# A proof of slow-roll local decay of inflaton fields in cosmology and axion
fields in cold dark matter models

Matías Morales  Departamento de Ingeniería Matemática, Universidad de Chile,
Casilla 170 Correo 3, Santiago, Chile<EMAIL_ADDRESS>and  Claudio
Muñoz  Departamento de Ingeniería Matemática and Centro de Modelamiento
Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3,
Santiago, Chile<EMAIL_ADDRESS>
###### Abstract.

We consider the long time behavior of solutions to scalar field models
appearing in the theory of cosmological inflation, oscillons and cold dark
matter, in presence or absence of the cosmological constant. These models are
not included in standard mathematical literature due to their unusual
nonlinearities, which model different features with respect to classical
fields. Here we prove that these models fit in the theory of dispersive decay
by computing new virials adapted to their setting. Several important examples,
candidates to model both effects are studied in detail.

###### Key words and phrases:

Cosmological Inflation, Axion, Cold Dark Matter, local decay

M.M. was partially funded by Chilean research grants FONDECYT 1191412 and
1231250, and Exploration ANID 13220060.

C.M. was partially funded by Chilean research grants FONDECYT 1191412,
1231250, and Basal CMM FB210005, and Exploration ANID 13220060.

##  1\. Introduction

This paper is concerned with the long time behavior of solutions to some
cosmological inflationary and Cold Dark Matter (CDM) models.

###  1.1. The theory of Inflation

Hubble [8], based on the observation of local galaxies, observed that the
universe expands and proposed the famous formula $v=H_{0}d$, where $v$ is the
radial velocity of nearby galaxies and $d$ its distance. The parameter
$H_{0}$, today known as the Hubble’s constant (or parameter) describes the
expansion of our universe. A posterior analysis, with the discovery of the
Cosmic Microwave Background (CMB) [15], a vestige of the decoupling between
matter and radiation, confirmed the idea of a primordial universe more compact
than the actual one. A classical assumption is that our large-scale universe
is governed by the cosmological principle, with a Universe homogeneous and
isotropic. In this setting, the Friedmann-Lemaitre-Robertson-Walker (FLRW)
metric

$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-Kr^{2}}+r^{2}d\Omega^{2}\right)$
(1.1)

is commonly used to describe the Universe at great scales and it is one of the
basis of the current Big-Bang theory, usually referred as the $\Lambda$CDM
cosmological model. The other key component of the model is the CDM theory.

A key component of the FLRW metric (1.1) is the _scale factor_ , usually
denoted as $a(t)$, which measures either the expansion or contraction of the
Universe with respect to a time scale variable. The parameter $K$ measures the
curvature of the universe, being $K=0$ a flat spacetime. Assuming this as a
perfect fluid leads to the Friedmann’s equations, one has an energy momentum
tensor

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},\quad
p=\hbox{pressure},\,\rho=\hbox{density},$

that through Einstein’s field equations couples the scalar factor with the
content of the universe, in the sense that

$\dot{\rho}+3H(\rho+p)=0,\quad H(t):=\frac{\dot{a}(t)}{a(t)}.$

Usually, and also in this paper, we shall assume that $H$ is constant. The
precise value of today’s Hubble parameter (and the consequent equipartition of
mass-energy of the universe) is matter of a hot controversy [17, 6], between
essentially two methods of measuring $H$ (among other important observables)
that differ in their outputs: measures using the local distance ladder and
those inferred using the CMB and galaxy surveys. Density $\rho$ in the
Universe is a tough question. Precisely, this last ingredient is another issue
in Cosmology, see Subsection 1.2.

However, the Big-Bang theory as it is does not explain several puzzling
observations of our current universe. These are the _horizon, the flatness,
and the initial conditions_ problems. The horizon problem refers to the
impressive homogeneity of our universe, given the lack of causal connection
among extreme sections of it. The flatness problem corresponds to the extreme
flatness of the Universe today, given the fact that expansion and time
evolution should make our universe even flatter. Finally, the Big-Bang model
does not explain the initial conditions required to fulfill today’s universe.
Precisely, the theory of inflation was introduced to repair these unsolved
issues of the model, and solves with great success the two first problems. For
the third one must consider perturbations of inflation.

Inflation is essentially a phenomenological theory and it is modeled via a
classical scalar field coupled to gravitation, usually called _inflaton_.
Since the period of inflation is the primordial Universe, perturbations of the
inflaton potential must also consider quantum effects, random perturbations of
gaussian type that recover with precision the observed data. The theory of
cosmological inflation was first introduced by Guth [7] and Starovinski [19].
The inflation field is characterized by a nonstandard well potential, and it
is suggested that it may be not related to the classical $\phi^{4}$ model
present in the Standard Model of particle Physics. Instead of tunneling out of
a false vacuum state, inflation occurred by the scalar field rolling down a
potential energy hill. When the field rolls very slowly compared to the
expansion of the Universe, inflation occurs. However, when the hill becomes
steeper, inflation ends and reheating can occur.

Since it is still an unproved theory, many potentials have been proposed to
describe this theory and its perturbations, some of them with better chances
due to current experimental observations of the CMB. Among these experimental
projects, Planck [16] has worked analyzing statistically anisotropies of the
CMB to understand, among other questions, which of these models could give a
better description of our universe.

###  1.2. A theory for Cold Dark Matter

The composition of the Universe is another puzzling problem in Cosmology, and
it is deeply related to the previous discussion of inflation. Today is widely
recognized that most of our universe is composed of radiation, baryonic
matter, dark matter and dark energy. The last two are essentially not well-
understood: dark matter was proposed to explain anomalous rotation speeds in
galaxies, and dark energy seeks to explain the acceleration of the universe.
Dark matter is detected only through its gravitational interactions with
ordinary matter and radiation.

There are several theories that proposes to explain dark matter. The primary
candidate for dark matter is some new kind of elementary particle that has not
yet been discovered, particularly weakly interacting massive particles
(WIMPs). One can also find Axions [20] and Primordial Black Holes (PBH) [10].
The latter are hypothesized compact objects of the early universe formed by
strong curvature deviations and not by accumulation of mass as astrophysical
black holes. They have been under review because recent detections of
gravitational waves by LIGO interferometers showed unusual ranges of masses.
It is also believed that PBH might have important implications in quantum
perturbations of the inflaton field [14]. In this paper we will concentrate
our efforts in working with Axion fields as the ones described in [3, 10, 9],
and in particular, in the theory of _oscillons_. Axions are proposed
elementary particles that under low mass constraint, are of interest as a
possible component of cold dark matter. Indeed, in recent years axions have
become one of the most promising candidates for dark matter [4].

###  1.3. Inflation’s mathematical description

Mathematically speaking, the theory goes as follows. The setting is the one
given by Einstein’s field equations for a Lorentzian metric $g$ in 1+3
dimensions. Let $\phi:\mathbb{R}^{1+3}\to\mathbb{R}$ be the perturbation of
the inflaton field. His action is given by

$\mathcal{S}=\int_{\mathbb{R}^{1+3}}\sqrt{-g}\left(\dfrac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-F(\phi)\right),$
(1.2)

where $g=g_{\mu\nu}$ the metric of the spacetime. Here we always assume that
the universe is spatially flat, homogeneous and isotropic. Specifically, we
consider a de Sitter universe, that is, the metric $(g_{\mu\nu})$ takes the
form

$ds^{2}=dt^{2}-e^{2Ht}(dx^{2}+dy^{2}+dz^{2}),$

where $H\geq 0$ represents the Hubble parameter, which we will assume
constant. The Euler Lagrange equation associated to (1.2) is

$\displaystyle\partial_{t}^{2}\phi+3H\partial_{t}\phi-e^{-2Ht}\Delta\phi+f(\phi)=0,$
(1.3)

with the notation $f=F^{\prime}$. The Laplacian $\Delta=\Delta_{x}$ here is
taken in the spatial variable $x$. The nonlinearity $f=f(s)$ is unusual, in
the sense that it will satisfy particular conditions natural for the study of
inflation [2] and or CDM [13, eqn. (15)]. This general setting is particularly
useful since one can also consider another important unsolved problem, the
dark matter existence, for which one of the most promising theories looks
quite similar, and the corresponding field is represented by another particle
called axion.

The equation (1.3) has no conserved energy in general, except when $H=0$.
Instead, it formally satisfies the following relation

$\displaystyle E[\phi,\phi_{t}]:=$
$\displaystyle~{}{}\int_{\mathbb{R}^{3}}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{|\nabla\phi|^{2}}{2e^{2Ht}}+F(\phi)\right),$
(1.4) $\displaystyle\dfrac{d}{dt}E[\phi,\phi_{t}]=$
$\displaystyle~{}{}-H\int_{\mathbb{R}^{3}}\left(3\phi_{t}^{2}+\dfrac{|\nabla\phi|^{2}}{e^{2Ht}}\right).$

This is a remarkable difference between the stationary ($H=0$) and non
stationary ($H>0$) universe scenarios. It is widely accepted in Physics that
today’s universe satisfies $H>0$. Finding the actual value of $H$ is an active
research topic, because there exists an inconsistency between the values
inferred by different experiments, ones using local measurements of the
current expansion and others using the $\Lambda$CDM model together with the
CMB data Notice that when $H=0$ equation (1.3) becomes the usual non linear
wave equation and we recover the classical conservation of energy. Previous
results in the case $H>0$ and focusing nonlinearities of F. John type reveal
that blow up exists under a sign condition on the initial data [21].

The Klein-Gordon model (1.3) has been worked many times in the literature,
specially in the case with constant coefficients and zero Hubble parameter. In
this case the literature is extensive. The reader may consult the review [12]
and references therein for a detailed account of results.

### Acknowledgments

We deeply thank Gonzalo Palma for several illuminating discussions and
comments that strongly helped us to better understand inflation theory.

### Organization of this paper

This paper is organized as follows: In Section 2 we discuss briefly the
potentials to be studied and the physical context where they appear, and in
Section 3 we enunciate the general main results that will be proved on this
work. In Section 4 some results about Sobolev spaces are recalled, as well as
classical existence and uniqueness theorems for nonlinear wave equations.
Section 5 is devoted to prove virial identities that later will be used in
Section 6 to prove Theorems 3.1, 3.4 and 3.6. Finally, in Section 7 we use the
theorems proved to study in detail the dynamics of the models discussed in 2.

##  2\. Slow-roll Inflationary and Cold Dark Matter models

In this section we will present the inflationary and CDM models to be studied
in this work. Some of them have been already studied in other works, but many
have never been considered in the mathematical literature.

###  2.1. $E$ and $T$ models

From Planck [16, Table 5] one can access to a selection of slow-roll
inflationary models of high interest in order to explain cosmological
inflation. Among the most favorable models we highlight the Starobinsky
$R^{2}$, $f(R)$ modified gravity or $E_{1}$ model, represented by the
potential

$F_{1,1}(\phi):=(1-e^{-\phi})^{2}.$ (2.1)

Notice that $F_{1,1}$ is a potential exponentially unbounded as
$\phi\to-\infty$ (see Fig. 1), with some very unpleasant features. Among them,
we can find

$F_{1,1}(0)=F_{1,1}^{\prime}(0)=0,\quad\hbox{but}\quad
F_{1,1}^{\prime\prime}(0)=2>0.$

This last positive sign makes mathematical treatment of the small data theory
not easy. Cosmological theory supposes that the initial configuration starts
with $\phi\gg 1$, and slowly decays in time towards the zero field value. This
process is called the “slow-roll” dynamics, and the exponential growth of the
scaling parameter of the universe ($a(t)\sim e^{2Ht}$) is described as the
“e-fold” procedure.

Figure 1. The potentials $F_{1,1}$ (2.1), $F_{1,2}$ and $F_{1,3}$ in (2.2).

The $F_{1,1}$ model is part of a family of inflationary potentials that gives
rise to the so-called $E_{n}$ theories:

$F_{1,n}(\phi):=(1-e^{-\phi})^{2n},\quad n\geq 1,$ (2.2)

see Fig. 1. For us the most interesting cases are the ones with $n=1,2$.
Additionally to the $E_{n}$ models, one has the $T_{n}$ ones, which are also
highly relevant in Planck data analysis. These are given by

$F_{2,n}(\phi):=\tanh^{2n}(\phi),\quad n\geq 1,$ (2.3)

see Fig. 2. The case $n=1$ is highly favorable in our setting, producing the
best result of this paper, but $n=2$ has some drawbacks due to the lack of a
sign condition. Only small data will be suitable to prove decay.

