# Local and Global Existence of Solutions to Scalar Equations on Spatially
Flat Universe as a Background with Non-minimal Coupling

Fiki T. Akbar1,2, Bobby E. Gunara1,2, Muhammad Iqbal2, Hadi Susanto3

1Indonesian Center for Theoretical and Mathematical Physics (ICTMP)
2Theoretical Physics Laboratory
Theoretical High Energy Physics and Instrumentation Research Group,
Faculty of Mathematics and Natural Sciences,
Institut Teknologi Bandung
Jl. Ganesha no. 10 Bandung, Indonesia, 40132
3Department of Mathematical Sciences, University of Essex,
Colchester, CO4 3SQ, United Kingdom

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###### Abstract

We prove the wellposedness of scalar wave equations on spatially flat universe
as a background with nonminimal coupling with the scalar potential turned on
by introducing the $k$-order linear energy and the corresponding energy norm.
In the local case, we show that both the $k$-order linear energy and the
energy norm are bounded for finite time with initial data in $H^{k+1}\times
H^{k}$. Whereas in the global case, we have to add three assumptions related
to the nonminimal coupling constant, the scale factor of spacetimes, and the
form of the scalar that has to be a polynomial with a small positive
parameter. Then, we show that the solution does globally exist with a
particular decay estimate that depends on the scale factor of the spacetimes.
Finally, we provide some physical models that support our general setup.

##  1 Introduction

It is of interest to study the Klein-Gordon equation because it describes the
dynamics of the spinless particle in our universe at quantum level or it can
be viewed as the scalar wave equation at classical level in our universe.
Moreover, to get a more realistic picture we have to include the influence of
geometrical properties of the universe which may also determine by its matter
distribution. It is extremely difficult, however, to solve the Klein-Gordon
equation on the family of four dimensional Friedmann-Robertson-Walker
spacetimes with general couplings even at the classical level. Therefore, we
have to specify both the spacetime and the coupling in order to get a solvable
model. Several simple models in four dimensions have been studied, for
example, in [1, 2, 3].

In this paper we prove the local and global existence of solutions of the
Klein-Gordon equation in higher dimensional spatially flat Friedmann-
Robertson-Walker spacetimes with non-minimal coupling between the scalar
curvature and the scalar field, and the scalar potential turned on. This
additional non-minimal coupling is the simplest generalization of the scalar
field theory on curved spacetimes [4, 5], which can be viewed, for example, as
a result of quantum corrections [6, 7].

The starting point of proving the local and global existence of solutions is
by introducing the $k$-order linear energy and the corresponding energy norm.
In the local case, we show that in order to admit a regular solution both
energy functionals must be bounded below a real constant $C$ for finite time
$T<\infty$ with initial data in $H^{k+1}\times H^{k}$. As we take
$T\to\infty$, namely the global existence, we give three additional
assumptions related to the nonminimal coupling constant, the scale factor of
spacetimes, and the form of the scalar that has to be a polynomial with small
positive parameter. Using these assumptions, we prove that the supremum of the
energy norm is bounded and thus, we could have a decay estimate.

We organize the paper as follows. In Section 2 we briefly review spatially
flat spacetimes in higher dimension. We discuss some local properties of the
real scalar field on higher dimensional spatially flat spacetimes by
introducing $k$-order linear energy and the corresponding energy norm in
Section 3. In Section 4 we provide a proof of the local existence and the
uniqueness of solutions together with a smoothness property. In Section 5 we
prove that the solutions could exist globally and they have a particular decay
estimate. Finally, we discuss some models, in which the global solution does
exist in Section 6.

##  2 Spatially Flat Spacetimes in Higher Dimension

In this section, we shortly review the higher dimensional conformally flat
spacetime which can be constructed by the $D$-dimensional spatially flat
Lorentzian manifold, $\mathcal{M}^{D}$, $D\geq 4$ with standard coordinates,
$x^{\mu}=(x^{0}=t,x^{i})$, $\mu=0,1,\dots,D-1$, $i=1,2,\dots,D-1$, and is
equipped by Lorentzian metric with signature $\\{-1,1,\dots,1\\}$. We can
write down the metric as

$ds^{2}=-dt^{2}+a^{2}(t)\sum_{i=1}^{D-1}dx_{i}^{2}\>,$ (2.1)

with $x^{i}$ being the usual Cartesian coordinates for $\mathrm{IR}^{D-1}$.
Defining a new time coordinate $\tau$ by

$\frac{d\tau}{dt}=\frac{1}{a(t)}\>,$ (2.2)

we can write (2.1) as

$ds^{2}=a^{2}(\tau)\left(-d\tau^{2}+\sum_{i=1}^{D-1}dx_{i}^{2}\right)\>.$
(2.3)

Thus, our spacetime $\mathcal{M}^{D}$ is conformal to flat Minkowski space
$M^{D+1}\simeq\mathrm{IR}\times\mathrm{IR}^{D-1}$. In terms of components, we
can write the metric of $\mathcal{M}^{D}$ as

$g_{\mu\nu}=a^{2}(\tau)\eta_{\mu\nu}\>,$ (2.4)

where $\eta_{\mu\nu}=\mathrm{diag}(-1,1,\dots,1)$ is the component of
Minkowski metric. Furthermore, it is of interest to write down the Ricci
tensor and the scalar curvature related to metric (2.4)

$\displaystyle R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\dot{H}\left(\eta_{\mu\nu}-(D-2)\delta^{0}_{\mu}\;\delta^{0}_{\nu}\right)+H^{2}(D-2)\left(\eta_{\mu\nu}+\delta^{0}_{\mu}\;\delta^{0}_{\nu}\right)\>,$
$\displaystyle R$ $\displaystyle=$
$\displaystyle(D-1)a^{-2}\left(2\dot{H}+(D-2)H^{2}\right)\>,$ (2.5)

respectively, assuming that the scale factor $a(\tau)$ belongs to
$C^{n}$-function with $n\geq 2$ for all $\tau>0$ where $\dot{a}\equiv
da/d\tau$ and we have defined the Hubble parameter $H\equiv\dot{a}/a$.

The higher dimensional Friedmann equations describing an accelerated universe
for single component matter are given by [8]

$\displaystyle\frac{(D-2)(D-1)}{2a^{4}}H^{2}$ $\displaystyle=$ $\displaystyle
8\pi\rho~{},$
$\displaystyle-\frac{(D-2)}{a^{3}}\dot{H}-\frac{(D-2)(D-5)}{2a^{4}}H^{2}$
$\displaystyle=$ $\displaystyle 8\pi P\>,$ (2.6)

together with equation of state $P=w\rho$. The general solutions of this
equation have the form

$a(\tau)=\begin{cases}(\tau+\tau_{0})^{\frac{2}{(D-1)(w+1)-2}}~{},&w\neq-(D-3)/(D-1)\\\
e^{\alpha(\tau+\tau_{0})}~{},&w=-(D-3)/(D-1)\end{cases}$ (2.7)

which will be useful for our analysis in the last part of this paper. Some
examples, which may be considered as the higher dimensional standard model of
cosmology, are listed in Table 1 [8].

Conditions | $w$ | $a(\tau)$
---|---|---
Matter Dominated | $w=0$ | $(\tau+\tau_{0})^{\frac{2}{D-3}}$
$\Lambda$-Dominated | $w=-1$ | $(\tau+\tau_{0})^{-1}$
Radiation Dominated | $w=\frac{1}{D-1}$ | $(\tau+\tau_{0})$

Table 1: Standard Models of Cosmology in higher dimension.

For a matter dominated universe, which is called higher dimensional Einstein-
de Sitter universe, the universe consists only of non-relativistic matter
(dust) and has zero cosmological constant. The Ricci tensor related to metric
(2.4) is given by

$R_{\mu\nu}=\frac{2(D-1)}{(D-3)^{2}\tau^{2}}\left(\eta_{\mu\nu}+(D-2)\delta^{0}_{\mu}\;\delta^{0}_{\nu}\right)\>,$
(2.8)

and the scalar curvature is given by

$R=\frac{4(D-1)}{(D-3)^{2}\tau^{2(D-1)/(D-3)}}\>.$ (2.9)

##  3 Real Scalar Field in Spatially Flat Universe

In this section, we discuss some local properties of the real scalar field on
the higher dimensional spatially flat universe as a background with additional
non-minimal coupling where the coupling interaction of the scalar field $\phi$
is proportional to the scalar curvature of the spacetime. We show that the
nonlinear terms and the $k$-order linear energy are bounded if the energy norm
is also bounded.

The action of our theory has the form

$\mathcal{S}=\int d\tau
dx\sqrt{-g}\left(\frac{1}{2}\partial_{\mu}\phi\>\partial^{\mu}\phi+\frac{\xi}{2}R\phi^{2}-V(\phi)\right)\>,$
(3.1)

where $g$ and $R$ are the determinant and the scalar curvature of the metric
(2.4), respectively. The non-minimal coupling is introduced in the second term
of the right hand side of (3.1) with positive constant $\xi$ 111Model with
positive $\xi$ is called canonical, while that with negative $\xi$ is called
phantom.. The real function $V(\phi)$ denotes the scalar potential which is
assumed to be smooth and satisfy the following conditions,

$\displaystyle(1)$ $\displaystyle V(0)=0~{},$ $\displaystyle(2)$
$\displaystyle\partial_{\phi}V(0)=0~{}.$ (3.2)

The conditions in (3.2) are satisfied by several known scalar potentials such
as the $\phi^{4}$ theory and the sine-Gordon theory.

