#  $L^{2}$-blowup estimates of the wave equation and its application to local
energy decay

Ryo Ikehata
Department of Mathematics
Division of Educational Sciences
Graduate School of Humanities and Social Sciences
Hiroshima University
Higashi-Hiroshima 739-8524, Japan<EMAIL_ADDRESS>
###### Abstract

We consider the Cauchy problems in ${\bf R}^{n}$ for the wave equation with a
weighted $L^{1}$-initial data. We derive sharp infinite time blowup estimates
of the $L^{2}$-norm of solutions in the case of $n=1$ and $n=2$. Then, we
apply it to the local energy decay estimates for $n=2$, which is not studied
so completely when the $0$-th moment of the initial velocity does not vanish.
The idea to derive them is strongly inspired from a technique used in [12,
15].

##  1 Introduction

000Keywords and Phrases: Wave equation; weighted $L^{1}$-data; local energy;
low dimensional case; blowup in infinite time; sharp estimates.0002010
Mathematics Subject Classification. Primary 35L05; Secondary 35B40, 35C20.

We consider the Cauchy problem of the wave equation:

$\displaystyle u_{tt}-\Delta u=0,\ \ \ (t,x)\in(0,\infty)\times{\bf R}^{n},$
(1.1) $\displaystyle u(0,x)=u_{0}(x),\quad u_{t}(0,x)=u_{1}(x),\ x\in{\bf
R}^{n}.$ (1.2)

Here, we assume, for the moment, $[u_{0},u_{1}]\in H^{1}({\bf R}^{n})\times
L^{2}({\bf R}^{n})$.
Concerning the existence of a unique energy solution to problem (1.1)-(1.2),
by the Lumer-Phillips Theorem one can find that the problem (1.1)-(1.2) has a
unique weak solution

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

satisfying the energy conservation law such that

$E(t)=E(0),\quad t\geq 0,$ (1.3)

where the total energy $E(t)$ for the solution to problem (1.1)-(1.2) can be
defined by

$E(t):=\frac{1}{2}\left(\|u_{t}(t,\cdot)\|^{2}+\|\nabla
u(t,\cdot)\|^{2}\right).$

Here, $\|u\|$ denotes the usual $L^{2}$-norm of $u\in L^{2}({\bf R}^{n})$.

Our main purpose is to establish the sharp estimates in time of the factor
$M(t):=\|u(t,\cdot)\|$. This observation is important from the view point of
the local energy decay, which is one of main topics in the wave equation
field. As is known from the work of C. Morawetz [22], the main task to get the
local energy decay is to control the behavior of the function $M(t)$. By the
way, as in a series of papers [1, 13] and so on, in the exterior mixed
problems on a star-shaped compliment (or more generally just on exterior
domains) one can control $M(t)$ for all $n\geq 2$. While if one considers the
local energy decay problems to (1.1) in the Euclidean space ${\bf R}^{n}$ (no
obstacle case!) one can proceed the same argument as in the exterior mixed
problem case when $n\geq 3$ (see [5]), however, in the case of $n=2$, one has
to impose stronger assumption such that $I_{0}:=\int_{{\bf
R}^{2}}u_{1}(x)dx=0$, which means the vanishing condition of the $0$th moment
of the initial data. Note that in these problems, the author images that one
never assumes the compactness of the support of the initial data, and low
regularity on them to control the function $M(t)$, and to get the local energy
decay. Therefore, it is highly desirable to remove the stronger condition
$I_{0}=0$ in the two dimensional case for the completeness of the theory. In
this connection, for $n\geq 3$ $L^{p}$-decay problems of the solution to
problem (1.1)-(1.2) including the Klein-Gordon type are already discussed by
[32] and [38], however, there the compactness of the support of the initial
data seems to be conscious, and their main concern is not in local energy
decay itself, but an application to nonlinear problems of the
$L^{p}$-boundedness.