Figure 2. The potentials $F_{2,n}$ in (2.3).

###  2.2. Natural Inflation and Axion potentials

There are other models of equal interest and high importance in the quest for
the inflaton potential. These are the so-called Natural inflation (- sign)
[16] and Axion potential (+ sign) [3, p. 4] (see Fig. 3)

$F_{3,\pm}(\phi):=1\pm\cos\phi.$ (2.4)

In both cases it is assumed that the field $\phi$ is no larger than
$\frac{\pi}{2}$ in absolute value. The potential $F_{3,\pm}$ is the classical
appearing in 1D sine-Gordon models, making the scalar field model integrable.
In 3D the situation is different, and in radial symmetry integrability seem
lost. In both cases we are able to give answers to the decay problem.

Figure 3. Left: The potential $F_{3,-}$ from (2.4). Right: The potentials
$F_{4,n}$, $n=1$ in (2.5). Note that the last potential is singular at
$\phi=0$.

###  2.3. The $D$-brane and Hilltop models

In addition to the sine-Gordon models, another important model is given by the
singular $D$-brane (Fig. 3):

$F_{4,n}(\phi):=1-\frac{1}{\phi^{2n}},\quad n=1,2,\quad\phi\neq 0.$ (2.5)

Notice that the physical problem here is the perturbation of a $\phi$ large
initial state. In this case we shall assume

$\phi=1+v,\quad|v|\ll 1,$

so that after renormalization (to have finite energy) we will work with the
modified potential

$\tilde{F}_{4,n}(v):=1-\frac{1}{(1+v)^{2n}}-2nv=\frac{(1+v)^{2n}-1-2nv(1+v)^{2n}}{(1+v)^{2n}}.$
(2.6)

Also considered in this paper will be the Hilltop models (Fig. 4):

$F_{5,n}(\phi):=-\phi^{2n},\quad n=1,2.$ (2.7)

The case $n=1$ is exactly linear Klein-Gordon and will not be studied in this
paper. However, the case $n=2$ is highly interesting because it behaves as one
of the most promising potentials to describe inflation.

###  2.4. Axion-Monodromy and log potentials

The last two examples that we will study here also appear when studying CDM.
These are the axion monodromy potential [23]

$F_{6,q}(\phi):=\frac{1}{q}\left((1+\phi^{2})^{q/2}-1\right),\quad
q\in[-1,1],\quad q\neq 0,$ (2.8)

and the logarithm potential (Fig. 4):

$F_{7}(\phi):=\frac{1}{2}\log(1+\phi^{2}).$ (2.9)

The potential $F_{6,q}$ formally converges to $F_{7}$ as $q\to 0$.

Figure 4. The potentials $F_{5,n}$ , $n=1,2$ in (2.7) and $F_{7}$ from (2.9).

These families have their own properties, usually not being part of the
standard local and global well-posedness theory appearing in the literature.
For each of these models, we will prove local and/or global well-posedness,
and provide a proof of decay under suitable assumptions on the initial data.

However, as far as we know, the long time behavior of the model (1.3) with
nonlinearities such as the ones considered here (and in practice, in cosmology
and dark matter) have not been considered before. In that sense, our results
show that strong decay is a common denominator in cosmologically motivated
scalar field models.

##  3\. Main results

We shall assume that $f:\mathbb{R}\to\mathbb{R}$ is a piecewise $C^{1}$
function such that $f(0)=0$, and the initial data
$(\phi,\partial_{t}\phi)\big{|}_{\\{t=0\\}}=(\phi_{0},\phi_{1})\in H^{1}\times
L^{2}$ in (1.3) is radially symmetric; consequently, from the local well-
posedness theory one has that the solution $(\phi,\partial_{t}\phi)$ is radial
for all time whenever it is well-defined.

Our first result is a sufficient condition to obtain local decay in the energy
space.

######  Theorem 3.1 (Large data case).

Let $f$ be globally Lipschitz and satisfying the sign conditions $F(\phi)\geq
0$ and $2F(\phi)-\phi f(\phi)\geq 0$. Then the solution
$(\phi,\partial_{t}\phi)$ of (1.3) with $H=0$ is defined for all times and

$\lim_{t\to\infty}\|(\phi,\partial_{t}\phi)(t)\|_{H^{1}\times
L^{2}(B(0,R))}=0,$ (3.1)

for any $R>0$.

Note that Theorem 3.1 is valid for any size of data and no presence of the
cosmological constant.

######  Corollary 3.2.

The $E$ model $F_{2,1}$ in (2.3), the Axion-monodromy model $F_{6,q}$ in
(2.8), and the logarithm mode $F_{7}$ in (2.9) satisfy (3.1).

Recall that the first potential appears as one of the most interesting
candidates to represent inflation according to Table 5 of [16]. See Section 7
for further details.

A second corollary from Theorem 3.1 is the following:

######  Corollary 3.3.

Under (3.1), the corresponding scalar field model has no standing waves nor
breathers.

By breathers, we mean localized in space, time periodic solutions. See also
[11] for similar results as Corollary 3.3 in the case of small odd data and
odd nonlinearities.

Our second result concerns the case where the condition $2F(\phi)-\phi
f(\phi)\geq 0$ is not satisfied, still in the case $H=0$. Several cosmological
models are in this class. Our result is now local decay under smallness
assumption and growth below a critical power.

######  Theorem 3.4.

If $f:\mathbb{R}\to\mathbb{R}$ is of class $C^{1}$ and satisfies that for some
$C,\delta>0$

$\displaystyle 0\leq\phi f(\phi)\leq
C\phi^{4},\quad\forall\phi\in(-\delta,\delta),$ (3.2)

or

$\displaystyle 2F(\phi)-\phi f(\phi)\geq
0,\quad\forall\phi\in(-\delta,\delta),$ (3.3)

then any global solution $\phi\in C([0,\infty);H^{1}\times L^{2})$ of (1.3)
with $H=0$ such that

$\sup_{t\geq 0}\|\phi(t)\|_{H^{1}\cap L^{\infty}}\leq\varepsilon,$ (3.4)

satisfies

$\lim_{t\to\infty}\|(\phi,\phi_{t})(t)\|_{H^{1}\times L^{2}(B(0,R))}=0,$ (3.5)

for any $R>0$ provided that $\varepsilon>0$ is small enough.

######  Remark 3.1.

Note that the condition $\phi f(\phi)\geq 0$ ensures that $F(\phi)\geq 0$, and
the model has defocusing character. Condition (3.2) describes that the model
has sufficient flatness at the origin to allow decay, and finally condition
(3.3) it is just an application of Theorem 3.1 to the case of small data and
needs no proof.

Among the potentials that satisfy (3.5) (under solutions satisfying (3.4)) we
find the following.

######  Corollary 3.5.

The $E$ model $F_{1,n}$ in (2.3) and the $T$ model $F_{2,n}$ in (2.3), for
$n\geq 2$, satisfy (3.5). The Natural Inflation model $F_{3,-}$ in (2.4)
satisfies (3.5). Finally, the Hilltop model $F_{5,2}$ also satisfies (3.5).

Notice that one of the most important models, the so-called $R^{2}$-model

$F_{2,1}(s)=(1-e^{-s})^{2},\quad s\in\mathbb{R},$

does not fit in the assumptions of this result. Additionally, the case of the
D-brane model (2.5), if $\phi$ is chosen of the form $\phi=1+v$, one should
expect decay, however that case has remain elusive to us. Finally, the Axion
model $F_{3,+}$ in (2.4) has proved to fall outside the scope of Theorems 3.1
and 3.4.

The case with cosmological constant. Now we turn into the case of our current
universe, assuming $H>0$. Here we have the following results:

######  Theorem 3.6.

Consider equation (1.3) with initial data $(u_{0},u_{1})=(\varepsilon
g,\varepsilon h)$, $g,h\in C_{0}^{\infty}(\mathbb{R}^{3})$. If $H>0$,
$F(x)\geq 0$ for all $x\in\mathbb{R}$ and there exist $\delta,M>0$ such that

$|f^{\prime}(\phi)|\leq M\phi^{2},\quad\forall\phi\in(-\delta,\delta),$

then there exist $\varepsilon>0$ such that the solution to (1.3) is global in
time. Moreover,

  1. (1)

The energy of any global solution decays to zero outside of the forward light
cone, that is

$\lim_{t\to\infty}\int_{R(t)}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{|\nabla\phi|^{2}}{2e^{2Ht}}+F(\phi)\right)=0,$
(3.6)

where $R(t)=\\{x\in\mathbb{R}^{3}\;|\;|x|>(1+b)t\\}$ for any $b>1$.

  2. (2)

If $R>0$ is fixed,

$\lim_{t\to\infty}\int_{B(0,R)}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{|\nabla\phi|^{2}}{2e^{2Ht}}+F(\phi)\right)=0.$
(3.7)

Estimate (3.7) shows that locally the energy must converge to zero, however,
the global energy, although decreasing, may not converge to zero in general.
Their main outcome should depend on the existence of moving solitary waves. In
the case of radial data, this is strongly unlikely, but solitary rings of
finite energy might exists.

######  Remark 3.2.

The global existence ensured in Theorem 3.6 can be weakened to the simple
assumption $(\phi,\phi_{t})\in H^{1}\times L^{2}$ radial and bounded in time
of finite energy; however this assumption seems not simple to prove in the
case of general nonlinearities and we have preferred to give a more
restrictive but evidently nonempty set of data for which Theorem 3.6 holds.

##  4\. Preliminaries

In this section we gather some results needed for the proof of our main
result. We first deal with Sobolev estimates in the case of radially defined
functions.

###  4.1. A density lemma

The following result is standard in the literature, but we include its proof
for completeness reasons.

######  Lemma 4.1.

If $n\geq 3$ then $C_{0}^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ is dense in
$H^{1}(\mathbb{R}^{n})$. Moreover, if $u\in H^{1}(\mathbb{R}^{n})$ is radial
and $u_{k}\in C_{0}^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ is such that
$u_{k}\to u$ in $H^{1}(\mathbb{R}^{n})$, then we can choose $(u_{k})_{k}$ also
radial.

######  Remark 4.1.

Notice that Lemma 4.1 is not valid in dimensions 1, because by Sobolev’s
embedding one has $H^{1}(\mathbb{R})\hookrightarrow C(\mathbb{R})$. Since
$H^{1}$ convergence implies uniform convergence, if the Lemma holds we could
conclude that $u(0)=0$ for every $u\in H^{1}(\mathbb{R})$. An analogous
argument works in the case $H^{2}(\mathbb{R}^{3})$. This Lemma is still true
in dimension 2, but the proof is more complicated and is not relevant to this
work.

###### Proof of Lemma 4.1.

Since $C_{0}^{\infty}(\mathbb{R}^{n})$ is dense in $H^{1}(\mathbb{R}^{n})$ is
enough to prove that

$C_{0}^{\infty}(\mathbb{R}^{n})\subseteq\overline{C_{0}^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})}^{\|\|_{H^{1}}}.$

Let $u\in C_{0}^{\infty}(\mathbb{R}^{n})$ and $\theta\in
C^{\infty}(\mathbb{R}^{n})$ such that

$\theta(x)=\begin{cases}0&\text{ if }|x|<1\\\ 1&\text{ if }|x|>2.\end{cases}$

Since $\theta(kx)\to 1$ a.e. we have that $u_{k}(x)=u(x)\theta(kx)\to u(x)$ in
$L^{2}(\mathbb{R}^{n})$ by dominated convergence theorem. For the derivative
we have that

$\partial_{i}(u(x)\theta(kx))=\partial_{i}u(x)\theta(kx)+ku(x)\partial_{i}\theta(kx)$

and the first term converges to $\partial_{i}u(x)$ by the same argument as
above. For the second term notice that $\partial_{i}\theta(kx)$ converges to 0
a.e. and

$\text{supp}(ku(x)\partial_{i}\theta(kx))\subseteq
B\left(0,\frac{2}{k}\right)\cap B\left(0,\frac{1}{k}\right)^{c}.$

This implies that

$|ku(x)\partial_{i}\theta(kx)|\leq\dfrac{2|u(x)\partial_{i}\theta(kx)|}{|x|},$

and thus $ku(x)\partial_{i}\theta(kx)\to 0$ a.e.. In addition, we have that

$|ku(x)\partial_{i}\theta(kx)|\leq\dfrac{2\|u\|_{\infty}\|\partial_{i}\theta\|_{\infty}}{|x|}\mathds{1}_{B(0,1)}\in
L^{2}(\mathbb{R}^{n}),$

and we conclude by dominated convergence theorem.