The equation of motions of scalar field in this case is given by

$\nabla_{\mu}\nabla^{\mu}\phi-\xi R\phi+\partial_{\phi}V(\phi)=0\>,$ (3.3)

where $\nabla_{\mu}$ is a covariant derivative with respect to the metric
(2.3). Using (2.3) and (2.5), we can write the equation of motion in form of
nonlinear waves equation

$\partial_{\tau}^{2}\phi-\Delta\phi=F(\phi,\partial_{\tau}\phi)\>,$ (3.4)

where

$F(\phi,\partial_{\tau}\phi)\equiv-(D-2)H~{}\partial_{\tau}\phi-\xi(D-1)\left(2\dot{H}+(D-2)H^{2}\right)\phi+a^{2}\partial_{\phi}V(\phi)\>,$
(3.5)

and $\Delta$ is Laplacian in $\mathrm{IR}^{D-1}$. The energy-momentum tensor
for system (3.1) is given by

$\displaystyle
T_{\mu\nu}=\nabla_{\mu}\phi\;\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\lambda}\phi\;\nabla^{\lambda}\phi-
g_{\mu\nu}V(\phi)-\xi
G_{\mu\nu}\phi^{2}-\xi\left(g_{\mu\nu}\nabla_{\lambda}\nabla^{\lambda}\phi^{2}-\nabla_{\mu}\nabla_{\nu}\phi^{2}\right)\>,$

where $G_{\mu\nu}$ is Einstein tensor of $\mathcal{M}^{D}$. Then, we could
define an energy functional $E=a^{2D-2}\int T_{00}~{}d^{D-1}x$. However, such
a functional covers only $L^{2}$-norm functions and it is difficult to obtain
a decay estimate of the scalar field from it in the global case (see Section
5) since it contains the nonlinear terms, namely the scalar potential and the
nonminimal coupling. To overcome the problem, we introduce the $k$-order
linear energy

$\mathcal{H}_{k}[\phi]=\frac{1}{2}\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}\>\left[(\partial^{\alpha}\partial_{\tau}\phi)^{2}+\left|\nabla\partial^{\alpha}\phi\right|^{2}\right]\>dx\>,$
(3.7)

with $\alpha$ being a multi index, which will be used in this paper. We also
define an energy norm as

$|\phi(\tau,\cdot)|_{k}:=\|\phi(\tau,\cdot)\|_{H^{k+1}(\mathrm{IR}^{D-1})}+\|\partial_{\tau}\phi(\tau,\cdot)\|_{H^{k}(\mathrm{IR}^{D-1})}$
(3.8)

This form of energy can be used as a bound for the nonlinear term. In the
following lemmas, we prove the properties of the nonlinear term and the linear
energy.

######  Lemma 1.

Let $\phi$ be a real function such that for all $\tau\in[\tau_{0},\tau_{0}+T]$
and $k\in\mathbb{N}_{0}$,

$|\phi(\tau,\cdot)|_{k}\leq C\>.$ (3.9)

Then for all $\tau\in[\tau_{0},\tau_{0}+T]$ and $k>(D-1)/2$, we have

$\left(\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}[\partial^{\alpha}F(\phi,\partial_{\tau}\phi)]^{2}\>dx\right)^{1/2}\leq
C\>,$ (3.10)

where $C$ depends only on the initial data, $T$, $k$ and the bound of the
scalar potential.

###### Proof.

First, we consider the case of $|\alpha|>0$. The spatial derivative of
Equation (3.5) gives

$\partial^{\alpha}F(\phi,\partial_{\tau}\phi)=-(D-2)H~{}\partial^{\alpha}\partial_{\tau}\phi-\xi(D-1)\left(2\dot{H}+(D-2)H^{2}\right)\partial^{\alpha}\phi+a^{2}\partial^{\alpha}\partial_{\phi}V\>.$
(3.11)

Since $\tau\in[\tau_{0},\tau_{0}+T]$, $a(\tau)$ is a regular $C^{n}$-function,
and using the hypothesis that the first and second terms are bounded, we thus
have

$\left(\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}[\partial^{\alpha}F(\phi,\partial_{\tau}\phi)]^{2}\>dx\right)^{1/2}\leq
C\left\\{1+\|\partial_{\phi}V\|_{H^{k}(\mathrm{IR}^{D-1})}\right\\}$ (3.12)

To estimate the derivative of the scalar potential, we write down
$\partial^{\alpha}\partial_{\phi}V$ as

$\partial^{\alpha}\partial_{\phi}V=\left(\partial_{\phi}^{\beta}\partial_{\phi}V\right)\;\partial^{\gamma_{1}}\phi\;\partial^{\gamma_{2}}\phi\ldots\partial^{\gamma_{i}}\phi\>,$
(3.13)

where $\gamma_{1}+\gamma_{2}+\ldots+\gamma_{i}=\alpha$. Since the scalar
potential is a smooth function, then by Sobolev embedding theorem for
$k>(D-1)/2$, the hypothesis of the lemma implies that
$\partial_{\phi}^{\beta}\partial_{\phi}V$ is bounded. Thus,

$\|\partial_{\phi}V\|_{H^{k}(\mathrm{IR}^{D-1})}\leq
C\|\phi\|_{H^{k}(\mathrm{IR}^{D-1})}\leq C\>,$ (3.14)

and we obtain Equation (3.10).

For $|\alpha|=0$, we have

$\left(\int_{\mathrm{IR}^{D-1}}\>[F(\phi,\partial_{\tau}\phi)]^{2}\>dx\right)^{1/2}\leq
C\left\\{\|\phi\|_{L^{2}(\mathrm{IR}^{D-1})}+\|\partial_{\tau}\phi\|_{L^{2}(\mathrm{IR}^{D-1})}+\|\partial_{\phi}V\|_{L^{2}(\mathrm{IR}^{D-1})}\right\\}\>.$
(3.15)

The first and second terms at the right hand side are bounded to a constant by
the hypothesis. Using the assumption that $\partial_{\phi}V(0)=0$, we obtain
the estimate

$\|\partial_{\phi}V\|_{L^{2}(\mathrm{IR}^{D-1})}\leq
C\|\phi\|_{L^{2}(\mathrm{IR}^{D-1})}\leq C\>,$ (3.16)

where the constant $C$ depends on the bound of the scalar potential. Hence,
the proof is complete. ∎

######  Lemma 2.

Let $\phi$ be a real function such that for all $\tau\in[\tau_{0},\tau_{0}+T]$
and $k\in\mathbb{N}_{0}$,

$|\phi(\tau,\cdot)|_{k}\leq C\>.$ (3.17)

Then for $\tau\in[\tau_{0},\tau_{0}+T]$ and $k>(D-1)/2$, we have

$\mathcal{H}_{k}^{1/2}[\phi](\tau)\leq\mathcal{H}_{k}^{1/2}[\phi](\tau_{0})+\frac{1}{2}CT\>,$
(3.18)

where the constant $C$ depends only on the initial data, $T$, $k$ and the
bound of the scalar potential.

###### Proof.

Let us consider,

$\displaystyle\frac{d\mathcal{H}_{k}[\phi]}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}\>\left[2(\partial^{\alpha}\partial_{\tau}\phi)(\partial^{\alpha}\partial_{\tau}^{2}\phi)+2(\nabla\partial^{\alpha}\phi)\cdot(\nabla\partial^{\alpha}\partial_{\tau}\phi)\right]dx$
(3.19) $\displaystyle=$ $\displaystyle\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}\partial^{\alpha}F(\phi,\partial_{\tau}\phi)\;\partial^{\alpha}\partial_{\tau}\phi\>dx\>.$

Using Schwartz and Hölder inequalities, we obtain

$\frac{d\mathcal{H}_{k}[\phi]}{d\tau}\leq\left(\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}[\partial^{\alpha}F(\phi,\partial_{\tau}\phi]^{2}dx\right)^{1/2}\left(\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}(\partial^{\alpha}\partial_{\tau}\phi)^{2}dx\right)^{1/2}\>.$
(3.20)

The second factor can be estimated by a constant times
$\mathcal{H}_{k}^{1/2}[\phi]$. Since $\phi$ satisfies the hypothesis of Lemma
1, then the first factor is bounded to a constant. Thus, we obtain

$\frac{d\mathcal{H}_{k}[\phi]}{d\tau}\leq C\mathcal{H}_{k}^{1/2}[\phi]\>.$
(3.21)

Since $\mathcal{H}_{k}[\phi]$ is positive, we can divide the inequality by
$\mathcal{H}_{k}^{1/2}[\phi]$ and integrate to obtain

$\mathcal{H}_{k}^{1/2}[\phi](\tau)\leq\mathcal{H}_{k}^{1/2}[\phi](\tau_{0})+\frac{1}{2}CT\>,$
(3.22)

and the proof is finished. ∎

A classical solution of a scalar field on spatially flat spacetimes with non-
minimal coupling is a real smooth function $\phi$ satisfying Equation (3.4).
Then, a generalized solution is a real function $\phi\in
C^{0}\left([\tau_{0},\tau_{0}+T],H^{k+1}(\mathrm{IR}^{D-1})\right)\cap
C^{1}\left([\tau_{0},\tau_{0}+T],H^{k}(\mathrm{IR}^{D-1})\right)$ such that
Equation (3.4) is satisfied in a distributional sense. In the rest of this
paper, we will prove the existence and uniqueness of both generalized and
classical solutions to Equation (3.4) together with the initial data

$\displaystyle\phi(\tau_{0},x)$ $\displaystyle=$ $\displaystyle f(x)\>,$
$\displaystyle\partial_{\tau}\phi(\tau_{0},x)$ $\displaystyle=$ $\displaystyle
g(x)\>,$ (3.23)

with $f\in H^{k+1}(\mathrm{IR}^{D-1})$ and $g\in H^{k}(\mathrm{IR}^{D-1})$ and
having compact support.

##  4 Local Existence, Uniqueness, and Smoothness

In this section, we prove the local existence and the uniqueness of the
solution of the scalar field Equation (3.4).

