leq CP(t)(\frac{m_{i}[\Delta
R](0)(t+1)}{e^{t}\cosh(2x)}+\mathcal{H}_{n}^{1/2})\lVert(\partial_{x}\widehat{A}_{1},\partial_{x}\widehat{B}_{1},\partial_{x}\widehat{C})\rVert.$

Using the same ideas, we see that

$\left\lvert(I)\right\rvert\leq CP(t)(C\frac{m_{i+1}[\Delta
R](0)(t+1)e^{-t}}{\cosh(2x)}+\lVert\partial_{x}\Delta\widehat{B}\rVert).$

Now, using Cauchy-Schwarz, and the fact that
$\int_{\mathbb{R}}\lVert(\partial_{x}\widehat{A}_{1},\partial_{x}\widehat{B}_{1},\partial_{x}\widehat{C})\rVert^{2}\
dx$ is bounded, we get

$\int_{\mathbb{R}}(\partial^{i}_{x}\Delta z)_{t}\partial^{i}_{x}(\Delta f)\
dx\leq CP(t)\left(\mathcal{H}_{n}^{1/2}m_{n+1}[\Delta
R](0)+\mathcal{H}_{n}\right)$

Putting all together, and using $ab\leq a^{2}+b^{2}$, we arrive at the
conclusion. ∎

Now we can enhance the results given by Corollary 4.11 and Corollary 5.9.

######  Theorem 6.3.

Let $k\geq 3$. Let $(R,W)$ be a $C^{2}$ solution of the system eqs. 2 and 3
with $q=0$. Suppose also that
$(R-R_{b}(0,\cdot),\partial_{t}(R-R_{b})(0,\cdot))\in C_{0}^{k}\times
C_{0}^{k-1}$, $m_{0}(0)<2R_{0}/3$ and
$((W-W_{b})(0,\cdot),\partial_{t}(W-W_{b})(0,\cdot))\in\widetilde{H}^{k}\times\widetilde{H}^{k-1}$.
Then, the solution is defined for every $t\geq 0$ and

$\widetilde{\mathcal{M}}_{k}(t)\leq
Ce^{-t}(t+1)\left(\widetilde{\mathcal{M}}_{k}(0)+m_{k}(0)\right).$

Moreover, the constant $C$ depends only on an upper bound on $m_{k}(0)$ and
$k$.

###### Proof.

Consider a sequence of initial data $(R_{i}(0,\cdot),{R_{i}}_{t}(0,\cdot))$
such that for each $i$,
$((R_{i}-R_{b})(0,\cdot),{R_{i}}_{t}-{R_{b}}_{t}(0,\cdot))$ are compactly
supported and converges to $((R-R_{b})(0,\cdot),(R-R_{b})_{t}(0,\cdot))$ in
$C_{0}^{k}\times C_{0}^{k-1}$. Similarly, we ask the same for the initial data
of $W$, but this time, we require that the sequence
$(z_{i}(0,\cdot),{z_{i}}_{t}(0,\cdot))$ converges to
$(z(0,\cdot),z_{t}(0,\cdot))$ in $H^{k}\times H^{k-1}$.