When $u$ is radial we can take

$\varphi(x)=\begin{cases}\dfrac{1}{|x|^{2}-1}&\text{if }|x|<1\\\ 0&\text{if
}|x|\geq 1,\end{cases}$

and define $\varphi_{k}(x)=Ck^{n}\varphi(kx)$, where $C>0$ is a normalization
constant. It’s well known that $(\varphi_{n}\star u)\to u$ in
$H^{1}(\mathbb{R}^{n})$, and the convolution of radial functions is also
radial. Thus, is enough to apply the previous part of the lemma to the
sequence $u_{n}=(\varphi_{n}\star u)$ to conclude. ∎

###  4.2. Estimates in radial Sobolev spaces

The following result is standard in the literature, for completeness we
include the proof.

######  Lemma 4.2.

If $u\in H^{1}(\mathbb{R}^{3})$ is radial; with abuse of notation,
$u(x)=u(r)$. Then $u(r)\in L^{2}(0,\infty)$ and $ru(r)\in L^{p}(0,\infty)$ for
all $p\in[2,\infty]$. Moreover, we have the estimate

$\sup_{r\geq 0}|ru(r)|\leq C\|u\|_{H^{1}(\mathbb{R}^{3})}.$

###### Proof.

Since $u\in H^{1}$, from Hardy’s inequality we have

$4\pi\int_{0}^{\infty}u^{2}(r)dr=\left\|\dfrac{u}{|x|}\right\|^{2}_{L^{2}(\mathbb{R}^{3})}\leq
4\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{3})}=16\pi\int_{0}^{\infty}r^{2}u_{r}^{2}(r)dr.$

Hence $u(r)\in L^{2}(0,\infty)$ and one has the inequality

$\int_{0}^{\infty}u^{2}(r)dr\leq 4\int_{0}^{\infty}r^{2}u_{r}^{2}(r)dr.$

In order to prove that $ru(r)\in L^{\infty}(0,\infty)$, notice first that
$(ru(r))^{\prime}=u(r)+ru_{r}(r)$ and since $\phi\in H^{1}(\mathbb{R}^{3})$
from the previous part we have that $ru(r)\in H^{1}(0,\infty)$. By the
Sobolev’s embedding we conclude that $ru(r)$ is continuous and bounded. Using
interpolation between $L^{2}$ and $L^{\infty}$, we conclude $ru(r)\in
L^{p}(0,\infty)$ for all $p\in[2,\infty]$. Moreover, we have from Sobolev’s
embedding that

$\displaystyle\sup_{r\geq 0}|ru(r)|\leq$
$\displaystyle~{}{}C\|ru(r)\|_{H^{1}(0,\infty)}$ $\displaystyle\leq$
$\displaystyle~{}{}C(\|ru(r)\|_{L^{2}(0,\infty)}+\|u+ru_{r}\|_{L^{2}(0,\infty)})$
$\displaystyle\leq$
$\displaystyle~{}{}C(\|u\|_{L^{2}(\mathbb{R}^{3})}+2\|u_{r}\|_{L^{2}(\mathbb{R}^{3})})\leq
2C\|u\|_{H^{1}(\mathbb{R}^{3})},$

proving the desired estimate. ∎

###  4.3. Estimates for the linear wave equation

Consider the linear equation associated with (1.3)

$\displaystyle\partial_{t}^{2}\phi+3H\partial_{t}\phi-e^{-2Ht}\Delta\phi=f(t,x)$
(4.1)

We can define some energy associated with this equation as

$E(t)=\int\dfrac{\phi_{t}^{2}}{2}+\dfrac{|\nabla\phi|^{2}}{2e^{2Ht}}$

To prove global existence for the nonlinear equation (1.3) we shall need the
following estimate.

######  Lemma 4.3.

Let $T>0$. If $f\in L^{1}([0,T],L^{2}(\mathbb{R}^{3}))$ then there exist
$c\in\mathbb{R}$ such that every solution $(\phi,\phi_{t})\in H^{1}\times
L^{2}$ satisfies

$\displaystyle E(t)\leq
e^{ct}E(0)+\int_{0}^{t}e^{c(t-s)}\|f(s)\|_{L^{2}}^{2}ds$

###### Proof.

Notice that multiplying the equation (4.1) by $\phi_{t}$ we obtain that

$\partial_{t}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{|\nabla\phi|^{2}}{2e^{2Ht}}\right)-\text{div}\left(\dfrac{\phi_{t}\nabla\phi}{e^{2Ht}}\right)+3H\phi_{t}^{2}+H\dfrac{|\nabla\phi|^{2}}{e^{2Ht}}=f(t,x)\phi_{t}$

Integrating on $\mathbb{R}^{3}$ we get

$\displaystyle\partial_{t}E(t)=$ $\displaystyle-H\int
3\phi_{t}^{2}+\dfrac{|\nabla\phi|^{2}}{e^{2Ht}}+\int f(t,x)\phi_{t}$
$\displaystyle\leq$ $\displaystyle-H\int
3\phi_{t}^{2}+\dfrac{|\nabla\phi|^{2}}{e^{2Ht}}+\int\dfrac{\phi_{t}^{2}}{2}+\int\dfrac{f(t,x)^{2}}{2}$
$\displaystyle\leq$ $\displaystyle cE(t)+\dfrac{\|f(t)\|_{L^{2}}^{2}}{2}$

Applying Gronwall’s inequality we finish the proof. ∎

######  Remark 4.2.

Notice that when $H\geq\frac{1}{6}$ we see in the proof that the constant
could be chosen $c\leq 0$, which would give us more control on the size of the
solution.

###  4.4. Local and global existence

Finally, we recall the following results.

This first result deals with global solutions in the energy space in the case
where the nonlinearity is Lipschitz.

######  Theorem 4.4 ([5]).

If $f:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous and $f(0)=0$ then for
initial data in $H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$ equation
(1.3) has a unique global solution such that

  1. (1)

$\phi\in C([0,\infty);H^{1}(\mathbb{R}^{n}))\cap
C^{1}([0,\infty);L^{2}(\mathbb{R}^{n})).$

  2. (2)

If the initial datum $(\phi_{0},\phi_{1})\in H^{1}\times L^{2}$ is radial,
then $(\phi,\partial_{t}\phi)$ is radial for all times.

  3. (3)

Equation (1.4) is satisfied for all times $t\geq 0$.

The following result follows closely Theorem 4.4, but deals with globally
defined small solutions in the case where the nonlinearity may have growth.

######  Theorem 4.5.

The following holds:

If $f:\mathbb{R}\to\mathbb{R}$ is of class $C^{2}$ and $f(0)=f^{\prime}(0)=0$,
then equation (1.3) has a unique maximal solution provided that the initial
data $(\phi_{0},\phi_{1})\in H^{2}\times H^{1}$ is small enough. This solution
satisfies

  1. (1)

$\phi\in C([0,T);H^{2}(\mathbb{R}^{3}))\cap
C^{1}([0,T);H^{1}(\mathbb{R}^{1})).$

  2. (2)

If the initial datum $(\phi_{0},\phi_{1})\in H^{2}\times H^{1}$ is radial,
then $(\phi,\partial_{t}\phi)$ is radial for all times.

  3. (3)

Equation (1.4) is satisfied along $0\leq t<T$.

###### Proof.

This result is standard, but for completeness we include a sketch of proof in
Appendix A. ∎

######  Theorem 4.6 ([18]).

Suppose that $f:\mathbb{R}\to\mathbb{R}$ is of class $C^{2}$,
$f(0)=f^{\prime}(0)=0$ and there exists $C>0$ such that

$|f^{\prime\prime}(s)|\leq C|s|^{p},\quad|s|\leq 1$

for $p\geq 1$. Then for smooth, compactly supported and small enough initial
data, the equation (1.3) for $H=0$ has a unique global solution $\phi\in
C^{2}(\mathbb{R}^{1+3})$.

##  5\. Virial identities

Now we prove some virial estimates needed for the proof of our main results.
These are similar to the ones proved in Kowalczyk et al. [11], and more
precisely Alejo and Maulén [1], but important differences appear in our case,
where we use some particular weights that make virials completely defocusing,
in a sense to be described below.

###  5.1. First computations

For locally integrable functions $\psi(r),\varphi(t,r)$ to be chosen later let

$\displaystyle\mathcal{P}(\phi)(t)$
$\displaystyle=\int_{0}^{\infty}\psi(r)\phi_{r}(t,r)\phi_{t}(t,r)dr,$ (5.1)
$\displaystyle\mathcal{R}(\phi)(t)$
$\displaystyle=\int_{0}^{\infty}\psi^{\prime}(r)\phi(t,r)\phi_{t}(t,r)dr,$
(5.2) $\displaystyle\mathcal{I}(\phi)(t)$
$\displaystyle=\mathcal{P}(\phi)(t)+\dfrac{1}{2}\mathcal{R}(\phi)(t),$ (5.3)
$\displaystyle\mathcal{J}(\phi)(t)$
$\displaystyle=\int_{0}^{\infty}r^{2}\varphi(t,r)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right).$
(5.4)

This functional has already been used in [11] and [1]. However, with respect
to these previous references, we introduced new weights that are better
adapted to the cosmological setting. In particular, our weights will also work
for large data.

The following is a standard but key computation:

######  Lemma 5.1.

If $(\phi,\phi_{t})\in H^{1}\times L^{2}$ is a radial solution of (1.3) with
$H=0$ then

$\displaystyle\dfrac{d\mathcal{P}(\phi)}{dt}=$
$\displaystyle~{}{}\int_{0}^{\infty}\dfrac{2\psi}{r}\phi_{r}^{2}-\int_{0}^{\infty}\psi^{\prime}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2}-F(\phi)\right),$
(5.5) $\displaystyle\dfrac{d\mathcal{R}(\phi)}{dt}=$
$\displaystyle~{}{}\int_{0}^{\infty}\psi^{\prime}\left(\phi_{t}^{2}-\phi_{r}^{2}-\phi
f(\phi)\right)+\int_{0}^{\infty}\left(\dfrac{\psi^{\prime}}{r^{2}}-\dfrac{\psi^{\prime\prime}}{r}+\dfrac{\psi^{\prime\prime\prime}}{2}\right)\phi^{2},$
(5.6)

and

$\dfrac{d\mathcal{I}(\phi)}{dt}=\int_{0}^{\infty}\left(\left(\dfrac{\psi^{\prime}}{r^{2}}-\dfrac{\psi^{\prime\prime}}{r}+\dfrac{\psi^{\prime\prime\prime}}{2}\right)\phi^{2}+\left(\dfrac{2\psi}{r}-\psi^{\prime}\right)\phi_{r}^{2}+\dfrac{\psi^{\prime}}{2}(2F(\phi)-\phi
f(\phi))\right).$ (5.7)

###### Proof of Lemma 5.1.

Thanks to Lemma 4.1, it is enough to compute all derivatives assuming data in
$C_{0}^{\infty}(\mathbb{R}^{3}\backslash\\{0\\})$.

Using equation (1.3) and the definition of
$F(s)=\int_{0}^{s}f(\sigma)d\sigma$, we have in (5.1):

$\displaystyle\dfrac{d\mathcal{P}(\phi)}{dt}=$
$\displaystyle\int_{0}^{\infty}\psi\left(\phi_{rt}\phi_{t}+\phi_{r}\phi_{rr}+\dfrac{2}{r}\phi_{r}^{2}-\phi_{r}f(\phi)\right)dr$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\psi\partial_{r}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2}-F(\phi)\right)+\dfrac{2\psi}{r}\phi_{r}^{2}$
$\displaystyle=$
$\displaystyle\psi\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2}-F(\phi)\right)\bigg{\rvert}_{0}^{\infty}+\int_{0}^{\infty}\dfrac{2\psi}{r}\phi_{r}^{2}-\psi^{\prime}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2}-F(\phi)\right).$

Thanks to Lemma 4.1, every boundary term at zero and infinity disappear. We
get (5.5).