###  4.1 Local Existence and Uniqueness

The scalar field equation in spatially flat universe is given by,

$\begin{cases}\partial_{\tau}^{2}\phi-\Delta\phi=F(\phi,\partial_{\tau}\phi)\\\
\phi(\tau_{0},\cdot)=f\in H^{k+1}(\mathrm{IR}^{D-1})\\\
\partial_{\tau}\phi(\tau_{0},\cdot)=g\in
H^{k}(\mathrm{IR}^{D-1})\>,\end{cases}$ (4.1)

where the nonlinear term is given by Equation (3.5). Let us consider the
sequence $\\{\phi_{l}\\}$ such that

$\begin{cases}\partial_{\tau}^{2}\phi_{0}-\Delta\phi_{0}=0\\\
\phi_{0}(\tau_{0},\cdot)=f_{0}\\\
\partial_{\tau}\phi_{0}(\tau_{0},\cdot)=g_{0}\>,\end{cases}$ (4.2)

and for $l\geq 0$,

$\begin{cases}\partial_{\tau}^{2}\phi_{l+1}-\Delta\phi_{l+1}=F(\phi_{l},\partial_{\tau}\phi_{l})\\\
\phi_{l}(\tau_{0},\cdot)=f_{l}\\\
\partial_{\tau}\phi_{l}(\tau_{0},\cdot)=g_{l}\>.\end{cases}$ (4.3)

Since the Schwartz space $\mathcal{S}(\mathrm{IR}^{D-1})$ is dense in
$H^{k}(\mathrm{IR}^{D-1})$, then we can choose the sequences $\\{f_{l}\\}$ and
$\\{g_{l}\\}$ such that $f_{l},g_{l}\in\mathcal{S}(\mathrm{IR}^{D-1})$ and
$f_{l}\rightarrow f$ in $H^{k+1}(\mathrm{IR}^{D-1})$ and $g_{l}\rightarrow g$
in $H^{k}(\mathrm{IR}^{D-1})$. Without loss of generality, we can choose
$f_{l}$ and $g_{l}$ such that,

$\displaystyle\|f_{l}\|_{H^{k+1}(\mathrm{IR}^{D-1})}$ $\displaystyle\leq$
$\displaystyle 2\|f\|_{H^{k+1}(\mathrm{IR}^{D-1})}$
$\displaystyle\|g_{l}\|_{H^{k}(\mathrm{IR}^{D-1})}$ $\displaystyle\leq$
$\displaystyle 2\|g\|_{H^{k}(\mathrm{IR}^{D-1})}\>.$ (4.4)

In the following lemma, we prove that $\mathcal{H}_{k}[\phi_{l}]$ is bounded
for all $l$.

######  Lemma 3.

Let $\\{\phi_{l}\\}$ be solutions of (4.2) and (4.3). Let $\mathcal{H}_{k}$ be
the linear energy defined in Equation (3.7). For $k>(D-1)/2$, there exist
constants $C$ and $T$ such that,

$\mathcal{H}_{k}[\phi_{l}](\tau)\leq C\>,$ (4.5)

for all $l\geq 0$ and $\tau\in[\tau_{0},\tau_{0}+T]$ . The constant $C$
depends on the initial data, $k$ and the bound of the scalar potential.

###### Proof.

We will prove the lemma by induction. Since $\phi_{0}$ is a solution of linear
wave equation, then $\mathcal{H}_{k}$ is conserved. Thus,

$\mathcal{H}_{k}[\phi_{0}](\tau_{0})=\mathcal{H}_{k}[\phi_{0}](\tau)\leq\tilde{C}\>,$
(4.6)

where the bound constant $\tilde{C}$ depends only on the initial data such
that,

$\mathcal{H}_{k}[\phi_{l}](\tau_{0})\leq\tilde{C}\>.$ (4.7)

Hence, Equation (4.5) is satisfied for $l=0$.

Now, we assume Equation (4.5) to be true for $l=n$. Then, we obtain

$\displaystyle\frac{\partial}{\partial\tau}\int_{\mathrm{IR}^{D-1}}\>|\phi_{n}|^{2}\>dx$
$\displaystyle=$ $\displaystyle
2\int_{\mathrm{IR}^{D-1}}\>\phi_{n}\;\partial_{\tau}\phi_{n}\>dx$ (4.8)
$\displaystyle\leq$ $\displaystyle
2\|\phi_{n}\|_{L^{2}(\mathrm{IR}^{D-1})}\|\partial_{\tau}\phi_{n}\|_{L^{2}(\mathrm{IR}^{D-1})}\leq
C\|\phi_{n}\|_{L^{2}(\mathrm{IR}^{D-1})}\mathcal{H}^{1/2}_{k}[\phi_{n}]\>,$

where we have used Hölder’s inequality and the definition of $\mathcal{H}_{k}$
in Equation (3.7). Integrating the inequality and using the induction
hypothesis, we obtain

$\|\phi_{n}(\tau,\cdot)\|_{L^{2}(\mathrm{IR}^{D-1})}\leq
C\left(\|f\|_{L^{2}(\mathrm{IR}^{D-1})}+T\right)\>.$ (4.9)

If we assume $T\leq 1$, then
$\|\phi_{n}(\tau,\cdot)\|_{L^{2}(\mathrm{IR}^{D-1})}$ is bounded for
$\tau\in[\tau_{0},\tau_{0}+T]$, which depends only on the initial data.

Using Sobolev embedding theorem for $k>(D-1)/2$, we have bounds on
$\sum_{j=1}^{D-1}\|\partial_{j}\phi_{l}(\tau,\cdot)\|_{C_{b}(\mathrm{IR}^{D-1})}$
and $\|\partial_{t}\phi_{l}(\tau,\cdot)\|_{C_{b}(\mathrm{IR}^{D-1})}$.
Furthermore, we also have

$|\phi_{n}(\tau,\cdot)|_{k}\leq C\>,$ (4.10)

where the constant only depends on initial data. Hence, $\phi_{n}$ satisfies
the hypothesis of Lemmas 1 and 2.

For $l=n+1$, we have

$\displaystyle\frac{d\mathcal{H}_{k}[\phi_{n+1}]}{d\tau}$ $\displaystyle=$
$\displaystyle\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}\partial^{\alpha}F(\phi_{n},\partial_{\tau}\phi_{n})\;\partial^{\alpha}\partial_{\tau}\phi_{n+1}\>dx$
(4.11) $\displaystyle\leq$ $\displaystyle\left(\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}[\partial^{\alpha}F(\phi_{n},\partial_{\tau}\phi_{n}]^{2}dx\right)^{1/2}\left(\sum_{|\alpha|\leq
k}\int_{\mathrm{IR}^{D-1}}(\partial^{\alpha}\partial_{\tau}\phi_{n+1})^{2}dx\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle C\mathcal{H}^{1/2}_{k}[\phi_{n+1}]\>,$

where we used Lemma 1 in the last inequality. Integrating the inequality, we
obtain

$\mathcal{H}^{1/2}_{k}[\phi_{n+1}](\tau)\leq\mathcal{H}^{1/2}_{k}[\phi_{n+1}](\tau_{0})+CT\>.$
(4.12)

Using Equation (4.6) and assuming $T\leq 1$, then
$\mathcal{H}_{k}[\phi_{n+1}](\tau)$ is bounded for
$\tau\in[\tau_{0},\tau_{0}+T]$.

∎

Let us define

$E_{l,k}=\sup_{\tau\in[\tau_{0},\tau_{0}+T]}\left[\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}](\tau)+\|(\phi_{l}-\phi_{l-1})(\tau,\cdot)\|_{L^{2}(\mathrm{IR}^{D-1})}\right]\>.$
(4.13)

Next, we derive an estimate for the difference of consecutive sequence which
is important to prove the convergence of the sequence.

######  Lemma 4.

Let $\\{\phi_{l}\\}$ be solutions of (4.2) and (4.3). Let $\mathcal{H}_{k}$ is
the linear energy defined in Equation (3.7). Then, for $k>(D-1)/2$, we have
the estimate

$\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}](\tau)\leq\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}](\tau_{0})+\frac{1}{2}CE_{l,k}T\>,$
(4.14)

for all $\tau\in[\tau_{0},\tau_{0}+T]$.

###### Proof.

First, we have the estimate

$\displaystyle\|F(\phi_{l},\partial_{\phi}\tau\phi_{l})-F(\phi_{l-1},\partial_{\phi}\tau\phi_{l-1})\|_{H^{k}(\mathrm{IR}^{D-1})}$
$\displaystyle\leq$ $\displaystyle
C\left[\|\phi_{l}-\phi_{l-1}\|_{H^{k}(\mathrm{IR}^{D-1})}+\|\partial_{\tau}\phi_{l}-\partial_{\tau}\phi_{l-1}\|_{H^{k}(\mathrm{IR}^{D-1})}\right.$
(4.15)
$\displaystyle\qquad\left.+\|\partial_{\phi}V(\phi_{l})-\partial_{\phi}V(\phi_{l-1})\|_{H^{k}(\mathrm{IR}^{D-1})}\right]\>.$

We can estimate the first and second terms as

$\displaystyle\|\phi_{l}-\phi_{l-1}\|_{H^{k}(\mathrm{IR}^{D-1})}$
$\displaystyle\leq$
$\displaystyle\|\phi_{l}-\phi_{l-1}\|_{L^{2}(\mathrm{IR}^{D-1})}+\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}]$
$\displaystyle\|\partial_{\tau}\phi_{l}-\partial_{\tau}\phi_{l-1}\|_{H^{k}(\mathrm{IR}^{D-1})}$
$\displaystyle\leq$
$\displaystyle\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}]\>.$ (4.16)

To estimate the third term, we write

$\partial_{\tau}V(\phi_{l})-\partial_{\tau}V(\phi_{l-1})=\int_{0}^{1}\partial_{\phi}^{2}V[\sigma\phi_{l}+(1-\sigma)\phi_{l-1}]d\sigma\;\left(\phi_{l}-\phi_{l-1}\right)\>.$
(4.17)

Using Lemma 3 and the fact that the scalar potential is a smooth function, for
$k>(D-1)/2$ we obtain

$\displaystyle\|\partial_{\phi}V(\phi_{l})-\partial_{\phi}V(\phi_{l-1})\|_{H^{k}(\mathrm{IR}^{D-1})}$
$\displaystyle\leq$ $\displaystyle
C\|\phi_{l}-\phi_{l-1}\|_{H^{k}(\mathrm{IR}^{D-1})}$ $\displaystyle\leq$
$\displaystyle
C\left(\|\phi_{l}-\phi_{l-1}\|_{L^{2}(\mathrm{IR}^{D-1})}+\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}]\right)\>,$

where the constant $C$ depends on the bound of the scalar potential. Hence, we
obtain the estimate