Now, in this paper we study the asymptotic behavior of the function $M(t)$ as
$t\to\infty$ under the non-compactly supported initial data framework. It
seems that this problem does not studied so far. We employ the simple Fourier
analysis to get the estimates of the function $M(t)$ by dividing the Fourier
space of the integrand of the Fourier transformed solution $\hat{u}(t,\xi)$
into two parts: one is low frequency zone ($|\xi|\leq q(t)$), and the other is
high frequency zone ($|\xi|\geq q(t)$), where $q(t)$ is a $t$-variable
function chosen suitably. So, one never relies on the Hardy type inequality,
which never holds in the low dimensional case (cf. [25]) because a core of the
idea in a series of papers [1, 5, 9, 10, 13, 16] is that the boundedness of
$M(t)$ can be derived by using the Hardy inequality. The problem to study the
behavior of $M(t)$ as $t\to\infty$ is important from the view point that the
Hardy type inequality does not hold in the low dimensional case.

Before going to introduce our Theorems, we set

$I_{0,n}:=\|u_{0}\|+\|u_{0}\|_{L^{1}({\bf
R}^{n})}+\|u_{1}\|+\|u_{1}\|_{L^{1}({\bf R}^{n})}.$

Our first result is concerned with the optimal infinite time blowup result in
the case of $n=1$.

###### Theorem 1.1

Let $n=1$. Let $[u_{0},u_{1}]\in H^{1}({\bf R})\times L^{2}({\bf R})$. Then,
the solution $u(t,x)$ to problem (1.1)-(1.2) satisfies the following
properties under the additional regularity on the initial data:

$[u_{0},u_{1}]\in L^{1}({\bf R})\times L^{1}({\bf
R})\quad\Rightarrow\quad\|u(t,\cdot)\|\leq C_{1}I_{0,1}\sqrt{t},$
$[u_{0},u_{1}]\in L^{1}({\bf R})\times L^{1,1}({\bf R})\quad\Rightarrow\quad
C_{2}\left|\int_{{\bf R}}u_{1}(x)dx\right|\sqrt{t}\leq\|u(t,\cdot)\|$

for $t\gg 1$, where $C_{j}>0$ ($j=1,2$) are constants depending only on the
space dimension.

Our next result is the case of $n=2$.

###### Theorem 1.2

Let $n=2$. Let $[u_{0},u_{1}]\in H^{1}({\bf R}^{2})\times L^{2}({\bf R}^{2})$.
Then, the solution $u(t,x)$ to problem (1.1)-(1.2) satisfies the following
properties under the additional regularity on the initial data:

$[u_{0},u_{1}]\in L^{1}({\bf R}^{2})\times L^{1}({\bf
R}^{2})\quad\Rightarrow\quad\|u(t,\cdot)\|\leq C_{1}I_{0,2}\sqrt{\log t},$
$[u_{0},u_{1}]\in L^{1}({\bf R}^{2})\times L^{1,1}({\bf
R}^{2})\quad\Rightarrow\quad C_{2}\left|\int_{{\bf
R}^{2}}u_{1}(x)dx\right|\sqrt{\log t}\leq\|u(t,\cdot)\|$

for $t\gg 1$, where $C_{j}>0$ ($j=1,2$) are constants depending only on the
space dimension.

###### Remark 1.1

Surprisingly if one imposes the additional $L^{1,1}$-regularity on the initial
velocity, one can get the sharp infinite time blowup result in the one and two
dimensional free waves. It seems that this crucial phenomenon is not so known.
Generally, people assume a vanishing moment condition $\int_{{\bf
R}^{n}}u_{1}(x)dx\ =0$ when one studies the $L^{2}$-boundedness of the
solution $u(t,x)$, for the low dimensional case $n=1,2$ (cf., [11]).

###### Remark 1.2

From the proof done in Section 3 for the upper bound estimates one can see
that $L^{1,1}$-assumption is no need to get upper bound estimates, and only
$L^{1}$-assumption works well. On the other hand, to obtain lower bound
estimates one needs to impose the $L^{1,1}$ assumption on the initial velocity
$u_{1}$ (see (2.14) and (2.15)).