We compute now $\dfrac{d\mathcal{R}(\phi)}{dt}$. We have from (5.2):

$\displaystyle\dfrac{d\mathcal{R}(\phi)}{dt}=$
$\displaystyle\int_{0}^{\infty}\psi^{\prime}\left(\phi_{t}^{2}+\phi\phi_{rr}+\dfrac{2}{r}\phi\phi_{r}-\phi
f(\phi)\right)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\psi^{\prime}(\phi_{t}^{2}-\phi
f(\phi))+\int_{0}^{\infty}\psi^{\prime}\phi\phi_{rr}+\int_{0}^{\infty}\psi^{\prime}\dfrac{2}{r}\phi\phi_{r}=:K_{1}+K_{2}+K_{3}.$

$K_{1}$ is left as it is. We compute $K_{2}$ first saving every boundary term:

$\displaystyle K_{2}=$
$\displaystyle\psi^{\prime}\phi\phi_{r}\bigg{\rvert}_{0}^{\infty}-\int_{0}^{\infty}\phi_{r}(\psi^{\prime\prime}\phi+\psi^{\prime}\phi_{r})$
$\displaystyle=$
$\displaystyle\psi^{\prime}\phi\phi_{r}\bigg{\rvert}_{0}^{\infty}-\int_{0}^{\infty}\psi^{\prime}\phi_{r}^{2}-\int_{0}^{\infty}\psi^{\prime\prime}\partial_{r}\left(\dfrac{\phi^{2}}{2}\right)$
$\displaystyle=$
$\displaystyle\psi^{\prime}\phi\phi_{r}\bigg{\rvert}_{0}^{\infty}-\int_{0}^{\infty}\psi^{\prime}\phi_{r}^{2}-\left(\dfrac{\psi^{\prime\prime}\phi^{2}}{2}\bigg{\rvert}_{0}^{\infty}-\int_{0}^{\infty}\psi^{\prime\prime\prime}\dfrac{\phi^{2}}{2}\right)$
$\displaystyle=$
$\displaystyle\psi^{\prime}\phi\phi_{r}\bigg{\rvert}_{0}^{\infty}-\dfrac{\psi^{\prime\prime}\phi^{2}}{2}\bigg{\rvert}_{0}^{\infty}+\int_{0}^{\infty}\psi^{\prime\prime\prime}\dfrac{\phi^{2}}{2}-\int_{0}^{\infty}\psi^{\prime}\phi_{r}^{2}.$

Similarly,

$\displaystyle K_{3}=$
$\displaystyle\int_{0}^{\infty}\dfrac{\psi^{\prime}}{r}\partial_{r}(\phi^{2})=\dfrac{\psi^{\prime}}{r}\phi^{2}\bigg{\rvert}_{0}^{\infty}-\int_{0}^{\infty}\partial_{r}\left(\dfrac{\psi^{\prime}}{r}\right)\phi^{2}$
$\displaystyle=$
$\displaystyle\dfrac{\psi^{\prime}}{r}\phi^{2}\bigg{\rvert}_{0}^{\infty}-\int_{0}^{\infty}\left(\dfrac{\psi^{\prime\prime}}{r}-\dfrac{\psi^{\prime}}{r^{2}}\right)\phi^{2}.$

Arranging all previous computations, we conclude that
$\mathcal{R}(\phi)=\int_{0}^{\infty}\psi^{\prime}\phi\phi_{t}dr$ satisfies:

$\displaystyle~{}{}\dfrac{d\mathcal{R}(\phi)}{dt}$
$\displaystyle~{}{}=\int_{0}^{\infty}\psi^{\prime}(\phi_{t}^{2}-\phi f(\phi))$
$\displaystyle~{}{}\quad+\psi^{\prime}\phi\phi_{r}\bigg{\rvert}_{0}^{\infty}-\dfrac{\psi^{\prime\prime}\phi^{2}}{2}\bigg{\rvert}_{0}^{\infty}+\int_{0}^{\infty}\psi^{\prime\prime\prime}\dfrac{\phi^{2}}{2}-\int_{0}^{\infty}\psi^{\prime}\phi_{r}^{2}+\dfrac{\psi^{\prime}}{r}\phi^{2}\bigg{\rvert}_{0}^{\infty}-\int_{0}^{\infty}\left(\dfrac{\psi^{\prime\prime}}{r}-\dfrac{\psi^{\prime}}{r^{2}}\right)\phi^{2}.$

Again thanks to Lemma 4.1, every boundary term disappears. We obtain

$\displaystyle\dfrac{d\mathcal{R}(\phi)}{dt}$
$\displaystyle~{}{}=\int_{0}^{\infty}\psi^{\prime}(\phi_{t}^{2}-\phi
f(\phi))+\int_{0}^{\infty}\psi^{\prime\prime\prime}\dfrac{\phi^{2}}{2}-\int_{0}^{\infty}\psi^{\prime}\phi_{r}^{2}-\int_{0}^{\infty}\left(\dfrac{\psi^{\prime\prime}}{r}-\dfrac{\psi^{\prime}}{r^{2}}\right)\phi^{2}$
$\displaystyle~{}{}=\int_{0}^{\infty}\psi^{\prime}\left(\phi_{t}^{2}-\phi_{r}^{2}-\phi
f(\phi)\right)+\int_{0}^{\infty}\left(\dfrac{\psi^{\prime}}{r^{2}}-\dfrac{\psi^{\prime\prime}}{r}+\dfrac{\psi^{\prime\prime\prime}}{2}\right)\phi^{2}.$

This proves (5.6). Finally, gathering (5.5) and the previous identity in the
definition of $\mathcal{I}(t)$ (5.3), we arrive to (5.7). ∎

######  Lemma 5.2.

If $(\phi,\phi_{t})\in H^{1}\times L^{2}$ is a globally defined radial
solution of (1.3), then

$\displaystyle\dfrac{d\mathcal{J}(\phi)}{dt}$
$\displaystyle=\int_{0}^{\infty}r^{2}\varphi_{t}(t,r)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)$
$\displaystyle-H\int_{0}^{\infty}r^{2}\varphi(t,r)\left(3\phi_{t}^{2}+\dfrac{\phi_{r}^{2}}{e^{2Ht}}\right)-\int_{0}^{\infty}r^{2}\varphi_{r}(t,r)\dfrac{\phi_{t}\phi_{r}}{e^{2Ht}}.$

###### Proof.

As in Lemma 5.1 we shall assume data in
$C_{0}^{\infty}(\mathbb{R}^{3}\setminus\\{0\\})$, and consequently, every
boundary term disappear. Deriving (5.4) we have

$\displaystyle\dfrac{d\mathcal{J}(\phi)}{dt}$
$\displaystyle=\int_{0}^{\infty}r^{2}\varphi_{t}(t,r)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)$
$\displaystyle+\int_{0}^{\infty}r^{2}\varphi(t,r)\left(\phi_{tt}\phi_{t}+\dfrac{\phi_{r}\phi_{rt}}{e^{2Ht}}-H\dfrac{\phi_{r}^{2}}{e^{2Ht}}+f(\phi)\phi_{t}\right)$
$\displaystyle=K_{1}+K_{2}$

For the second term, using (1.3) we get

$\displaystyle K_{2}$
$\displaystyle=\int_{0}^{\infty}r^{2}\varphi(t,r)\left(-3H\phi_{t}^{2}+\dfrac{\phi_{t}\Delta\phi}{e^{2Ht}}-f(\phi)\phi_{t}+\dfrac{\phi_{r}\phi_{rt}}{e^{2Ht}}-H\dfrac{\phi_{r}^{2}}{e^{2Ht}}+f(\phi)\phi_{t}\right)$
$\displaystyle=-H\int_{0}^{\infty}r^{2}\varphi(t,r)\left(3\phi_{t}^{2}+\dfrac{\phi_{r}^{2}}{e^{2Ht}}\right)+\int_{0}^{\infty}r^{2}\varphi\left(\dfrac{\phi_{r}\phi_{rt}}{e^{2Ht}}+\dfrac{\phi_{t}\Delta\phi}{e^{2Ht}}\right)$
$\displaystyle=-H\int_{0}^{\infty}r^{2}\varphi(t,r)\left(3\phi_{t}^{2}+\dfrac{\phi_{r}^{2}}{e^{2Ht}}\right)-\int_{0}^{\infty}r^{2}\varphi_{r}(t,r)\dfrac{\phi_{r}\phi_{t}}{e^{2Ht}},$

where in the last equality we have used the Green identity on
$\mathbb{R}^{3}$. Arranging the previous calculations we got

$\displaystyle\dfrac{d\mathcal{J}(\phi)}{dt}=$
$\displaystyle~{}{}\int_{0}^{\infty}r^{2}\varphi_{t}(t,r)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)$
$\displaystyle~{}{}-H\int_{0}^{\infty}r^{2}\varphi(t,r)\left(3\phi_{t}^{2}+\dfrac{\phi_{r}^{2}}{e^{2Ht}}\right)-\int_{0}^{\infty}r^{2}\varphi_{r}(t,r)\dfrac{\phi_{t}\phi_{r}}{e^{2Ht}}.$

∎

###  5.2. Choice of the weight function

We will use Lemma 5.1 and 5.2 with a particular choice of $\psi$ and
$\varphi$.

######  Corollary 5.3.

Consider the weight

$\psi(r):=\dfrac{r^{2}}{1+r}.$ (5.8)

Then the following are satisfied:

  1. (1)

One has that

$\displaystyle\mathcal{I}(\phi)=\int_{0}^{\infty}\dfrac{r^{2}}{1+r}\phi_{r}\phi_{t}+\dfrac{r(r+2)}{2(1+r)^{2}}\phi\phi_{t}$
(5.9)

is well-defined and bounded uniformly in time by the energy of the solution:

$\sup_{t\geq 0}\left|\mathcal{I}(\phi)(t)\right|\lesssim
H[\phi,\partial_{t}\phi](t=0).$

  2. (2)

Also,

$\displaystyle\dfrac{d\mathcal{P}(\phi)}{dt}$
$\displaystyle=\int_{0}^{\infty}\dfrac{r(2+3r)}{2(1+r)^{2}}\phi_{r}^{2}-\dfrac{r(r+2)}{(1+r)^{2}}\left(\dfrac{\phi_{t}^{2}}{2}-F(\phi)\right)$
(5.10) $\displaystyle\dfrac{d\mathcal{R}(\phi)}{dt}$
$\displaystyle=\int_{0}^{\infty}\dfrac{r(r+2)}{(1+r)^{2}}(\phi_{t}^{2}-\phi
f(\phi))+\dfrac{r(r+4)}{(1+r)^{4}}\phi^{2}-\dfrac{r(r+2)}{(1+r)^{2}}\phi_{r}^{2}$
(5.11) $\displaystyle\dfrac{d\mathcal{I}(\phi)}{dt}$
$\displaystyle=\int_{0}^{\infty}r^{2}\left(\dfrac{1}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r+4}{2r(1+r)^{4}}\phi^{2}\right)+\dfrac{r(r+2)}{2(1+r)^{2}}(2F(\phi)-\phi
f(\phi)).$ (5.12)

###### Proof.

The proof of (5.9), (5.10), (5.11) and (5.12) are direct from Lemma 5.1 and
(5.8). We check now that $\mathcal{I}(\phi)$ is well-defined. Indeed,

$\left|\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)}\phi_{r}\phi_{t}\right|\leq\int_{0}^{\infty}r^{2}\left|\phi_{t}\phi_{r}\right|\leq
H(\phi,\phi_{t}).$

Additionally,

$\left|\int_{0}^{\infty}\dfrac{r(r+2)}{2(1+r)^{2}}\phi\phi_{t}\right|\leq\int_{0}^{\infty}r|\phi\phi_{t}|\leq
H(\phi,\phi_{t})^{1/2}\left(\int_{0}^{\infty}\phi^{2}dr\right)^{1/2}.$

Finally, Lemma 4.2 gives the desired uniform in time proof. ∎

######  Corollary 5.4.

For $\sigma,b\in\mathbb{R}$ consider the weight

$\displaystyle\varphi(t,r)=1+\tanh(r+\sigma t+b).$ (5.13)

Then we have the estimate

$\displaystyle\dfrac{d\mathcal{J}(\phi)}{dt}\leq(1+\sigma)\int_{0}^{\infty}r^{2}\,\text{sech}\,^{2}(r+\sigma
t+b)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right).$
(5.14)

###### Proof.

Replacing (5.13) in Lemma 5.2 we get

$\displaystyle\dfrac{d\mathcal{J}(\phi)}{dt}=$
$\displaystyle~{}{}\sigma\int_{0}^{\infty}r^{2}\,\text{sech}\,^{2}(r+\sigma
t+b)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)$
$\displaystyle-H\int_{0}^{\infty}r^{2}(1+\tanh(r+\sigma
t+b))\left(3\phi_{t}^{2}+\dfrac{\phi_{r}^{2}}{e^{2Ht}}\right)-\int_{0}^{\infty}r^{2}\,\text{sech}\,^{2}(r+\sigma
t+b)\dfrac{\phi_{t}\phi_{r}}{e^{2Ht}}.$