$\|F(\phi_{l},\partial_{\phi}\tau\phi_{l})-F(\phi_{l-1},\partial_{\phi}\tau\phi_{l-1})\|_{H^{k}(\mathrm{IR}^{D-1})}\leq
C\left(\|\phi_{l}-\phi_{l-1}\|_{L^{2}(\mathrm{IR}^{D-1})}+\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}]\right)\leq
CE_{l}\>.$ (4.19)

Now, similar to the proof of Lemma 3, we have

$\displaystyle\frac{d}{d\tau}\mathcal{H}_{k}[\phi_{l}-\phi_{l-1}]$
$\displaystyle\leq$ $\displaystyle
C\|F(\phi_{l},\partial_{\tau}\phi_{l})-F(\phi_{l-1},\partial_{\tau}\phi_{l-1})\|_{H^{k}(\mathrm{IR}^{D-1})}\;\|\partial_{\tau}\phi_{l}-\partial_{\tau}\phi_{l-1}\|_{H^{k}(\mathrm{IR}^{D-1})}$
(4.20) $\displaystyle\leq$ $\displaystyle
CE_{l,k}\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}]\>.$

Integrating the inequality, we obtain

$\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}](\tau)\leq\mathcal{H}^{1/2}_{k}[\phi_{l}-\phi_{l-1}](\tau_{0})+\frac{1}{2}CE_{l,k}T\>,$
(4.21)

for all $\tau\in[\tau_{0},\tau_{0}+T]$ and the proof is finished.

∎

Next, we prove the estimate of $E_{l}$.

######  Lemma 5.

Let $\\{\phi_{l}\\}$ be solutions of (4.2) and (4.3). Let $E_{l}$ be given by
Equation (4.13). For $k>(D-1)/2$, there exist constants $C_{0}>1$ and $T$ such
that,

$E_{l,k}\leq\frac{C_{0}}{2^{l}}\>,$ (4.22)

for all $l$.

###### Proof.

The lemma is true for $l=1$ by assuming that the constant $C_{0}$ is big
enough. Let us consider

$\displaystyle\frac{\partial}{\partial\tau}\int_{\mathrm{IR}^{D-1}}\>|\phi_{l+1}-\phi_{l}|^{2}\>dx$
$\displaystyle=$ $\displaystyle
2\int_{\mathrm{IR}^{D-1}}\>|\phi_{l+1}-\phi_{l}|\;\partial_{\tau}|\phi_{l+1}-\phi_{l}|\>dx$
(4.23) $\displaystyle\leq$ $\displaystyle
2\|\phi_{l+1}-\phi_{l}\|_{L^{2}(\mathrm{IR}^{D-1})}\|\partial_{\tau}(\phi_{l+1}-\phi_{l})\|_{L^{2}(\mathrm{IR}^{D-1})}$
$\displaystyle\leq$ $\displaystyle
2^{3/2}\|\phi_{l+1}-\phi_{l}\|_{L^{2}(\mathrm{IR}^{D-1})}\mathcal{H}^{1/2}_{k}[\phi_{l+1}-\phi_{l}]\>,$

Integrating this inequality and since $k>(D-1)/2$, we obtain

$\|\phi_{l+1}-\phi_{l}\|_{L^{2}(\mathrm{IR}^{D-1})}(\tau)\leq\|\phi_{l+1}-\phi_{l}\|_{L^{2}(\mathrm{IR}^{D-1})}(\tau_{0})+2^{1/2}\left|\int_{\tau_{0}}^{\tau}\mathcal{H}^{1/2}_{k}[\phi_{l+1}-\phi_{l}](s)\>ds\right|\>.$
(4.24)

Assuming $T<1/2$ and using Lemma 4, we have the estimate

$E_{l+1,k}\leq\|\phi_{l+1}-\phi_{l}\|_{L^{2}(\mathrm{IR}^{D-1})}(\tau_{0})+2\mathcal{H}^{1/2}_{k}[\phi_{l+1}-\phi_{l}](\tau_{0})+CE_{l,k}T\>.$
(4.25)

The first and second terms depend only on the initial data. However, we can
choose them as such that,

$\|\phi_{l+1}-\phi_{l}\|_{L^{2}(\mathrm{IR}^{D-1})}(\tau_{0})+2\mathcal{H}^{1/2}_{k}[\phi_{l+1}-\phi_{l}](\tau_{0})\leq\frac{1}{2^{l+2}}\>.$
(4.26)

Hence, by assuming $CT<1/4$, Equation (4.25) and the induction hypothesis give

$E_{l+1,k}\leq\frac{1}{2^{l+2}}+\frac{C_{0}}{2^{l+2}}\leq\frac{C_{0}}{2^{l+1}}\>.$
(4.27)

Thus, Equation (4.22) is true for all $l$ and the proof is finished.

∎

Let us consider, for all $|\alpha|\leq k+1$,

$\displaystyle\|\nabla\partial^{\alpha}(\phi_{l}-\phi_{l-1})(\tau,\cdot)\|_{L^{2}(\mathrm{IR}^{D-1})}$
$\displaystyle=$
$\displaystyle\left(\int_{\mathrm{IR}^{D-1}}\left|\nabla\partial^{\alpha}(\phi_{l}-\phi_{l-1})(\tau,\cdot)\right|dx\right)^{1/2}$
(4.28) $\displaystyle\leq$ $\displaystyle
C\mathcal{H}_{k}[\phi_{l}-\phi_{l-1}](\tau)$ $\displaystyle\leq$
$\displaystyle CE_{l,k}\>.$

In other words, $\phi_{l}-\phi_{l-1}\in H^{k+1}(\mathrm{IR}^{D-1})$. However,
using Lemma 5, we have

$\sup_{\tau\in[\tau_{0},\tau_{0}+T]}\|\phi_{l}-\phi_{l-1}\|_{H^{k+1}(\mathrm{IR}^{D-1})}\leq\frac{C_{0}}{2^{l}}\>,$
(4.29)

which show that $\\{\phi_{l}\\}$ is a Cauchy sequence on
$C\left([\tau_{0},\tau_{0}+T],H^{k+1}(\mathrm{IR}^{D-1})\right)$. Using a
similar method, we can show that $\\{\partial_{\tau}\phi_{l}\\}$ is also a
Cauchy sequence on
$C\left([\tau_{0},\tau_{0}+T],H^{k}(\mathrm{IR}^{D-1})\right)$. Thus, we have
proven the existence of generalized solutions of Equation (3.4) such that

$\phi\in C\left([\tau_{0},\tau_{0}+T],H^{k+1}(\mathrm{IR}^{D-1})\right)\cap
C^{1}\left([\tau_{0},\tau_{0}+T],H^{k}(\mathrm{IR}^{D-1})\right)\>.$ (4.30)

Furthermore, using Lemma 5, we also get that
$\\{\partial_{i}\partial_{j}\phi_{l}\\}$ and
$\\{\partial_{i}\partial_{\tau}\phi_{l}\\}$ are Cauchy sequences on
$C\left([\tau_{0},\tau_{0}+T],H^{k-1}(\mathrm{IR}^{D-1})\right)$. In fact, we
have

$\displaystyle\|\partial_{\tau}^{2}(\phi_{l+1}-\phi_{l})\|_{H^{k-1}(\mathrm{IR}^{D-1})}$
$\displaystyle\leq$
$\displaystyle\|F(\phi_{l},\partial_{\tau}\phi_{l})-F(\phi_{l-1},\partial_{\tau}\phi_{l-1})\|_{H^{k-1}(\mathrm{IR}^{D-1})}$
(4.31)
$\displaystyle\qquad+\|\partial_{i}\partial^{i}(\phi_{l+1}-\phi_{l})\|_{H^{k-1}(\mathrm{IR}^{D-1})}\>.$

On the right hand side, the first term is bounded by $E_{l,k}$ and the second
term is bounded by a constant since it is a Cauchy sequence on
$H^{k-1}(\mathrm{IR}^{D-1})$. Then using Lemma 5 , we have that
$\\{\partial_{\tau}^{2}\phi_{l}\\}$ is also a Cauchy sequence in
$H^{k-1}(\mathrm{IR}^{D-1})$. Thus, for $k>(D-1)/2$, there exists a real
function $\phi$ such that,

$\sup_{\tau\in[\tau_{0},\tau_{0}+T]}\|\partial^{2}_{\tau}\phi\|_{H^{k-1}(\mathrm{IR}^{D-1})}\leq
C\>.$ (4.32)

The above inequality implies that $\partial_{\tau}^{2}\phi\in
C\left([\tau_{0},\tau_{0}+T],H^{k-1}(\mathrm{IR}^{D-1})\right)$.