A quick computation using D’Alembert shows that for each fixed $T>0$,
$(R_{i},{R_{i}}_{t})$ is a Cauchy sequence in $C([0,T],C_{0}^{k})\times
C([0,T],C_{0}^{k-1})$, converging to $(R,R_{t})$. Regarding $z_{i}$, first
note that eq. 42 holds for every $i$ with the same constant. Accordingly, we
are in the hypothesis of Proposition 6.2, with the same $K$ for each $i$. By
applying this with $n=k-1$, we see that $\mathcal{H}_{k-1}\leq
C^{\ast}(\mathcal{H}_{k-1}(0)+m_{k}(0)),0\leq t\leq T$, where $C^{\ast}$ now
depends also on $T$. Again, we can use the same constant, $C^{*}$, each time
we apply this proposition. It follows that $(z_{i},{z_{i}}_{t})$ is a Cauchy
sequence in $C([0,T],H^{k})\times C([0,T],H^{k-1})$. Since $k\geq 3$, we have
that $(z_{i},{z_{i}}_{t})$ is a Cauchy sequence in $C([0,T],C^{2})\times
C([0,T],C^{1})$. Finally, using the equation (73), we see that we also have
control of $\partial_{t}^{2}z_{i}$ and then, we have that $z_{i}$ is a Cauchy
sequence in $C^{2}([0,T]\times\mathbb{R},\mathbb{R})$. As result, $z_{i}$
converges in $C^{2}([0,T]\times\mathbb{R})$ to a function
$h:[0,T]\times\mathbb{R}\to\mathbb{R}$ which satisfies the equation (26) with
initial data $(z(0,\cdot),z_{t}(0,\cdot))$. By uniqueness, $h$ is the solution
with this initial data, defined on $[0,T]\times\mathbb{R}$. From now on, we
shall call it $z$ instead of $h$. Now, by eq. 43, we have

$\widetilde{\mathcal{M}}_{k}[z_{i}](t)\leq
Ce^{-t}(t+1)\left(\widetilde{\mathcal{M}}_{k}[z_{i}](0)+m_{k}[R_{i}](0)\right)\
\forall t\geq 0,$

where we have used the same constant for each $i$. Since $(z_{i},{z_{i}}_{t})$
converges in $C([0,T],H^{k})\times C([0,T],H^{k-1})$ to $(z,z_{t})$ and
$(R_{i},{R_{i}}_{t})$ converges in $C([0,T],C_{0}^{k})\times
C([0,T],C_{0}^{k-1})$ to $(R,R_{t})$, taking limit in the above estimate
yields

$\mathcal{M}_{k}[z](t)\leq
Ce^{-t}(t+1)\left(\mathcal{M}_{k}[z](0)+m_{k}[R](0)\right)\ \ 0\leq t\leq T.$

Since this is valid for all $T$, by uniqueness, we have constructed a smooth
solution $z$ to the eq. 26 which satisfies this estimate for all $t\geq 0$ and
which has the desired initial data, namely, that of $W$ but seen in the
$z-$variable. By the passage Lemma 4.10, this $z$ corresponds to the smooth
solution $W$ given in the statement. The estimate in the statement is a
consequence of this lemma as well. ∎

######  Theorem 6.4.

There is a number $\delta>0$ such that, any $C^{2}$ solution of the system
eqs. 2, 3 and 4, $(R,W,q)$, with initial data satisfying
$(R-R_{b}(0,\cdot),\partial_{t}(R-R_{b})(0,\cdot))\in C_{0}^{k}\times
C_{0}^{k-1}$, $m_{0}(0)<2R_{0}/3$,
$((W-W_{b})(0,\cdot),\partial_{t}(W-W_{b})(0,\cdot))\in\widetilde{H}^{k}\times\widetilde{H}^{k-1}$,
$(q(0,\cdot),\partial_{t}q(0,\cdot))\in\widetilde{H}^{k}\times\widetilde{H}^{k-1}$,
and $m_{3}(0)<\delta$, $\widetilde{\mathcal{M}}_{3}(0)<\delta$, is defined for
all $t\geq 0$ and

$\widetilde{\mathcal{M}}_{3}(t)\leq
Ce^{-t}(t+1)(\widetilde{\mathcal{M}}_{3}(0)+m_{3}(0)).$

Moreover, the constant $C$ depends only on an upper bound on $m_{3}(0)$ and on
$\widetilde{\mathcal{M}}_{3}(0)$.

###### Proof.

The argument is essentially the same, the difference being that now we work
with $z_{i}$ and $v_{i}$, that it is Theorem 5.8 which allows us to use
Proposition 6.2 with $n=2$, and also that we use the equation given by
Corollary 5.9, version $\mathcal{M}_{3}$, instead of the Equation 43. ∎

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