Noticing that the second term is strictly negative and

$-\,\text{sech}\,^{2}(r+\sigma
t+b)\dfrac{\phi_{r}\phi_{t}}{e^{2Ht}}\leq\,\text{sech}\,^{2}(r+\sigma
t+b)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}\right),$

we have that

$\displaystyle\dfrac{d\mathcal{J}(\phi)}{dt}\leq$
$\displaystyle~{}{}\sigma\int_{0}^{\infty}r^{2}\,\text{sech}\,^{2}(r+\sigma
t+b)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)$
$\displaystyle~{}{}+\int_{0}^{\infty}r^{2}\,\text{sech}\,^{2}(r+\sigma
t+b)\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}\right).$

Using that $F(\phi)\geq 0$ we obtain the desired estimate (5.14). ∎

If we define

$\|\phi\|_{H^{1}_{w}}^{2}=\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{4}}(\phi^{2}+\phi_{r}^{2}),\qquad\|\phi\|_{L^{2}_{w}}^{2}=\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{4}}\phi^{2}$

we can see that

$\dfrac{d\mathcal{I}(\phi)}{dt}\geq\|\phi\|^{2}_{H^{1}_{w}}+\dfrac{r(r+2)}{2(1+r)^{2}}(2F(\phi)-\phi
f(\phi))$

By other side, if we choose $\psi(r)=-\dfrac{3r^{2}+3r+1}{(1+r)^{3}}$ we have

$\tilde{\mathcal{R}}=\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{4}}\phi\phi_{t},$

and

$\dfrac{d\tilde{\mathcal{R}}}{dt}(\phi)=\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{4}}(\phi_{t}^{2}-\phi_{r}^{2}-\phi
f(\phi))+\dfrac{3r(3r-2)}{(1+r)^{6}}\phi^{2}.$

This allows us to prove the following propositions:

######  Proposition 5.5.

Under the hypothesis of Theorem 3.1 the solution $(\phi,\phi_{t})$ of (1.3)
satisfies

$\int_{0}^{\infty}(\|\phi\|_{H^{1}_{w}}^{2}+\|\phi_{t}\|_{L^{2}_{w}}^{2})dt<+\infty.$

###### Proof.

From Corollary 5.3 and the previous calculations we see that

$\dfrac{d\mathcal{I}(\phi)}{dt}\geq\|\phi\|_{H_{w}^{1}}^{2},$

and then

$\displaystyle\int_{0}^{\infty}\|\phi\|_{H^{1}_{w}}^{2}dt\leq$
$\displaystyle~{}{}\lim_{t\to\infty}\mathcal{I}(\phi(t))-\mathcal{I}(\phi(0))$
$\displaystyle\lesssim$
$\displaystyle~{}{}H(\phi,\phi_{t})+|\mathcal{I}(\phi(0))|.$

On the other hand, since

$\dfrac{d\tilde{\mathcal{R}}}{dt}(\phi)=\int_{0}^{\infty}\left(\dfrac{r^{2}}{(1+r)^{4}}(\phi_{t}^{2}-\phi_{r}^{2}-\phi
f(\phi))+\dfrac{3r(3r-2)}{(1+r)^{6}}\phi^{2}\right),$

we have

$\displaystyle\|\phi_{t}\|_{L^{2}_{w}}^{2}=$
$\displaystyle~{}{}\dfrac{d\tilde{\mathcal{R}}}{dt}+\|\phi_{r}\|_{L^{2}_{w}}^{2}+\int_{0}^{\infty}\left(\dfrac{r^{2}}{(1+r)^{4}}\phi
f(\phi)+\dfrac{3r(2-3r)}{(1+r)^{6}}\phi^{2}\right)$ $\displaystyle\leq$
$\displaystyle~{}{}\dfrac{d\tilde{\mathcal{R}}}{dt}+\|\phi_{r}\|_{L^{2}_{w}}^{2}+\int_{0}^{\infty}\left(\dfrac{Mr^{2}}{(1+r)^{4}}+\dfrac{3r(2-3r)}{(1+r)^{6}}\right)\phi^{2}$
$\displaystyle\leq$
$\displaystyle~{}{}\dfrac{d\tilde{\mathcal{R}}}{dt}+\|\phi_{r}\|_{L^{2}_{w}}^{2}+C(M)\int_{0}^{\infty}\dfrac{r(4+r)}{(1+r)^{4}}\phi^{2}$
$\displaystyle\leq$
$\displaystyle~{}{}\dfrac{d\tilde{\mathcal{R}}}{dt}+\|\phi_{r}\|_{L^{2}_{w}}^{2}+C(M)\dfrac{d\mathcal{I}}{dt},$

where $M$ is the Lipschitz constant of $f$, and we have used (5.12) and the
previous observations. We note that

$|\tilde{R}(\phi)|\leq\int_{0}^{\infty}\dfrac{r^{2}}{1+r}|\phi\phi_{t}|\leq\int_{0}^{\infty}r|\phi\phi_{t}|,$

and from the proof of Corollary 5.3 we see that $\tilde{R}(\phi)$ is uniformly
bounded in time by the energy of the solution. Integrating the last inequality
the result follows. ∎

######  Proposition 5.6.

Under the hypothesis of Theorem 3.4 the solution $(\phi,\phi_{t})$ of (1.3)
satisfies

$\int_{0}^{\infty}(\|\phi\|_{H^{1}_{w}}^{2}+\|\phi_{t}\|_{L^{2}_{w}}^{2})dt<+\infty.$

###### Proof.

From (3.2) we see that $F(\phi)\geq 0$ an using the inequality (3.2) we obtain
that

$\dfrac{d\mathcal{I}(\phi)}{dt}\geq\int_{0}^{\infty}r^{2}\left(\dfrac{1}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r+4}{2r(1+r)^{4}}\phi^{2}\right)-C\dfrac{r(r+2)}{2(1+r)^{2}}\phi^{4}.$

Since we have supposed that $\sup_{t\geq 0}\|\phi(t)\|_{H^{1}\cap
L^{\infty}}\leq\varepsilon$ we have

$\displaystyle|\phi(t,r)|\leq$ $\displaystyle~{}{}\varepsilon\quad\forall
t,r\geq 0,$ $\displaystyle\|\phi(t)\|_{H^{1}}\leq$
$\displaystyle~{}{}\varepsilon\quad\forall t\geq 0,$

and from Lemma 4.2 we have that

$r|\phi(r)|\leq C\|\phi\|_{H^{1}}.$

Gathering both inequalities we obtain

$\phi(r)^{2}\leq\dfrac{(1+C)\varepsilon^{2}}{(1+r)^{2}},$

and hence

$\displaystyle\dfrac{d\mathcal{I}(\phi)}{dt}\geq$
$\displaystyle~{}{}\int_{0}^{\infty}r^{2}\left(\dfrac{1}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r+4}{2r(1+r)^{4}}\phi^{2}\right)-(1+C)\dfrac{r(r+2)}{2(1+r)^{4}}\varepsilon^{2}\phi^{2}$
$\displaystyle\geq$
$\displaystyle~{}{}\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r(r+2)}{2(1+r)^{4}}(1-(1+C)\varepsilon^{2})\phi^{2}$
$\displaystyle\geq$ $\displaystyle~{}{}\|\phi\|^{2}_{H_{w}^{1}},$

provided that $\varepsilon$ is small enough. Notice that, since $\phi(t,r)$ is
uniformly bounded and $f$ is $C^{1}$ there existe a constant $M>0$ such that

$|f(\phi(t))|\leq M|\phi(t)|,\quad\forall t\geq 0,$

which implies that $\phi f(\phi)\leq M\phi^{2}$, and we conclude as in the
previous proposition. ∎

##  6\. Proof of main results

Now we can prove Theorem 3.1, Theorem 3.4 and Theorem 3.6.

###  6.1.  Proof of Theorem 3.1 and 3.4

Let

$\mathcal{H}(t)=\int_{0}^{\infty}\psi(\phi^{2}+\phi_{r}^{2}+\phi_{t}^{2}),$

then, we can see that

$\displaystyle\dfrac{d}{dt}\mathcal{H}(t)=$
$\displaystyle\int_{0}^{\infty}2\psi(\phi\phi_{t}+\phi_{r}\phi_{rt}+\phi_{t}\phi_{tt})$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}2\psi\left(\phi\phi_{t}+\phi_{r}\phi_{rt}+\phi_{t}\phi_{rr}+\dfrac{2}{r}\phi_{t}\phi_{r}-\phi_{t}f(\phi)\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}2\psi\left(\phi\phi_{t}+\phi_{r}\phi_{rt}+\dfrac{2}{r}\phi_{t}\phi_{r}-\phi_{t}f(\phi)\right)dr+\int_{0}^{\infty}2\psi\phi_{t}\phi_{rr}.$

Using Lemma 4.1 we have

$\int_{0}^{\infty}2\psi\phi_{t}\phi_{rr}=-\int_{0}^{\infty}2\phi_{r}(\psi\phi_{tr}+\psi^{\prime}\phi_{t}),$

and hence

$\displaystyle\dfrac{d}{dt}\mathcal{H}(t)=2\int_{0}^{\infty}\psi(\phi\phi_{t}-\phi_{t}f(\phi))+\left(\dfrac{2\psi}{r}-\psi^{\prime}\right)\phi_{t}\phi_{r}$

Since $\psi(r)=\dfrac{r^{2}}{(1+r)^{4}}$, we have that

$\mathcal{H}(t)=\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{4}}(\phi^{2}+\phi_{r}^{2}+\phi_{t}^{2})=\|\phi\|_{H^{1}_{w}}^{2}+\|\phi_{t}\|_{L^{2}_{w}}^{2},$

and

$\dfrac{d}{dt}\mathcal{H}(t)=2\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{4}}(\phi\phi_{t}-\phi_{t}f(\phi))+\dfrac{4r^{2}}{(1+r)^{5}}\phi_{r}\phi_{t}.$

As we see, whether $f$ is globally Lipschitz or $f$ is $C^{1}$ and
$\|\phi(t)\|_{L^{\infty}}$ is uniformly bounded in time then we have

$|f(u)|\lesssim|u|.$

Thus, we see that

$\displaystyle\left|\dfrac{d}{dt}\mathcal{H}(t)\right|\lesssim$
$\displaystyle\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{4}}(|\phi||\phi_{t}|+|\phi_{t}||f(\phi)|)+\dfrac{4r^{2}}{(1+r)^{5}}\phi_{r}\phi_{t}$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{\infty}\dfrac{r^{2}}{(1+r)^{4}}(\phi^{2}+\phi_{t}^{2}+\phi^{2})+\dfrac{4r^{2}}{(1+r)^{4}}(\phi_{r}^{2}+\phi_{t}^{2})$
$\displaystyle\lesssim$ $\displaystyle~{}{}\mathcal{H}(t).$

From Proposition 5.5 and 5.6 there exists a sequence $t_{n}\to\infty$ such
that $H(t_{n})\to 0$. Integrating the inequality above on $[t,t_{n}]$we see
that

$\displaystyle|\mathcal{H}(t_{n})-\mathcal{H}(t)|=$
$\displaystyle\left|\int_{t}^{t_{n}}\dfrac{d}{dt}\mathcal{H}(s)ds\right|$
$\displaystyle\leq$
$\displaystyle\int_{t}^{t_{n}}\left|\dfrac{d}{dt}\mathcal{H}(s)ds\right|\lesssim\int_{t}^{t_{n}}\mathcal{H}(s)ds,$

and passing to limit as $n\to\infty$ we have

$\mathcal{H}(t)\leq\int_{t}^{\infty}\mathcal{H}(s)ds,$

and hence
$\displaystyle\lim_{t\to\infty}\mathcal{H}(t)=\displaystyle\lim_{t\to\infty}\left(\|\phi\|_{H^{1}_{w}}^{2}+\|\phi_{t}\|_{L^{2}_{w}}^{2}\right)=0$.
To conclude the proof is enough to note that for any $R>0$ we have

$\displaystyle\|(\phi,\phi_{t})\|_{H^{1}\times L^{2}(B(0,R))}^{2}=$
$\displaystyle\int_{B(0,R)}(\phi^{2}+\phi_{r}^{2}+\phi_{t}^{2})dx$
$\displaystyle=$
$\displaystyle~{}{}4\pi\int_{0}^{R}r^{2}(\phi^{2}+\phi_{r}^{2}+\phi_{t}^{2})dr$
$\displaystyle\leq$
$\displaystyle~{}{}4\pi(1+R)^{4}\int_{0}^{R}\dfrac{r^{2}}{(1+r)^{4}}(\phi^{2}+\phi_{r}^{2}+\phi_{t}^{2})dr$
$\displaystyle\leq$ $\displaystyle~{}{}4\pi(1+R)^{4}\mathcal{H}(t),$

and the result follows.