Let us consider a fixed point $(\tau,x)\in\mathcal{M}^{D}$. We make a sequance
$(t_{l},x_{l})\rightarrow(\tau,x)$, where
$\tau_{0}\leq\tau_{l},\tau\leq\tau_{0}+T$. Now, we have the estimate

$\left|\partial_{\tau}^{2}\phi(\tau,x)-\partial_{\tau}^{2}\phi(\tau_{l},x_{l})\right|\leq\left|\partial_{\tau}^{2}\phi(\tau,x)-\partial_{\tau}^{2}\phi(\tau,x_{l})\right|+\left|\partial_{\tau}^{2}\phi(\tau,x_{l})-\partial_{\tau}^{2}\phi(\tau_{l},x_{l})\right|\>.$
(4.33)

By Sobolev embedding theorem, for $k>(D-1)/2$ we have
$\partial_{\tau}^{2}\phi$ to be a continuous function in $x$, and thus, the
first term on the right hand side vanishes as $l\rightarrow\infty$. The second
term can be estimated by a constant times
$\|\partial_{\tau}^{2}\phi(\tau,\cdot)-\partial_{\tau}^{2}\phi(\tau_{l},\cdot)\|_{H^{k-1}(\mathrm{IR}^{D-1})}$.
Since $\partial_{\tau}^{2}\phi\in
C\left([\tau_{0},\tau_{0}+T],H^{k-1}(\mathrm{IR}^{D-1})\right)$, then the
second term also goes to zero as $l\rightarrow\infty$. Hence, we conclude
that, $\partial_{\tau}^{2}\phi\in
C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)$, thus

$\phi\in C^{2}\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)\>.$
(4.34)

To show the uniqueness, consider $\phi,\phi^{\prime}$ as solutions of Equation
(4.1) with the same initial data. Similar to the proof of Lemma 4, we have the
estimate

$\displaystyle\frac{d}{d\tau}\mathcal{H}_{k}[\phi^{\prime}-\phi]$
$\displaystyle\leq$ $\displaystyle
C\|F(\phi^{\prime},\partial_{\tau}\phi^{\prime})-F(\phi,\partial_{\tau}\phi)\|_{H^{k}(\mathrm{IR}^{D-1})}\;\|\partial_{\tau}\phi^{\prime}-\partial_{\tau}\phi\|_{H^{k}(\mathrm{IR}^{D-1})}$
(4.35) $\displaystyle\leq$ $\displaystyle
C\mathcal{H}_{k}[\phi^{\prime}-\phi]\>.$

Using Gronwall lemma and the fact that $\phi,\phi^{\prime}$ have the same
initial data, then for all $\tau\in[\tau_{0},\tau_{0}+T]$, we conclude that
$\phi^{\prime}(\tau)=\phi(\tau)$ and the uniqueness follows.

Thus, we have proven,

######  Theorem 1.

Let $f\in H^{k+1}(\mathrm{IR}^{D-1})$ and $g\in H^{k}(\mathrm{IR}^{D-1})$ be
initial data with compact support. Assume that the scalar potential is a
smooth function satisfying $V(0)=0$ and $\partial_{\phi}V(0)=0$ and that
$k>(D-1)/2$. Then, there exist $T>0$ and a unique $\phi\in
C^{2}\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)$ being a local
solution to the Equation (3.4) such that

$\phi\in C\left([\tau_{0},\tau_{0}+T],H^{k+1}(\mathrm{IR}^{D-1})\right)\cap
C^{1}\left([\tau_{0},\tau_{0}+T],H^{k}(\mathrm{IR}^{D-1})\right)\>.$ (4.36)

###  4.2 Smoothness Properties

The local solution, which we have discussed above, is $\phi\in
C^{2}\left([\tau_{0},\tau_{0}+T],\mathrm{IR}^{D-1}\right)$ throughout their
interval of existence. In general, however they are actually smoother than
this. In this section, we prove the smoothness properties of the solution of
equation (4.1).

We claim that the solution is
$C^{(m-1)}\left([\tau_{0},\tau_{0}+T],\mathrm{IR}^{D-1}\right)$ for $m\geq 3$.
We will prove the statement using the induction argument. For $m=3$, the
statement is true. Now, assume that it is also true for $m=n$, Then, we have

$\partial^{r}_{\tau}\partial_{j_{1}}\partial_{j_{2}}\cdots\partial_{j_{i-r-1}}\phi\in
C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)\>,$ (4.37)

where $r=0,1,\cdots,n-1$. For $m=n+1$, we need to prove that the limit of
sequence defined by equations (4.2) and (4.3) satisfies

$\partial_{j_{1}}\partial_{j_{2}}\cdots\partial_{j_{i-p}}\partial^{p}_{\tau}\phi\in
C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)\>,$ (4.38)

for $p=0,1,\cdots,n-1$.

Since $n+1>3$, using Lemma 5, we have

$E_{n+1}[\phi_{l}-\phi_{l-1}]\leq C\>,$ (4.39)

for some constant $C$. In other words, we have the estimate

$\|\partial_{j}(\phi_{l}-\phi_{l-1})(\tau,\cdot)\|_{H^{n+1}(\mathrm{IR}^{D-1})}+\|\partial_{\tau}(\phi_{l}-\phi_{l-1})(\tau,\cdot)\|_{H^{n+1}(\mathrm{IR}^{D-1})}\leq
C\>.$ (4.40)

From the first term on the right hand side, we obtain

$\|\partial_{j_{1}}\cdots\partial_{j_{n-1}}\partial_{j}(\phi_{l}-\phi_{l-1})(\tau,\cdot)\|_{H^{2}(\mathrm{IR}^{D-1})}\leq
C\|\partial_{j}(\phi_{l}-\phi_{l-1})(\tau,\cdot)\|_{H^{n+1}(\mathrm{IR}^{D-1})}\leq
C\>.$ (4.41)

Using Sobolev embedding theorem, we conclude

$\partial_{j_{1}}\cdots\partial_{j_{n}}\phi\in
C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)\>.$ (4.42)

For the second term, using similar methods, we have

$\partial_{j_{1}}\cdots\partial_{j_{n-1}}\partial_{\tau}\phi\in
C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)\>.$ (4.43)

Hence, we have shown that Equation (4.38) is satisfied for $p=0$ and $p=1$.

From Equation (4.3), we have

$\displaystyle\partial^{p}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}(\phi_{l}-\phi_{l-1})=\partial^{p-2}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}\hat{F}_{l}+\partial^{p-2}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}\partial_{k}\partial^{k}(\phi_{l}-\phi_{l-1})\>,$
(4.44)

with

$\hat{F}_{l}=F(\phi_{l},\partial_{\tau}\phi_{l})-F(\phi_{l-1},\partial_{\tau}\phi_{l-1})\>.$
(4.45)

The first term of Equation (4.44) can be written as

$\displaystyle\partial^{p-2}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}\hat{F}_{l}$
$\displaystyle=$
$\displaystyle-\frac{12\xi}{\tau^{2}}\partial^{p-2}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}(\phi_{l}-\phi_{l-1})-\frac{4}{\tau}\partial^{p-1}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}(\phi_{l}-\phi_{l-1})$
(4.46)
$\displaystyle-~{}\tau^{4}\partial^{p-2}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}\left[\partial_{\phi}V(\phi_{l})-\partial_{\phi}V(\phi_{l-1})\right]\>.$

From the induction hypothesis, the first and second terms are
$C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)$. To estimate the
last term, define

$G(\phi_{l},\phi_{l-1})=\int_{0}^{1}\partial_{\sigma}\partial_{\phi}V[\sigma\phi_{l}+(1-\sigma)\phi_{l-1}]\>d\sigma\>,$
(4.47)

and we have the estimate,

$\displaystyle\partial^{p-2}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}\left[G(\phi_{l},\phi_{l-1})\hat{\phi}_{l-1}\right]$
$\displaystyle=$ $\displaystyle\sum_{c+d\leq i-p}~{}~{}\sum_{s+t\leq
p-2}(\partial^{s}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{c}}G)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\partial^{t}_{\tau}\partial_{j_{c+1}}\cdots\partial_{j_{c+d}}(\phi_{l}-\phi_{l-1}))\>,$

for $c\geq 1$ and $d\geq 0$. Since the scalar potential is a smooth function
and $E_{c}[\phi_{l}]$ is bounded for $c\leq n-p$ by Lemma 3, therefore we can
bound $(\partial^{s}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{c}}G)$ by a
constant. Because the last factor is
$C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)$ by induction
hypothesis, then
$\partial^{p-2}_{\tau}\partial_{j_{1}}\cdots\partial_{j_{i-p}}\hat{F}_{l}$ is
also $C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)$, and we
conclude

$\displaystyle\partial^{p}_{\tau}\partial_{j_{1}}\partial_{j_{2}}\cdots\partial_{j_{i-p}}\phi\in
C\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)\>,$ (4.49)

for $p=0,1,\cdots,n-1$.

Hence, we have proven

######  Theorem 2.

Let $f\in H^{k+1}(\mathrm{IR}^{D-1})$ and $g\in H^{k}(\mathrm{IR}^{D-1})$ be
the initial data with compact support. Assume that the scalar potential is a
smooth function satisfying $V(0)=0$ and $\partial_{\phi}V(0)=0$ and that
$k>(D-1)/2$. Then, there exist $T>0$ and a unique $\phi\in
C^{(k-1)}\left([\tau_{0},\tau_{0}+T]\times\mathrm{IR}^{D-1}\right)$, which is
a local solution to the Equation (3.4), such that

$\phi\in C\left([\tau_{0},\tau_{0}+T],H^{k+1}(\mathrm{IR}^{D-1})\right)\cap
C^{1}\left([\tau_{0},\tau_{0}+T],H^{k}(\mathrm{IR}^{D-1})\right)\>.$ (4.50)

##  5 Global Existence

###  5.1 General Setup

In Section 4, we proved the existence a unique local solution of Equation
(3.4), i.e.

$\begin{cases}\partial_{\tau}^{2}\phi+(D-2)H~{}\partial_{\tau}\phi-\Delta\phi=-\xi(D-1)\left(2\dot{H}+(D-2)H^{2}\right)\phi+a^{2}\partial_{\phi}V(\phi)\\\
\phi(\tau_{0},\cdot)=f\in H^{k+1}(\mathrm{IR}^{D-1})\\\
\partial_{\tau}\phi(\tau_{0},\cdot)=g\in
H^{k}(\mathrm{IR}^{D-1})\>.\end{cases}$ (5.1)

Here, we show that it is possible to have a set of global solutions of (3.4)
for $T\to+\infty$. To proceed, let us define a new field,
$\psi=a^{(D-2)/2}\phi$ where $a\equiv a(\tau,\tau_{0})$ such that Equation
(3.4) can be written down as

$\begin{cases}\partial_{\tau}^{2}\psi-\Delta\psi=h(\tau)\psi+a^{D}\partial_{\psi}V(\psi)\\\
\psi(0,\cdot)=a(\tau_{0})^{(D-2)/2}f\\\
\partial_{\tau}\psi(0,\cdot)=\frac{1}{2}(D-2)a(\tau_{0})^{(D-4)/2}\dot{a}(\tau_{0})f+a(\tau_{0})^{(D-2)/2}g\>,\end{cases}$
(5.2)

where

$h(\tau)=\left[\frac{1}{2}(D-2)-2\xi(D-1)\right]\left[\dot{H}+\frac{D-2}{2}H^{2}\right]~{}.$
(5.3)

Note that Equation (5.3) is Riccati’s form of the Hubble parameter $H(\tau)$.
In particular, $H(\tau)$ could be thought of as a solution of Riccati’s
equation, see for example, [9]. In the rest of the paper we simply take
several assumptions as follows.