Let us introduce one example to make sure the reliability of Theorems in the
paper about only in the one dimensional case.
Example. Let $n=1$, $u_{0}\equiv 0$, and we choose $u_{1}\in(L^{1,1}({\bf
R})\cap L^{2}({\bf R}))$ as follows:

$u_{1}(x)=\left\\{\begin{array}[]{ll}\displaystyle{2}&\qquad(|x|\leq
1),\\\\[5.69046pt] \displaystyle{0}&\qquad(|x|>1).\end{array}\right.$

Then, one easily gets the formula:

$u(t,x)=\frac{1}{2}\int_{x-t}^{x+t}u_{1}(r)dr=-\chi(x-t)+\chi(x+t),$

where

$\chi(x):=\frac{1}{2}\int_{0}^{x}u_{1}(r)dr.$

One can soon check that

$-\chi(x-t)=\left\\{\begin{array}[]{ll}\displaystyle{-(x-t)}&\qquad(|x-t|\leq
1),\\\\[5.69046pt] \displaystyle{-1}&\qquad(x-t>1),\\\\[5.69046pt]
\displaystyle{1}&\qquad(x-t<-1),\end{array}\right.$
$\chi(x+t)=\left\\{\begin{array}[]{ll}\displaystyle{(x+t)}&\qquad(|x+t|\leq
1),\\\\[5.69046pt] \displaystyle{1}&\qquad(x+t>1),\\\\[5.69046pt]
\displaystyle{-1}&\qquad(x+t<-1).\end{array}\right.$

Therefore, if one fixes a sufficiently large $t_{0}>1$, then one can finally
arrive at the following situation for the solution $u(t_{0},x)$:

$u(t_{0},x)=\left\\{\begin{array}[]{ll}\displaystyle{2}&\qquad(1-t_{0}\leq
x\leq t_{0}-1),\\\\[5.69046pt]
\displaystyle{x+t_{0}+1}&\qquad(-t_{0}-1<x<1-t_{0}),\\\\[5.69046pt]
\displaystyle{-x+t_{0}+1}&\qquad(t_{0}-1<x<t_{0}+1).\\\\[5.69046pt]
\displaystyle{0}&\qquad(x<-t_{0}-1,\,t_{0}+1<x).\end{array}\right.$

Therefore, one can compute

$\int_{{\bf R}}|u(t_{0},x)|^{2}dx=8(t_{0}-1)+\frac{16}{3},$

which implies the credibility of the result

$\|u(t_{0},\cdot)\|\sim\sqrt{t_{0}},\quad t_{0}\gg 1.$

In this connection, one naturally has the condition $\displaystyle{\int_{{\bf
R}}}u_{1}(x)dx=4\neq 0$. This example can be stated in the fundamental text
book of PDE (e.g. [17]).

###### Remark 1.3

In a sense, the infinite time blowup results in $L^{2}$-sense may express
quantitatively failure of the Huygens principle.

Notation. Throughout this paper, $\|\cdot\|_{q}$ stands for the usual
$L^{q}({\bf R}^{n})$-norm. For simplicity of notation, in particular, we use
$\|\cdot\|$ instead of $\|\cdot\|_{2}$. We also introduce the following
weighted functional spaces.

$L^{1,\gamma}({\bf R}^{n}):=\left\\{f\in L^{1}({\bf
R}^{n})\;\bigm{|}\;\|f\|_{1,\gamma}:=\int_{{\bf
R}^{n}}(1+|x|^{\gamma})|f(x)|dx<+\infty\right\\}.$

One denotes the Fourier transform ${\cal F}_{x\to\xi}(f)(\xi)$ of $f(x)$ by

${\cal F}_{x\to\xi}(f)(\xi)=\hat{f}(\xi):=\displaystyle{\int_{{\bf
R}^{n}}}e^{-ix\cdot\xi}f(x)dx,\quad\xi\in{\bf R}^{n},$

as usual with $i:=\sqrt{-1}$, and ${\cal F}_{\xi\to x}^{-1}$ expresses its
inverse Fourier transform. Finally, we denote the surface area of the
$n$-dimensional unit ball by
$\omega_{n}:=\displaystyle{\int_{|\omega|=1}}d\omega$.

The paper is organized as follows. In Section 2 we derive the lower bound
estimates of the $L^{2}$-norm of solutions, and in Section 3 we obtain the
upper bound estimates of the $L^{2}$-norm of solutions, and by combining the
results obtained in Sections $2$ and $3$ one can prove Theorems 1.1 and 1.2 at
a stroke. Section 4 is devoted to apply Theorem 1.2 to the local energy decay
results of the wave equation (1.1) in the two dimensional Euclidean space
${\bf R}^{2}$.