###  6.2.  Proof of Theorem 3.6: Existence

Let $\phi(t,x)$ be the local solution to (1.3) on $[0,T)\times\mathbb{R}^{3}$
given by Theorem 4.5. We shall prove that for $\varepsilon>0$ small enough we
can extend this solution smoothly to $[0,T]\times\mathbb{R}^{3}$. First notice
that we have the identity

$\partial_{t}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{|\nabla\phi|^{2}}{2e^{2e^{2Ht}}}+F(\phi)\right)-\text{div}\left(\dfrac{\phi_{t}\nabla\phi}{e^{2Ht}}\right)=-3H\phi_{t}^{2}-H\dfrac{|\nabla\phi|^{2}}{e^{2Ht}},$

where the divergence is taken in the spatial variables. We denote

$K(t_{0},x_{0})=\\{(t,x)\in\mathbb{R}^{4}\;|\;t\leq t_{0},\quad H|x-x_{0}|\leq
e^{-Ht}-e^{-Ht_{0}}\\},$

the backward light cone and

$M(t_{0},x_{0})=\\{(t,x)\in\mathbb{R}^{4}\;|\;t\leq t_{0},\quad
H|x-x_{0}|=e^{-Ht}-e^{-Ht_{0}}\\},$

its lateral boundary. Then, integrating on

$K_{s}^{t}(t_{0},x_{0})=\\{(t,x)\in\mathbb{R}^{4}\;|\;H|x-x_{0}|\leq
e^{-Ht}-e^{-Ht_{0}}\\}\cap[s,t]\times\mathbb{R}^{3};$

and applying the divergence theorem we see that

$\displaystyle\int_{B(x_{0},R(t))}e(t)dx+\int_{M_{s}^{t}}\dfrac{1}{\sqrt{1+e^{2Ht}}}\left(\dfrac{\left|\phi_{t}\frac{x-x_{0}}{|x-x_{0}|}-\nabla\phi\right|^{2}}{2}+F(\phi)\right)dS=$
$\displaystyle\int_{B(x_{0},R(s))}e(s)dx-\int_{M_{s}^{t}}3H\phi_{t}^{2}+H\dfrac{|\nabla\phi|^{2}}{e^{2Ht}},$

where $R(t)=\frac{e^{-Ht}-e^{-Ht_{0}}}{H}$, $M_{s}^{t}$ is the lateral
boundary of the truncated cone $K_{s}^{t}$ and

$e(t)=\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{|\nabla\phi|^{2}}{2e^{2e^{2Ht}}}+F(\phi)\right)(t,x)$

is the energy density. The last identity implies that if $\phi=0$ on
$B(x_{0},R(s))$ then $\phi=0$ on $B(x_{0},R(t))$ for every $t\in[s,t_{0}]$,
that is, $\phi$ have finite speed of propagation and then, if the initial data
have compact support then $\phi$ has it too for every time where it is
defined. From the well posedness theory we have in addition that the solution
is smooth in the spatial variables for every time.

Now we must to show that

$\sup_{t\in[0,T)}\|\phi(t)\|_{\infty}<\infty.$

To do this we shall estimate the $H^{2}$ norm of $\phi$. Notice that from the
hypothesis on the initial data there exist some constant such that

$\|(\phi,\phi_{t})(0)\|_{H^{2}\times H^{1}}\leq\dfrac{C_{0}\varepsilon}{4}.$

To estimate $\|\phi(t)\|_{H^{2}}$ suppose that

$\sup_{t\in[0,T)}\|\phi(t)\|_{H^{2}}\leq C_{0}\varepsilon$

for the same constant as above. We will show that we can improve this estimate
to obtain that

$\sup_{t\in[0,T)}\|\phi(t)\|_{H^{2}}\leq\dfrac{C_{0}\varepsilon}{2}.$

For this we note that from (1.4) if $\varepsilon>0$ is small enough we have
that

$\dfrac{\|\phi_{t}\|^{2}_{L^{2}}}{2}+\dfrac{\|\nabla\phi\|_{L^{2}}^{2}}{2e^{2Ht}}\leq\varepsilon^{2}\left(\dfrac{\|h\|_{L^{2}}^{2}}{2}+\dfrac{\|\nabla
g\|^{2}_{L^{2}}}{2e^{2Ht}}\right)+\varepsilon^{4}\int g^{4},$

and in consequence

$\sup_{t\in[0,T)}\|\phi(t)\|_{H^{1}}\leq(1+T)\dfrac{C_{0}(\varepsilon+\varepsilon^{2})}{4}.$

To estimate the $H^{2}$ norm note that $u_{i}=\partial_{x_{i}}\phi$ satisfies
the equation

$\partial_{t}^{2}u_{i}+3H\partial_{t}u_{i}-\dfrac{\Delta
u_{i}}{e^{2Ht}}+f^{\prime}(\phi)u_{i}=0,$

and from Lemma 4.3 we have

$\displaystyle\dfrac{\|u_{it}\|_{L^{2}}^{2}}{2}+\dfrac{\|\nabla
u_{i}\|^{2}}{2e^{2Ht}}\leq$
$\displaystyle\varepsilon^{2}e^{ct}\left(\dfrac{\|\partial_{x_{i}}h\|_{L^{2}}^{2}}{2}+\dfrac{\|\partial_{x_{i}}\nabla
g\|^{2}}{2e^{2Ht}}\right)+\int_{0}^{t}e^{c(t-s)}\|f^{\prime}(\phi)u_{i}\|_{L^{2}}^{2}ds$
$\displaystyle\leq$
$\displaystyle\varepsilon^{2}e^{ct}\left(\dfrac{\|\partial_{x_{i}}h\|_{L^{2}}^{2}}{2}+\dfrac{\|\partial_{x_{i}}\nabla
g\|^{2}}{2e^{2Ht}}\right)+\varepsilon^{4}\int_{0}^{t}e^{c(t-s)}\|\phi^{2}u_{i}\|_{L^{2}}^{2}ds,$

and in the same way as before we obtain that

$\sup_{t\in[0,T)}\|\partial_{x_{i}}\phi(t)\|_{H^{1}}\leq
C(T)\dfrac{C_{0}(\varepsilon+\varepsilon^{2})}{4}.$

This implies, for $\varepsilon>0$ small enough, that

$\sup_{t\in[0,T)}\|\phi(t)\|_{H^{2}}\leq\dfrac{C_{0}\varepsilon}{2},$

as we wanted. Note that applying the same argument as above, in addition to
Lemma 4.3 we obtain that all the spatial derivatives of $\phi$ are bounded in
time. Additionally, since $u=\phi_{t}$ satisfies

$\partial_{t}^{2}u+3H\partial_{t}u-\dfrac{\Delta
u}{e^{2Ht}}+2H\dfrac{\Delta\phi}{e^{2Ht}}+f^{\prime}(\phi)u=0$

we can apply the same idea to prove that the $H^{2}$ norm of $\phi_{t}$ is
uniformly bounded in time, due to since $\phi$ is uniformly bounded in $H^{2}$
then $\Delta\phi\in L^{1}([0,T];L^{2})$. This allow us to extend $\phi$ to a
smooth function on $[0,T]$, and from the finite speed of propagation property
we have that $\phi(T,\cdot)$ is a smooth function with compact support. In
consequence we can extend the solution globally in time.

###  6.3.  Proof of Theorem 3.6: Decay

We first prove (3.6). For $t_{0}>0$ $\sigma<-1$ and $b=1+\sigma$ in Corollary
5.4, which yields

$\mathcal{J}(\phi)(t)=\int_{0}^{\infty}r^{2}(1+\tanh(r+\sigma
t+(1+\sigma)t_{0}))\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right),$

and

$\dfrac{d\mathcal{J}(\phi)}{dt}(t)\leq(1+\sigma)\int_{0}^{\infty}r^{2}\,\text{sech}\,^{2}(r+\sigma
t+(1+\sigma)t_{0})\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right).$

Since $\sigma<-1$ we have that $\mathcal{J}(\phi)$ is decreasing on
$[2,t_{0}]$. In particular

$\displaystyle\int_{0}^{\infty}r^{2}(1+\tanh(r+(1+2\sigma)t_{0}))\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)(t_{0},r)$
$\displaystyle\leq\int_{0}^{\infty}r^{2}(1+\tanh(r+2\sigma+(1+\sigma)t_{0}))\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)(2,r).$

Since $1+\sigma<0$ we have that

$\lim_{t_{0}\to\infty}(1+\tanh(r+2\sigma+(1+\sigma)t_{0}))=0.$

By dominated convergence theorem we have that the right hand side converges to
0 as $t_{0}\to\infty$, and consequently we get

$\lim_{t\to\infty}\int_{0}^{\infty}r^{2}(1+\tanh(r+(1+2\sigma)t))\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)(t,r)=0.$

To conclude it is enough to note that if $r>-(1+2\sigma)t$ then

$1+\tanh(r+(1+2\sigma)t)\geq 1,$

and then

$\displaystyle\int_{R(t)}\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)$
$\displaystyle\leq\int_{-(1+2\sigma)t}^{\infty}r^{2}(1+\tanh(r+(1+2\sigma)t))\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right)$
$\displaystyle\leq\int_{0}^{\infty}r^{2}(1+\tanh(r+(1+2\sigma)t))\left(\dfrac{\phi_{t}^{2}}{2}+\dfrac{\phi_{r}^{2}}{2e^{2Ht}}+F(\phi)\right).$

Since we have proved that the right hand side converges to 0 we got the
desired decay (3.6).

Now we prove (3.7). Let $R>0$ fixed. Lemma 5.2 with
$\varphi(t,r)=\varphi(r)\geq 0$ yields for some $C>0$

$\displaystyle\dfrac{d\mathcal{J}(\phi)}{dt}=$
$\displaystyle~{}{}-H\int_{0}^{\infty}r^{2}\varphi(r)\left(3\phi_{t}^{2}+\dfrac{\phi_{r}^{2}}{e^{2Ht}}\right)-\int_{0}^{\infty}r^{2}\varphi^{\prime}(r)\dfrac{\phi_{t}\phi_{r}}{e^{2Ht}}$
$\displaystyle\leq$
$\displaystyle~{}{}-3H\int_{0}^{\infty}r^{2}\phi_{t}^{2}\left(\varphi(r)-Ce^{-Ht}\varphi^{\prime}(r)\right)-H\int_{0}^{\infty}r^{2}\dfrac{\phi_{r}^{2}}{e^{2Ht}}\left(\varphi(r)-\varphi^{\prime}(r)\right).$

Choosing $\varphi_{0}(r),\varphi(r)$ satisfying

$\varphi_{0}(r)=1,\quad r\leq 1,\quad\varphi_{0}^{\prime}\leq
0,\quad\varphi_{0}(r)=e^{-r},\quad r\geq 2,\quad\varphi(r)=\varphi_{0}(r/R),$

If $R$ and $t$ are large,

$\dfrac{d\mathcal{J}(\phi)}{dt}\leq-\frac{3}{2}H\int_{0}^{\infty}r^{2}\varphi(r)\phi_{t}^{2}-\frac{1}{2}H\int_{0}^{\infty}r^{2}\varphi(r)\dfrac{\phi_{r}^{2}}{e^{2Ht}}.$

Consequently, for some $t_{0}>0$ large,

$H\int_{t_{0}}^{\infty}\int_{0}^{\infty}r^{2}\varphi(r)\left(\phi_{t}^{2}+\dfrac{\phi_{r}^{2}}{e^{2Ht}}\right)dt<+\infty.$

Again, we have now

$\dfrac{d\mathcal{J}(\phi)}{dt}\lesssim
H\int_{0}^{\infty}r^{2}\varphi(r)\left(3\phi_{t}^{2}+\dfrac{\phi_{r}^{2}}{e^{2Ht}}\right),$

following the same ideas as in previous proofs, we get the desired result.