######  Assumption 1.

$h(\tau)\leq 0$ and $\partial_{\tau}h(\tau)\geq 0$ for all
$\tau\in\mathrm{IR}^{+}$.

This assumption follows that $h(\tau)$ tends to vanish as $\tau\to+\infty$.
For example, the function $h(\tau)$ may have the form of either
$-m\mathrm{e}^{-n\tau}$ or $-m\tau^{-n}$ with $m,n\in\mathrm{IR}^{+}$. The
latter function for $n=2$ could be related to a cosmological model where the
scale factor $a(\tau)$ has a polynomial form. This occurs, for example, in the
standard cosmological models discussed in Section 2.

######  Assumption 2.

The scalar potential has the form

$V(\phi)=-\frac{\epsilon}{p+1}\>\phi^{p+1}\>,$ (5.4)

with $\epsilon$ is a small positive parameter and $p\in\mathrm{IR}^{+}$.

######  Assumption 3.

$\int_{0}^{\infty}a^{[D+2-(D-2)p]/2}d\tau=A(\tau_{0})<+\infty$ for all
$\tau_{0}\in\mathrm{IR}^{+}$.

Thus, we can write the differential equation as

$\partial_{\tau}^{2}\psi-\Delta\psi=h(\tau)\psi+\epsilon P(\tau,\psi)\>,$
(5.5)

where $h(\tau)$ is given by (5.3) and
$P(\tau,\psi)=-a^{[D+2-(D-2)p]/2}~{}\psi^{p}$.

Now, suppose we have the nonhomogenous linear equation

$\begin{cases}\partial_{\tau}^{2}\eta-\Delta\eta=h(\tau)\eta+P(\tau,x)\\\
\eta(0,\cdot)=a(\tau_{0})^{(D-2)/2}f\\\
\partial_{\tau}\eta(0,\cdot)=\frac{1}{2}(D-2)a(\tau_{0})^{(D-4)/2}\dot{a}(\tau_{0})f+a(\tau_{0})^{(D-2)/2}g\>,\end{cases}$
(5.6)

where $h(\tau)$ is given by (5.3) and $\tau_{0}>0$ is arbitary real number.
First, we prove the following lemma,

######  Lemma 6.

Let $\eta$ be a solution of linear Equation (5.6) with compact support. Let
$P(\tau,x)$ be a $C^{0}\left(\mathrm{IR}^{+},H^{k}(\mathrm{IR}^{D-1})\right)$
function such that
$\int_{0}^{\infty}\>\|P(s)\|_{H^{k}(\mathrm{IR}^{D-1})}\>ds<\infty$. If
Assumption 1 holds, then we have the following inequality,

$\sup_{\tau\in\mathrm{IR}^{+}}\>|\eta(\tau)|_{k}\leq
C\left(\|g\|_{H^{k}(\mathrm{IR}^{D-1})}+\|f\|_{H^{k+1}(\mathrm{IR}^{D-1})}+\int_{0}^{\infty}\|P(s)\|_{H^{k}(\mathrm{IR}^{D-1})}\>ds\right)\leq
M\>,$ (5.7)

where $M$ depends on the initial data, $k$, $h$ and $P$.

###### Proof.

Taking spatial Fourier transform of Equation (5.6), we have

$\partial_{\tau}^{2}\tilde{\eta}+|\lambda|^{2}\tilde{\eta}=h(\tau)\tilde{\eta}+\tilde{P}\>.$
(5.8)

Multipying by $\partial_{\tau}\tilde{\eta}$, integrating over $\tau$ and using
partial integration, we obtain

$\displaystyle|\partial_{\tau}\tilde{\eta}(\tau)|^{2}+|\lambda|^{2}|\tilde{\eta}(\tau)|^{2}$
$\displaystyle=$
$\displaystyle|\tilde{g}|^{2}+|\lambda|^{2}|\tilde{f}|^{2}+h(\tau)|\tilde{\eta}(\tau)|^{2}-h(0)|\tilde{f}|^{2}$
(5.9)
$\displaystyle\qquad-\int_{0}^{\tau}\partial_{\tau}h(s)|\tilde{\eta}(s)|^{2}\;ds+2\int_{0}^{\tau}\tilde{P}(s)\partial_{\tau}\tilde{\eta}(s)\;ds\>.$

Since we have $h(\tau)\leq 0$ and $\partial_{\tau}h(\tau)\geq 0$ for all
$\tau\in\mathrm{IR}^{+}$, then we have

$|\partial_{\tau}\tilde{\eta}(\tau)|^{2}+|\lambda|^{2}|\tilde{\eta}(\tau)|^{2}\leq|\tilde{g}|^{2}+|\lambda|^{2}|\tilde{f}|^{2}-h(0)|\tilde{f}|^{2}+2\int_{0}^{t}\tilde{P}(s)\partial_{\tau}\tilde{\eta}(s)\;ds\>.$
(5.10)

Multiplying by $(1+\lambda^{2})^{k}$ and integrating over $\lambda$-space, we
obtain

$\mathcal{H}_{k}[\eta](\tau)\leq
C\left(\|g\|^{2}_{H^{k}(\mathrm{IR}^{D-1})}+\|f\|^{2}_{H^{k+1}(\mathrm{IR}^{D-1})}+2\int_{0}^{\tau}\int_{\mathrm{IR}^{D-1}}\>(1+\lambda^{2})^{k}\tilde{P}(s)\partial_{\tau}\tilde{\eta}(s)\;d\lambda\;ds\right)\>,$
(5.11)

where $C$ depends only on $k$ and $h$. Using Hölder inequality,

$\int_{\mathrm{IR}^{D-1}}\>(1+\lambda^{2})^{k}\tilde{P}(s)|\partial_{\tau}\tilde{\eta}(s)|^{2}\;d\lambda\leq\left[\int_{\mathrm{IR}^{D-1}}\>(1+\lambda^{2})^{k}|\tilde{P}(s)|^{2}\;d\lambda\right]^{1/2}\left[\int_{\mathrm{IR}^{D-1}}\>(1+\lambda^{2})^{k}|\partial_{\tau}\tilde{\eta}(s)|^{2}\;d\lambda\right]^{1/2}$
(5.12)

we have,

$\mathcal{H}_{k}[\eta](\tau)\leq
C\left(\|g\|^{2}_{H^{k}(\mathrm{IR}^{D-1})}+\|f\|^{2}_{H^{k+1}(\mathrm{IR}^{D-1})}+\sup_{s\in[0,\tau]}\|\partial_{\tau}\eta(s)\|_{H^{k}(\mathrm{IR}^{D-1})}\int_{0}^{\tau}\>\|P(s)\|_{H^{k}(\mathrm{IR}^{D-1})}\;ds\right)\>.$
(5.13)

Let us consider,

$\displaystyle\frac{\partial}{\partial\tau}\int_{\mathrm{IR}^{D-1}}\>|\eta|^{2}\>dx$
$\displaystyle=$ $\displaystyle
2\int_{\mathrm{IR}^{D-1}}\>\eta\;\partial_{\tau}\eta\>dx$ (5.14)
$\displaystyle\leq$ $\displaystyle
2\|\eta\|_{L^{2}(\mathrm{IR}^{D-1})}\|\partial_{\tau}\eta\|_{L^{2}(\mathrm{IR}^{D-1})}\leq
C\|\eta\|_{L^{2}(\mathrm{IR}^{D-1})}\mathcal{H}^{1/2}_{k}[\eta]\>.$

Integrating the inequality, we obtain

$\|\eta(\tau)\|^{2}_{L^{2}(\mathrm{IR}^{D-1})}\leq\|\eta(0)\|^{2}_{L^{2}(\mathrm{IR}^{D-1})}+C\int_{0}^{\tau}\|\eta(s)\|_{L^{2}(\mathrm{IR}^{D-1})}\mathcal{H}^{1/2}_{k}[\eta](s)\>ds\>.$
(5.15)

Adding up inequalities (5.13) and (5.15) and taking the supremum for
$\tau\in\mathrm{IR}^{+}$ will yield (5.7) and the proof is finished. ∎

Since $\psi=a^{(D-2)/2}\phi$, by Theorems 1 and 2, there exists a unique local
solution of Equation (5.2). Furthermore, by Lemma (6), for $\tau\in[0,T]$ we
have the following inequality,

$|\psi(\tau)|_{k}\leq
C\left(\|g\|_{H^{k}(\mathrm{IR}^{D-1})}+\|f\|_{H^{k+1}(\mathrm{IR}^{D-1})}+\int_{0}^{\tau}\|P(s,\psi(s))\|_{H^{k}(\mathrm{IR}^{D-1})}\>ds\right)\>.$
(5.16)

We show that the result can be extended to $T\rightarrow\infty$, hence the
solution globally exists.