##  7\. Applications

In this section we shall see how Theorems 3.1 and 3.4 can be used to study the
space-time dynamics of cosmological models. In particular, we shall prove here
Corollaries 3.2 and 3.5.

###  7.1. Slow-roll in strongly defocusing models

First of all, we will need the following lemma about the potentials associated
to certain models. Recall that from Section 2 we have

$\displaystyle F_{1,n}(s)=$ $\displaystyle~{}{}(1-e^{-s})^{2n},\quad
n=1,2,3,\ldots$ $\displaystyle F_{2,1}(s)=$
$\displaystyle~{}{}\tanh^{2}(s),\quad s\in\mathbb{R},$ $\displaystyle
F_{6,q}(s)=$ $\displaystyle~{}{}\dfrac{1}{q}[(1+s^{2})^{q/2}-1],\quad
q\in[-1,1],\quad q\neq 0.$

######  Lemma 7.1.

There exists a constant $C>0$ such that

$\displaystyle sf_{1,n}(s)\leq$ $\displaystyle~{}{}Cs^{4},\quad
s\in[-1,1],\quad n\geq 2,$ $\displaystyle 2F_{2,1}(s)-sf_{2,1}(s)\geq$
$\displaystyle~{}{}0,\quad s\in\mathbb{R},$ $\displaystyle
2F_{6,q}(s)-sf_{6,q}(s)\geq$ $\displaystyle~{}{}0,\quad
q\in[-1,1],s\in\mathbb{R}.$

###### Proof.

For $F_{2,1}$ by the symmetry of the function is enough to prove the
inequality for $s\geq 0$. We have

$\displaystyle 2F_{2,1}(s)-sf_{2,1}(s)$
$\displaystyle=2\tanh^{2}(s)-2s\tanh(s)\,\text{sech}\,^{2}(s)$
$\displaystyle=2\tanh(s)(\tanh(s)-s\,\text{sech}\,^{2}(s)),$

and since $|\tanh(s)|\leq|s|$ we have the inequality. For $F_{1,n}$ notice
that for $s\in[-1,1]$

$\displaystyle sf_{1,n}(s)=$ $\displaystyle~{}{}2ns(1-e^{-s})^{2n-1}e^{-s}$
$\displaystyle\leq$
$\displaystyle~{}{}\left(\sup_{y\in[-1,1]}2n(1-e^{-y})^{2n-4}\right)s(1-e^{-s})^{3},$

and therefore, from the classic inequality $1+s\leq e^{s}$ we see that for
$s\in[0,1]$ we have

$sf_{1,n}(s)\leq\left(\sup_{y\in[-1,1]}2n(1-e^{-y})^{2n-4}\right)s^{4}.$

For $s\in[-1,0]$ we define

$g_{a}(s)=s(1-e^{-s})^{3}-as^{4},$

and we prove that there exists some $a>0$ such that this $g_{a}(s)\leq 0$ on
$[-1,0]$. Indeeed, a straightforward calculation gives us that

$g_{a}(0)=g_{a}^{\prime}(0)=g_{a}^{\prime\prime}(0)=g_{a}^{\prime\prime\prime}(0)=0,$

and

$g_{a}^{(iv)}(s)=h(s)-24a,$

for some function $h(s)$. We can take $a>0$ as large as we need to guarantee
that $g^{(iv)}(s)\leq 0$ on $[-1,0]$, and this implies that $g_{a}(s)\leq 0$
on $[-1,0]$, giving the desired inequality.

For $F_{6,q}$ we have that

$g(s)=2F_{6,q}(s)-sf_{6,q}(s)=(1+s^{2})^{q/2-1}\left[\dfrac{2}{q}+s^{2}\left(\dfrac{2}{q}-1\right)\right]-\dfrac{2}{q},$

and we notice that $g(0)=0$. By the symmetry of the function it is sufficient
to prove that $g^{\prime}(s)\geq 0$ for $s\geq 0$

$\displaystyle g^{\prime}(s)=$ $\displaystyle f_{6,q}(s)-sf^{\prime}_{6,q}(s)$
$\displaystyle=$ $\displaystyle\left(2-q\right)s^{3}(1+s^{2})^{q/2-2},$

and since $q\in[-1,1]$ we conclude. ∎

######  Remark 7.1.

For the $T_{n}$-models and Axion Monodromy potential $F_{6,q}$ we always have
global solutions for any initial data in $H^{1}\times L^{2}$ due to Theorem
4.4.

######  Remark 7.2.

The potentials in Lemma 7.1 correspond to the $E$-model, $T$-model and Axion-
Monodromy models, respectively.

######  Corollary 7.2 (Nonexistence of standing solitons).

The scalar field models of finite energy solutions associated to $F_{1,1}$ and
$F_{6,q}$, $q\in[-1,1]$, $q\neq 0,$ does not allow for nontrivial stationary
solutions.

###### Proof.

This is a standard result. First note that from Pohozaev’s indentity [22],
every solution $\phi\in H^{1}(\mathbb{R}^{n})$ to

$-\Delta\phi+f(\phi)=0,$

such that $F(\phi)\in L^{1}(\mathbb{R}^{n})$ must satisfy

$\dfrac{n-2}{2}\int|\nabla\phi|^{2}+n\int F(\phi)=0,$

and consequently it does not exist nontrivial stationary solutions of finite
energy precisely when potential is positive. ∎

###  7.2. The general $T$ model

For the $T$-model, when $n=1$ we already know from Lemma 7.1 and Theorem 3.1
that every solution to (1.3) with $H=0$ satisfies (3.1). For $n\geq 2$ the
situation is different.

######  Lemma 7.3.

We have $sf_{2,n}(s)\leq 2ns^{4}$, $s\in\mathbb{R}.$

###### Proof.

We have

$\displaystyle sf_{2,n}(s)$
$\displaystyle=2ns\tanh^{2n-1}(u)\,\text{sech}\,^{2}(s)$
$\displaystyle=2ns\tanh^{3}(s)\tanh^{2n-4}(s)\,\text{sech}\,^{2}(s)\leq
2ns^{4}.$

∎

Hence we can apply Theorem 3.4 to conclude that the same decay occurs as long
as $\|\phi(t)\|_{H^{1}\cap L^{\infty}}$ is small enough. This shows Corollary
3.5 in the case of the $F_{2,n}$ model.

######  Remark 7.3.

Notice that from Theorem 4.4 if we set initial data in $H^{2}\times H^{1}$ we
have a unique global solution to (1.3) with $(\phi,\phi_{t})\in H^{2}\times
H^{1}$ for every time.

######  Remark 7.4.

For the E-model we can apply again Theorem 3.4 to conclude the decay of small
enough solutions to (1.3) when $n\geq 2$. This is the missing part of
Corollary 3.5. In this case it has not been possible to ensure that solutions
are global in time for any initial data. However, Theorem 4.6 assures the
existence of global solutions when initial data is regular and small enough.

When $n=1$ we cannot apply neither Theorem 3.1 nor 3.4 for $F_{1,1}$ because
in this case we have no longer the required sign condition and
$f_{1,1}^{\prime}$ is not well behaved near the origin.

###  7.3. The non-minimal coupling and Hilltop models

For the non-minimal coupling model

$F(\phi)=\lambda^{2}\phi^{4}+\beta^{2}\phi^{2}$

we give a slightly modified argument to the presented in Theorem 3.4 to
conclude the decay of small enough solutions. Since $2F(\phi)-\phi
f(\phi)=-\lambda^{2}\phi^{4}$ we have in (5.12) that

$\dfrac{d\mathcal{I}(\phi)}{dt}=\int_{0}^{\infty}r^{2}\left(\dfrac{1}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r+4}{2r(1+r)^{4}}\phi^{2}\right)-\lambda^{2}\dfrac{r(r+2)}{2(1+r)^{2}}\phi^{4}$

Supposing that $\sup_{t\geq 0}\|\phi(t)\|_{H^{1}\cap L^{\infty}}<\varepsilon$
and using that

$\phi(r)^{2}\leq\dfrac{(1+C)\varepsilon^{2}}{(1+r)^{2}},$

we have

$\displaystyle\dfrac{d\mathcal{I}(\phi)}{dt}=$
$\displaystyle~{}{}\int_{0}^{\infty}r^{2}\left(\dfrac{1}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r+4}{2r(1+r)^{4}}\phi^{2}\right)-(1+C)\lambda^{2}\varepsilon^{2}\dfrac{r(r+2)}{2(1+r)^{4}}\phi^{2}$
$\displaystyle\geq$
$\displaystyle~{}{}\int_{0}^{\infty}\left(\dfrac{r^{2}}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r(r+2)}{2(1+r)^{4}}\phi^{2}(1-(1+C)\lambda^{2}\varepsilon^{2})\right)$
$\displaystyle\geq$ $\displaystyle~{}{}\|\phi\|_{H_{w}^{1}}^{2},$

provided that $\varepsilon$ is small enough. Thus, we can conclude as in
Theorem 3.4. Notice that the hypothesis (3.4) is also needed to prove that

$\int_{0}^{\infty}\|\phi_{t}\|_{L^{2}_{w}}dt<\infty$

as in Proposition 5.6. For the Hilltop model with $n=2$ we have a simpler
situation. We have that $2F(\phi)-\phi f(\phi)=2\phi^{4}$ and therefore

$\displaystyle\dfrac{d\mathcal{I}(\phi)}{dt}=$
$\displaystyle\int_{0}^{\infty}r^{2}\left(\dfrac{1}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r+4}{2r(1+r)^{4}}\phi^{2}\right)+2\dfrac{r(r+2)}{2(1+r)^{2}}\phi^{4}$
$\displaystyle\geq$
$\displaystyle\int_{0}^{\infty}r^{2}\left(\dfrac{1}{(1+r)^{2}}\phi_{r}^{2}+\dfrac{r+4}{2r(1+r)^{4}}\phi^{2}\right)$
$\displaystyle\geq$ $\displaystyle\|\phi\|_{H^{1}_{w}}.$

Supposing (3.4) we can conclude as in Theorem 3.4.

###  7.4. The Natural Inflation and Axion potentials

Assume small data $\phi$. Observe that in this case $F_{3,\pm}\geq 0$ in
(2.4), and

$2F_{3,-}(\phi)-\phi f_{3,-}(\phi)=\frac{1}{12}\phi^{4}+O(\phi^{6}).$

In the other case, we must take out the nonzero value of the potential at
infinity to get finite energy, considering $F_{3,+}(\phi)=\cos\phi-1$. We get

$2F_{3,+}(\phi)-\phi f_{3,+}(\phi)=-\frac{1}{12}\phi^{4}+O(\phi^{6}).$

In the former case, natural inflation, by virtue of (5.12) we get the desired
result applying the same ideas as in the previous subsection. The case of
Axion potential remains an interesting open problem.

###  7.5. The D-brane model

Finally for the D-brane model $F_{4,n}$ in (2.5), since the potential and his
derivative are singular in the origin we look for solutions of the form
$\phi=1+v$, where we suppose that $v(t)\in H^{2}$ and

$\|v(t)\|_{H^{2}}<1,$

for all times. Notice that this bound ensures control of the $L^{\infty}$ norm
of $v$. In this case we have that the function $v$ satisfies (see (2.6))

$\partial_{t}^{2}v-\partial_{r}^{2}v-\dfrac{2}{r}\partial_{r}v+2n\left(\dfrac{1}{(1+v)^{2n+1}}-1\right)=0.$

We conclude that

$2\tilde{F}_{4,n}(v)-v\tilde{f}_{4,n}(v)=\begin{cases}\frac{-2v^{3}(v+2)}{(1+v)^{3}}&n=1\\\
-\frac{2v^{3}(10+15v+9v^{2}+2v^{3})}{(1+v)^{5}}&n=2.\end{cases}$

Consequently, Theorems 3.1 and 3.4 remain inconclusive in this setting.

###  7.6. The potential $F_{7}$

Now we consider the logarithm potential present in Corollary 3.2. Recall that

$F_{7}(s)=\frac{1}{2}\log(1+s^{2}).$

Its derivative is clearly Lipschitz and from the well-known inequality (valid
for $x>0$)

$1-\dfrac{1}{x}\leq\log(x),$

we have that

$2F_{7}(\phi)-\phi
f_{7}(\phi)=\log(1+\phi^{2})-\dfrac{\phi^{2}}{1+\phi^{2}}\geq 0,$

and in consequence for any initial data in $H^{1}\times L^{2}$ the solution is
global in time and satisfies (3.1).

##  Appendix A Proof of local and global existence

In this section, we sketch the proof of Theorem 4.4 and 4.5. This proof is
essentially taken from Evans [5].

###### Proof of Theorem 4.4.

It is well known that for $H\geq 0$, $h\in L^{1}([0,T];H^{s})$ and $(f,g)\in
H^{s+1}\times H^{s}$ the linear wave equation

$\displaystyle\partial_{t}^{2}\phi+3H\partial_{t}\phi-\dfrac{\Delta\phi}{e^{2Ht}}=$
$\displaystyle~{}{}h$ $\displaystyle(\phi(0),\phi_{t}(0))=$
$\displaystyle~{}{}(f,g),$

has a unique solution $\phi\in C([0,T];H^{s+1})\cap C^{1}([0,T],H^{s})$ such
that

$\|\phi(t)\|_{H^{s+1}}+\|\phi_{t}(t)\|_{H^{s}}\leq
C(1+T)\left(\|f\|_{H^{s+1}}+\|g\|_{H^{s}}+\int_{0}^{t}\|h(r)\|_{H^{s}}dr\right).$
(A.1)

See for example [18]. Using this, consider the Banach space

$X=C([0,T];H^{1})\cap C^{1}([0,T],L^{2}),$

with norm

$\|\phi\|_{X}=\sup_{t\in[0,T]}(\|\phi(t)\|_{H^{1}}+\|\phi_{t}\|_{L^{2}}),$

for some $T>0$ to be chosen later. We define the operator $A:X\to X$ by
$Av=\phi$, where $\phi$ is the unique solution to

$\displaystyle\partial_{t}^{2}\phi+3H\partial_{t}\phi-\dfrac{\Delta\phi}{a}=$
$\displaystyle f(v),$ $\displaystyle(\phi(0),\phi_{t}(0))=$
$\displaystyle(h,g).$

Note that $A$ is well defined because, since $f$ is globally Lipschitz and
$f(0)=0$, we have that $|f(v)|\leq M|v|$ and hence $f(v)\in
L^{1}([0,T],L^{2})$. Thus, given $v_{1},v_{2}\in X$ we have that
$w=Av_{1}-A_{v}2$ satisfies

$\displaystyle\partial_{t}^{2}w+3H\partial_{t}w-\dfrac{\Delta w}{a}=$
$\displaystyle~{}{}f(v_{1})-f(v_{2})$ $\displaystyle(w(0),w_{t}(0))=$
$\displaystyle~{}{}(0,0)$

and so we can use the estimate (A.1), obtaining

$\displaystyle\|w(t)\|_{H^{1}}+\|w_{t}(t)\|_{L^{2}}\leq$
$\displaystyle~{}{}C(1+T)\int_{0}^{t}\|f(v_{1}(s))-f(v_{2}(s))\|_{L^{2}}ds$
$\displaystyle\leq$
$\displaystyle~{}{}CM(1+T)t\sup_{s\in[0,T]}\|v_{1}(s)-v_{2}(s)\|_{L^{2}},$

where $M>0$ is the Lipschitz constant of $f$. Thus we have that

$\|w\|_{X}=\|Av_{1}-Av_{2}\|_{X}\leq CM(1+T)T\|v_{1}-v_{2}\|_{X},$

and taking $T$ small enough such that $CM(1+T)T<1$ by Banach’s fixed point
theorem we have that there exist $\phi\in X$ such that

$\displaystyle\partial_{t}^{2}\phi+3H\partial_{t}\phi-\dfrac{\Delta\phi}{e^{2Ht}}=$
$\displaystyle~{}{}f(\phi),$ $\displaystyle(\phi(0),\phi_{t}(0))=$
$\displaystyle~{}{}(h,g).$

Since $T$ does not depend on $h,g$ we can extend this solution for all times,
concluding the proof. ∎

###### Proof of Theorem 4.5.

Proceeding as above, consider the Banach space

$X=C([0,T];H^{2})\cap C^{1}([0,T];H^{1}),$

with norm

$\|\phi\|_{X}=\sup_{t\in[0,T]}(\|\phi(t)\|_{H^{2}}+\|\phi_{t}(t)\|_{H^{1}}),$

and consider the subset $Y=\\{\phi\in X\;|\;\|\phi\|_{X}\leq R\\}$ with the
metric induced by the norm of $X$. Define the operator $A:Y\to X$ by
$Av=\phi$, where $\phi$ is the unique solution to

$\displaystyle\partial_{t}^{2}\phi+3H\partial_{t}\phi-\dfrac{\Delta\phi}{e^{2Ht}}=$
$\displaystyle~{}{}f(v)$ $\displaystyle(\phi(0),\phi_{t}(0))=$
$\displaystyle~{}{}(h,g),$

where $(h,g)\in H^{2}\times H^{1}(\mathbb{R}^{3})$. In order to see that
$f(v)\in H^{1}$ notice that from Sobolev’s embedding we have that
$H^{2}(\mathbb{R}^{3})\hookrightarrow C(\mathbb{R}^{3})\cap
L^{\infty}(\mathbb{R}^{3})$, and then $|v(t,x)|\leq M$ for all
$(t,x)\in[0,T]\times\mathbb{R}^{3}$. Note that this constant is independent of
$v$, because $\|v\|_{X}\leq R$. Since $f$ is of class $C^{2}$ (and in
particular locally Lipschitz continuous) we have

$|f(v)|\lesssim|v|,$

and $f(v)\in L^{2}$. The same argument allows us to prove that

$|\nabla f(v)|=|f^{\prime}(v)\nabla v|\lesssim|\nabla v|,$

and thus $f(v)\in L^{1}([0,T];H^{1})$ and $A$ is well defined. Using (A.1) for
$s=1$ we see that

$\displaystyle\|w\|_{X}\leq$
$\displaystyle~{}{}C(1+T)\left(\|h\|_{H^{2}}+\|g\|_{H^{1}}+\int_{0}^{T}\|f(v(s))\|_{H^{1}}ds\right)$
$\displaystyle\leq$
$\displaystyle~{}{}C(1+T)\left(\|h\|_{H^{2}}+\|g\|_{H^{1}}+TM\|v\|_{X}\right),$

where $M>0$ depends only on $f$ and $R$. Thus, taking $R\geq
C(1+T)\|(h,g)\|_{H^{2}\times H^{1}}$ and $T>0$ small enough we have that
$\|w\|_{X}\leq R$. Now, for $v_{1},v_{2}\in Y$ we have that $w=Av_{1}-Av_{2}$
satisfies

$\displaystyle\partial_{t}^{2}w-\Delta w=$
$\displaystyle~{}{}f(v_{1})-f(v_{2})$ $\displaystyle(w(0),w_{t}(0))=$
$\displaystyle~{}{}(0,0),$

and using (A.1) again we have that

$\displaystyle\|w(t)\|_{H^{2}}+\|w_{t}(t)\|_{H^{1}}\leq$
$\displaystyle~{}{}C(1+t)\int_{0}^{t}\|f(v_{1}(s))-f(v_{2}(s))\|_{H^{1}}ds$
$\displaystyle\leq$
$\displaystyle~{}{}C(1+t)\int_{0}^{t}(\|f(v_{1})-f(v_{2})\|_{L^{2}}+\|f^{\prime}(v_{1})\nabla
v_{1}-f^{\prime}(v_{2})\nabla v_{2}\|_{L^{2}})ds.$

Since both $f$ and $f^{\prime}$ are locally Lipschitz continuous we have that

$\|f(v_{1})-f(v_{2})\|_{L^{2}}\lesssim\|v_{1}-v_{2}\|_{L^{2}},$

and

$\displaystyle\|f^{\prime}(v_{1})\nabla v_{1}-f^{\prime}(v_{2})\nabla
v_{2}\|_{L^{2}}\leq$ $\displaystyle~{}{}\|f^{\prime}(v_{1})\nabla
v_{1}-f^{\prime}(v_{1})\nabla
v_{2}\|_{L^{2}}+\|(f^{\prime}(v_{1})-f^{\prime}(v_{2}))\nabla v_{2}\|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle~{}{}\|\nabla v_{1}-\nabla
v_{2}\|_{L^{2}}+\|v_{1}-v_{2}\|_{L^{2}}.$

This implies that

$\|w\|_{X}\leq CN(1+T)T\|v_{1}-v_{2}\|_{X}.$

and taking $T$ small enough we have that $A:Y\to Y$ is a contraction, and we
conclude as in the proof of Theorem 4.4. By standard arguments this solution
can be extended to a maximal interval $[0,T)$. ∎

## References

  * [1] M. A. Alejo, and C. Maulén, _Decay for Skyrme wave maps_ , Lett. Math. Phys. 112 (2022), no. 5, Paper No. 90, 33 pp.
  * [2] M. A. Amin, R. Easther, H. Finkel, R. Flauger, and M. P. Hertzberg, _Oscillons after Inflation_ , Phys. Rev. Lett. 108, 241302 – Published 14 June 2012.
  * [3] E. Braaten, and H. Zhang, _Axion stars_ , preprint arXiv https://arxiv.org/abs/1810.11473 (2018).
  * [4] F. Chadha-Day; J. Ellis; D. J. E. Marsh, _Axion dark matter: What is it and why now?_. Science Advances. 8 (8): eabj3618 (23 February 2022), https://arxiv.org/abs/2105.01406.
  * [5] L. C. Evans, _Partial Differential Equations_ , Second Edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp. ISBN: 978-0-8218-4974-3.
  * [6] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, J. Silk, _In the Realm of the Hubble tension–a Review of Solutions_ , preprint 2021 https://arxiv.org/pdf/2103.01183.pdf.
  * [7] A. H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D, vol. 23, pp. 347–356, (1981).
  * [8] E. Hubble, _A relation between distance and radial velocity among extra-galactic nebulae_ , PNAS vol. 15 pp. 168–173, 1929.
  * [9] Hong-Yi Zhang et al JCAP07 (2020) 055.
  * [10] M. Kawasaki, N. Kitajima, and T. T. Yanagida, _Primordial black hole formation from an axionlike curvaton model_ , Phys. Rev. D, vol. 87, no. 6, p. 063519, 2013.
  * [11] M. Kowalczyk, Y. Martel, and C. Muñoz, Nonexistence of small, odd breathers for a class of nonlinear wave equations, Letters in Mathematical Physics, (2017) Vol. 107, Issue 5, 921–931.
  * [12] M. Kowalczyk, Y. Martel, and C. Muñoz, On asymptotic stability of nonlinear waves, Laurent Schwartz seminar notes (2017), see url at http://slsedp.cedram.org/slsedp-bin/fitem?id=SLSEDP_2016-2017____A18_0.
  * [13] D. H. Lyth, _Axions and Inflation: Vaccuum Fluctuations_ , Phys. Rev. D. Vol. 45 no. 10, May 15 1992, p. 3394, https://doi.org/10.1103/PhysRevD.45.3394.
  * [14] G. Palma, S. Sypsas, and C. Zenteno, _Seeding Primordial Black Holes in Multifield Inflation_ , PRL 125, 121301 (2020).
  * [15] A. A. Penzias and R. W. Wilson, _A Measurement of excess antenna temperature at 4080-Mc/s,_ Astrophys. J., vol. 142, pp. 419–421, 1965.
  * [16] Planck Collaboration, _Planck 2018 results. X. Constraints on inflation_ , A&A Volume 641, September 2020.
  * [17] N. Schöneberg, G. F. Abellán, A. Pérez Sánchez, S. J. Witte, V. Poulin, J. Lesgourgues, _The $H0$ Olympics: A fair ranking of proposed models_, Physics Reports Volume 984, 26 October 2022, Pages 1-55 https://arxiv.org/abs/2107.10291.
  * [18] C. D. Sogge, _Lectures on Non-Linear Wave Equations_ , Second edition. International Press, Boston, MA, 2008. x+205 pp. ISBN: 978-1-57146-173-5.
  * [19] A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91, 99 (1980).
  * [20] P. Svrček, and E. Witten, _Axions in string theory_. J. High Energy Phys. 2006, no. 6, 051, 52 pp.
  * [21] K. Tsutaya, Y. Wakasugi, _Blow up of solutions of semilinear wave equations related to nonlinear waves in de Sitter spacetime_ , Partial Differential Equations and Applications 3, Article number: 6 (2022).
  * [22] Mi. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp. ISBN: 0-8176-3913-6.
  * [23] H. Zhang, M. A. Amin, E. J. Copeland, P. M. Saffin and K. D. Lozanov, Classical decay rates of oscillons Journal of Cosmology and Astroparticle Physics (2020)