Similar with proving the local existence, we construct a sequence
$\\{\psi_{l}(\tau)\\}$ such that,

$\begin{cases}\partial_{\tau}^{2}\psi_{0}-\Delta\psi_{0}=h(\tau)\psi_{0}+\epsilon
P(\tau,0)\\\
\partial_{\tau}^{2}\psi_{l+1}-\Delta\psi_{l+1}=h(\tau)\psi_{l+1}+\epsilon
P(\tau,\psi_{l})\\\ \psi_{l}(0,\cdot)=a^{(D-2)/2}f\\\
\partial_{\tau}\psi_{l}(0,\cdot)=\frac{1}{2}(D-2)a(\tau_{0})^{(D-4)/2}\dot{a}(\tau_{0})f+a^{(D-2)/2}g\>,\end{cases}$
(5.17)

We show by induction that there exists a positive constant $\epsilon_{0}$ such
that for $0<\epsilon<\epsilon_{0}$, we have

$\displaystyle\sup_{\tau\in\mathrm{IR}^{+}}|\psi_{l}(\tau)|_{k}$
$\displaystyle\leq$ $\displaystyle M$ (5.18)
$\displaystyle\sup_{\tau\in\mathrm{IR}^{+}}|\psi_{l+1}(\tau)-\psi_{l}(\tau)|_{k}$
$\displaystyle\leq$
$\displaystyle\kappa\sup_{\tau\in\mathrm{IR}^{+}}|\psi_{l}(\tau)-\psi_{l-1}(\tau)|_{k}\>,$
(5.19)

for some positive constant $\kappa\in(0,1)$. For $l=0$, the inequalities are
true due to Lemma 6. Assume that these are true for $l=n$, thus similar to the
proof of Lemma 1, we have

$\displaystyle\|P(\tau,\psi_{n}(\tau))\|_{H^{k}(\mathrm{IR}^{D-1})}$
$\displaystyle\leq$ $\displaystyle
Ca^{[D+2-(D-2)p]/2}\|\psi_{n}(\tau)\|_{H^{k+1}(\mathrm{IR}^{D-1})}$ (5.20)
$\displaystyle\leq$ $\displaystyle Ca^{[D+2-(D-2)p]/2}|\psi_{n}(\tau)|_{k}$
$\displaystyle\leq$ $\displaystyle CMa^{[D+2-(D-2)p]/2}$

Then, from (5.7) we obtain

$\displaystyle\sup_{\tau\in\mathrm{IR}^{+}}\>|\psi_{n+1}(\tau)|_{k}$
$\displaystyle\leq$ $\displaystyle
C_{0}\left(\|g_{n+1}\|_{H^{k}(\mathrm{IR}^{D-1})}+\|f_{n+1}\|_{H^{k+1}(\mathrm{IR}^{D-1})}+\epsilon\int_{0}^{\infty}\|P(s,\psi_{n}(s))\|_{H^{k}(\mathrm{IR}^{D-1})}\>ds\right)$
(5.21) $\displaystyle=$ $\displaystyle C_{0}\left(L+\epsilon
CM\int_{0}^{\infty}a^{[D+2-(D-2)p]/2}ds\right)$ $\displaystyle\leq$
$\displaystyle C_{0}\left(L+\epsilon CMA(\tau_{0})\right)\>.$

Defining $\epsilon_{0}=(M-C_{0}L)/C_{0}CMA(\tau_{0})$, it implies

$\sup_{\tau\in\mathrm{IR}^{+}}\>|\psi_{n+1}(\tau)|_{k}\leq M\>,$ (5.22)

and, hence the inequality (5.18) holds for all nonnegative integers.

Similar to the proof of Lemma 4, we have

$\|P(\tau,\psi_{l}(\tau))-P(\tau,\psi_{l-1}(\tau))\|_{H^{k}(\mathrm{IR}^{D-1})}\leq
Ca^{[D+2-(D-2)p]/2}|\psi_{l}(\tau)-\psi_{l-1}(\tau)|_{k}\>,$ (5.23)

which follows that

$\displaystyle\sup_{\tau\in\mathrm{IR}^{+}}\>|\psi_{l+1}(\tau)-\psi_{l}(\tau)|_{k}$
$\displaystyle\leq$ $\displaystyle
C_{0}\epsilon\int_{0}^{\infty}\|P(s,\psi_{l}(s))-P(s,\psi_{l-1}(s))\|_{H^{k}(\mathrm{IR}^{D-1})}\>ds$
(5.24) $\displaystyle\leq$ $\displaystyle
C_{0}C\epsilon\int_{0}^{\infty}a^{[D+2-(D-2)p]/2}|\psi_{l}(\tau)-\psi_{l-1}(\tau)|_{k}~{}ds$
$\displaystyle\leq$ $\displaystyle C_{0}C\epsilon
A(\tau_{0})\sup_{\tau\in\mathrm{IR}^{+}}\>|\psi_{l}(\tau)-\psi_{l-1}(\tau)|_{k}\>.$

This inequality proves that $\\{\psi_{l}\\}$ converges to $\psi\in
C\left(\mathrm{IR}^{+},{H^{k+1}(\mathrm{IR}^{D-1})}\right)$. Furthermore,
similar to the proof of local existence, we conclude that there exists a
unique global solution of Equation (5.5) such that $\psi\in
C^{2}\left(\mathrm{IR}^{+},{H^{k+1}(\mathrm{IR}^{D-1})}\right)$ and
$\sup_{\tau\in\mathrm{IR}^{+}}|\psi(\tau)|\leq M$.

Hence, we have proven,

######  Theorem 3.

Let $f\in H^{k+1}(\mathrm{IR}^{D-1})$ and $g\in H^{k}(\mathrm{IR}^{D-1})$ be
the initial data with compact support for $k>(D-1)/2$. Suppose that
Assumptions 1-3. hold. For any positive constant $M$ that depends on the
initial data, $k$ and $\tau_{0}$, there exists a positive number
$\epsilon_{0}$ that also depends on the initial data, $k$ and $\tau_{0}$, such
that for any $0<\epsilon<\epsilon_{0}$, Equation (3.4) admits unique classical
global solutions

$\phi\in C\left([\tau_{0},\infty],H^{k+1}(\mathrm{IR}^{D-1})\right)\cap
C^{1}\left([\tau_{0},\infty],H^{k}(\mathrm{IR}^{D-1})\right)\>,$ (5.25)

satisfying the following decay estimate

$\left\|\phi(\tau)\right\|_{H^{k+1}(\mathrm{IR}^{D-1})}+\frac{1}{2}(D-2)\left\|\frac{\dot{a}}{a}\phi(\tau)+\partial_{\tau}\phi(\tau)\right\|_{H^{k}(\mathrm{IR}^{D-1})}\leq
Ma^{-(D-2)/2}~{},$ (5.26)

where $a\equiv a(\tau,\tau_{0})$.

##  6 Some Models

In this section, we consider some specific models related to the scale factor
$a(\tau)$. These models may have a global regular solution in the sense of our
setup in the preceding section.

###  6.1 Power Form

First, let us take the scale factor $a(\tau)$ to be of the form

$a(\tau)=(\tau+\tau_{0})^{\alpha}\>,$ (6.1)

where $\alpha$ is a real constant. In particular, for a single component
universe the constant $\alpha$ is given by

$\alpha=\frac{2}{(D-1)(w+1)-2}\>,$ (6.2)

where $w\neq-(D-3)/(D-1)$ is related to the equation of state discussed in
Section 2. Thus we have,

$h(\tau)=\frac{\alpha}{(\tau+\tau_{0})^{2}}\left[\frac{1}{2}(D-2)-2\xi(D-1)\right]\left[-1+\frac{\alpha(D-2)}{2}\right]$
(6.3)

In Table 2 we list four cases single component of higher dimensional
cosmological models where Assumptions 1 and 3 are fulfilled.

Case I | $\alpha<0$ | $w<-\frac{D-3}{D-1}$ | $\xi>\frac{D-2}{4(D-1)}$ | $p<\frac{1}{D-2}\left(D+2-\frac{2}{|\alpha|}\right)$
---|---|---|---|---
Case II | $0<\alpha<\frac{2}{D-2}$ | $w>\frac{1}{D-1}$ | $\xi<\frac{D-2}{4(D-1)}$ | $p>\frac{1}{D-2}\left(D+2+\frac{2}{\alpha}\right)$
Case III | $\alpha>\frac{2}{D-2}$ | $-\frac{D-3}{D-1}<w<\frac{1}{D-1}$ | $\xi>\frac{D-2}{4(D-1)}$ | $p>\frac{1}{D-2}\left(D+2+\frac{2}{\alpha}\right)$
Case IV | $\alpha=\frac{2}{D-2}$ | $w=\frac{1}{D-1}$ | $\xi\in\mathrm{IR}$ | $p>\frac{2D}{D-2}$
Table 2: Four cases in a single component higher dimensional universe.

Moreover, it is of interest to consider, for examples, the standard cosmology
in four dimensions as listed in Table 3.

Matter Dominated | $w=0$ | $\alpha=2$ | $\xi>\frac{1}{6}$ | $p>\frac{7}{2}$
---|---|---|---|---
$\Lambda$-Dominated | $w=-1$ | $\alpha=-1$ | $\xi>\frac{1}{6}$ | $p<2$
Radiation Dominated | $w=\frac{1}{3}$ | $\alpha=1$ | $\xi\in\mathrm{IR}$ | $p>4$
Table 3: Three cases in a single component four dimensional universe.

###  6.2 Exponential Form

Finally, we take a case where the scale factor $a(\tau)$ has the form

$a(\tau)=e^{\alpha(\tau+\tau_{0})}\>,$ (6.4)

with $\alpha\in\mathrm{IR}^{+}$. This exponential form is related to the
single component universe for $w=-(D-3)/(D-1)$. Then we have

$h(\tau)=\frac{\alpha^{2}(D-2)}{2}\left[\frac{1}{2}(D-2)-2\xi(D-1)\right].$
(6.5)

In order to satisfy Assumptions 1 and 3, we should have $w=-\frac{D-3}{D-1}$,
$\xi>\frac{D-2}{4(D-1)}$, and $p>\frac{D+2}{D-2}$.

##  7 Acknowledgments

The work of this paper is supported by Riset Unggulan PT Kemenristekdikti
2017.

## References

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# Binary Image Skeletonization Using 2-Stage U-Net

Mohamed A. Ghanem
<EMAIL_ADDRESS>Alaa A. Anani
<EMAIL_ADDRESS>
###### Abstract

Object Skeletonization is the process of extracting skeletal, line-like
representations of shapes. It provides a very useful tool for geometric shape
understanding and minimal shape representation. It also has a wide variety of
applications, most notably in anatomical research and activity detection.
Several mathematical algorithmic approaches have been developed to solve this
problem, and some of them have been proven quite robust. However, a lesser
amount of attention has been invested into deep learning solutions for it. In
this paper, we use a 2-stage variant of the famous U-Net architecture to split
the problem space into two sub-problems: shape minimization and corrective
skeleton thinning. Our model produces results that are visually much better
than the baseline SkelNetOn model. We propose a new metric, M-CCORR, based on
normalized correlation coefficients as an alternative to F1 for this challenge
as it solves the problem of class imbalance, managing to recognize skeleton
similarity without suffering from F1’s over-sensitivity to pixel-shifts.

##  1 Introduction

Skeletonization is an interesting task because it collapses object shapes into
their backbone structure, allowing for constructing a minimal representation
of it. Such skeletal representation are useful for many purposes. One simple
example of that would be object detection by skeleton matching. Another more
involved application is anatomical labelling of different parts of the body
such as pulmonary airway trees which requires extracting and identifying
different branches of such airways. As such, extracting skeletons can be
utilized in shape modelling, compression, and analysis [1]. One thing to note
about skeletons is that they contained condensed local as well as global
features. That is, they contain information about the main shape along with
peripheral details.

In spite of the conceptual differences between segmentation and
skeletonization, a few famous segmentation models have been observed to
perform well when trained for skeletonization. One of these models is the
U-Net architecture used for fast segmentation [7]. U-Net is a fully
convolutional neural network that takes its U-shape by virtue of its two main
paths: the contraction path, and the expansion path. Passing through the
contracting path, the network keeps acquiring features (i.e., information)
about what the shape its but loses track of where they came from. To remedy
that, the expansive path tries to reconstruct the image again from the
condensed features along with the residual connections with their origin
layers. Naturally, the reconstructed image will be a reduced version of the
original one, which is nicely aligned with the idea of skeletonization.

##  2 Related Work

The skeletonization problem has various approaches including heuristical
algorithms and using deep neural nets. For the first approach, numerous
mathematical algorithms are demonstrated in [8]. The problem can also be
modelled as a pixel-level binary classification problem known as semantic
segmentation. In [1], the SkelNetOn 2019 Challenge three datasets were
proposed to serve as a benchmark for shape understanding approaches, one of
which is the Pixel SkelNetOn, which poses the challenge in extracting the
skeleton pixels from a given binary image. The authors propose a baseline
model, which is a vanilla pix2pix model [2]. They apply distance transform on
the input images and L1 loss scoring an F1 score of 0.6244. In [3], a feature
hourglass network (FHN) is used, using a backbone based on VGGNet [9] and
side-output integration following it [4]. They score an F1 score of 0.6477. In
[6], a modified U-Net based CNN is used alongside the weighted focal loss to
overcome the class imbalance problem. Using an ensemble of 8 models they
scored an F1 score of 0.75 on the validation set. The same idea of binary
classification was used by [10] in their Holistically-Nested Edge Detection
(HED) method. The authors use deep supervision on fully connected neural
networks and side-output integration. Inspired by the HED method, authors of
[5], propose their SkeletonNet encoder-decoder architecture. Similar to HED,
the side layers are fused into final output layers. To improve the HED
architecture performance, another type of layers is introduced after the up-
sampling stage and their outputs are considered side layers. The loss function
used is the sum of the binary cross-entropy and Dice Loss. The best F1 score
of SkeletonNet is 0.7877.

Figure 1: High-level view of the 2-stage U-Net architecture composed of the
skeletonizer and the corrector models

##  3 Dataset

In this work, we focus on the Pixel SkelNetOn dataset containing a total of
1,725 binary images on the PNG format of size $256\times 256$ pixels. Objects
within each image have been pre-segmented and cleaned. They also contain
exactly one object per image. The ground truth was generated algorithmically
by [1] with some manual intervention to remove unnecessary branches. Note that
the results stated in this paper are preliminary (on a reserved subset of the
training set) as we do not yet have access to the validation set ground truth
since it is an ongoing competition.

##  4 Methodology

###  4.1 Two-stage Problem Split

In our method, we split the problem space into two consecutive problems
handled by two U-Nets that were trained accordingly. The first sub-problem is
concerned with condensing or minimizing input shapes which typically results
in a thick skeleton-like shape. The second sub-problem expects a bad skeleton
and aims to correct as well as thin it. Our training pipeline consists of two
such U-Nets in series. Note that the two models are not trained together, that
is, the first-stage model is trained on the original data, then the second-
stage one is trained on the outputs of stage 1 and the original target
skeletons. This is was experimentally observed to produce better results than
training both models together.

###  4.2 Modified U-Net Architecture

Both models have the exact same structure and hyper-parameters. A high-level
view of our modified U-Net architecture is shown in Figure 1. As briefly
explained before, a U-Net consists of a contracting path and an expansive one,
separated by a bottleneck region. Each contraction (i.e., down-sampling) block
contains two convolutional layers with ReLU activation, ending with a $3\times
3$ stride-2 max pooling layer. On the other hand, each expansive (i.e., up-
sampling) block contains one deconvolutional layer followed by two
convolutional layers and a concatenation layer. The last concatenation layer
combines the last output

###  4.3 Weighted Categorical Cross-entropy Loss

Due to the fact that background pixels heavily outnumber skeleton pixels in
target images, mistakes in each of them cannot be weighted equally. This is a
classic issue of class imbalance which we address in our loss by giving higher
weight to misclassifications on the positive class (i.e., skeleton). This
incentivizes the model to be more careful with the skeletal region. On that
basis, our loss function is a weighted categorical cross-entropy (CCE)
computed as:

$\mathcal{L}(p_{t},p_{r})=1-w*p_{t}*p_{r}$

where $w$ is the class weights ($w_{0}\ll w_{1}$), $p_{t}$ is the true class
(0 for background, 1 for skeleton), and $p_{r}$ is the predicted probability.

Figure 2: Output example for the f1 and correlation scores (M-CCORR=corr) of a
good, but manually shifted skeleton output. Though the skeletons look alike,
the F1 score is too low while M-CCORR shows a reasonable number

##  5 Results

###  5.1 Experimental Setup

We have experimented with multiple U-Net based architectures using different
losses and weights. Although there are differences among the architectures,
there are some parameters that are fixed across all experiments. These
parameters are shown in Table 1.

Parameter | Value
---|---
Optimizer | Adam
Learning Rate | 0.001
$\beta_{1}$ | 0.9
$\beta_{2}$ | 0.999
Batch Size | 32
loss | Weighted CCE
Loss weights | [1, 25]
Metrics | [F1, M-CCORR]
Table 1: Hyper-parameters across architectural experiments

###  5.2 New Metric Proposal

Though the F1 score is used widely on this dataset as an evaluation metric [1,
3, 6, 5], one issue that arises with it is that it is very sensitive to small
pixel-offset errors. If the skeleton image and the ground truth are partially
translated (even by few pixels), the F1 score decreases dramatically (check
Figure 2 for illustration). This pixel-offset sensitivity poorly evaluates
good-looking skeletons though they logically match the ground truth. Hence, we
propose the usage of template matching as a new translation-aware metric for
this problem. We have developed a variant of the normalized cross-correlation
coefficient which we call the Matching Cross-correlation (M-CCORR) as defined
in the following equation:

$\textit{M-CCORR}(y_{t},y_{p})=\frac{max(CCORR(y_{t},y_{p}))}{log_{2}(\mathcal{D}(y_{t},y_{p})+2)}$

where $y_{t}$ and $y_{p}$ are the ground truth and prediction output images
respectively and $\mathcal{D}$ is a function that return the distance between
the bounding box center of the ground truth skeleton and that calculated on
the predicted skeleton. This denominator ensures that the Matching CCORR-based
metric we are proposing is not only aware of pixel-offsets but also of the
logical resemblance between the target and prediction.

###  5.3 Experiments

We have experimented with multiple stages of our U-Net based architectures,
through varying the number of corrector after the skeletonizer models as shown
in Figre 1. The architectures are the following:

  * •

One-Stage (Skeletonizer): The entire pipeline consists of a signle model that
gets trained to perform the whole skeletonization process.

  * •

Two-Stage (Skeletonizer + Corrector): The training pipeline consists of two
models: a skeletonizer that produces rough skeletons and a corrector which
enhances rough skeletons.

  * •

Three-Stage (Skeletonizer + 2 Correctors): The training pipeline consists of
three models: a skeletonizer and two correctors in series. This adds an extra
enhancement layer on its 2-stage predecessor.

The following Table 2 shows our $F1$ and M-CCORR scores corresponding to every
$n$-stage experiment:

Model | F1 Score | M-CCORR
---|---|---
One-Stage | 0.4866 | 0.5604
Two-Stage | 0.5968 | 0.6399
Three-Stage | 0.5802 | 0.6271
Table 2: F1 and M-CCORR scores corresponding to different number of stages of
the $U$-$Net$ like model

The two-stage model outperforms both one-stage and three-stage models in both
F1 and M-CCORR scores. It has an F1 score of 0.5968 and an M-CCORR score of
0.6399 on the validation set. A single skeletonizer produces thick skeletons
causing many false positives reducing the F1 score, which is fine-tuned by the
corrector following it in the two-stage model. Adding more than one corrector
(i.e, the three-stage model) does not further improve the results.

In addition to the above setups, we have explored multiple other modifications
that did not further improve the F1 and M-CCORR scores. These include (but are
not limited to): L1, hinge and focal losses, adding a dilation layer before
the output, and applying a distance transform on the input images.

##  6 Conclusion

In summary, our preliminary results have shown that splitting the problem of
skeletonization into two stage does indeed bring significant improvement of
accuracy. We have also shown that there is a limit to the split-level beyond
which performance will not improve. The literature has several solutions that
perform very well in one stage, so incorporating them into a similar 2-stage
workflow can boost their accuracy even further. Last but not least, we have
demonstrated how sensitivity to translation in the F1 score undermines its
representativeness of skeleton quality - proposing normalized correlation
coefficients in its stead.

## References

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