##  2 $L^{2}$-lower bound estimates of the solution

In this section, let us derive the lower bound estimates of the $t$-function
$\|u(t,\cdot)\|$ by using the Plancherel Theorem and low frequency estimates.
This technique is well-developed in the damped wave equation field (cf. [12]).
We first prepare two basic facts which will be frequently used in the proof.
Set

$L:=\sup_{\theta\neq 0}\left|\frac{\sin\theta}{\theta}\right|<+\infty.$ (2.1)

Furthermore, since

$\lim_{\theta\to+0}\frac{\sin\theta}{\theta}=1,$

one can find a real number $\delta_{0}\in(0,1)$ such that

$\left|\frac{\sin\theta}{\theta}\right|\geq\frac{1}{2}$ (2.2)

for all $\theta\in(0,\delta_{0}]$. One also prepares the fundamental
inequality:

$|a+b|^{2}\geq\frac{1}{2}|a|^{2}-|b|^{2}$ (2.3)

for all $a,b\in{\bf C}$.
In order to get the lower bound estimate for the quantity $\|u(t,\cdot)\|$, it
suffices to treat $\|w(t,\cdot)\|$ with $w(t,\xi):={\cal
F}_{x\to\xi}(u(t,\cdot))(\xi)=\hat{u}(t,\xi)$ because of the Plancherel
Theorem.
Now we decompose the quantity $\|w(t,\cdot)\|$ as follows: for each $n\geq 1$

$\|w(t,\cdot)\|^{2}=\left(\int_{|\xi|\leq\frac{\delta_{0}}{t}}+\int_{|\xi|\geq\frac{\delta_{0}}{t}}\right)|w(t,\xi)|^{2}d\xi=I_{low}^{(n)}(t)+I_{high}^{(n)}(t).$
(2.4)

Here we have just chosen $t>0$ large enough such that

$\frac{\delta_{0}}{t}\leq 1.$

By the way, as is well-known in the text book of PDE, in the Fourier space
${\bf R}_{\xi}^{n}$ the problem (1.1)-(1.2) and its solution $u(t,x)$ can be
transformed into the following ODE with parameter $\xi\in{\bf R}_{\xi}^{n}$:

$\displaystyle w_{tt}+|\xi|^{2}w=0,\ \ \ t>0,\quad\xi\in{\bf R}_{\xi}^{n},$
(2.5) $\displaystyle w(0,\xi)=w_{0}(\xi),\quad w_{t}(0,\xi)=w_{1}(\xi),\ \ \
\xi\in{\bf R}^{n},$ (2.6)

where $w_{0}(\xi):=\hat{u}_{0}(\xi)$ and $w_{1}(\xi):=\hat{u}_{1}(\xi)$.
Moreover, one can easily solve the problem (2.5)-(2.6) as follows:

$w(t,\xi)=\frac{\sin(t|\xi|)}{|\xi|}w_{1}(\xi)+\cos(t|\xi|)w_{0}(\xi).$ (2.7)

Now, let us give the lower bound estimates for the quantity $I_{low}^{(n)}(t)$
because of $\|w(t,\cdot)\|^{2}\geq I_{low}^{(n)}(t)$. The technique developed
below is strongly inspired from the idea in [12]. Indeed, it follows from
(2.7) and (2.3) that

$I_{low}^{(n)}(t)=\int_{|\xi|\leq\frac{\delta_{0}}{t}}\left|\frac{\sin(t|\xi|)}{|\xi|}w_{1}(\xi)+\cos(t|\xi|)w_{0}(\xi)\right|^{2}d\xi$
$\geq\frac{1}{2}\int_{|\xi|\leq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi-\int_{|\xi|\leq\frac{\delta_{0}}{t}}\cos^{2}(t|\xi|)|w_{0}(\xi)|^{2}d\xi$
$=:\frac{1}{2}J_{1}(t)-J_{2}(t).$ (2.8)

Let us first estimate $J_{1}(t)$ by using the decomposition of the initial
data $w_{1}(\xi)$ in the Fourier space:

$w_{1}(\xi)=P+(A(\xi)-iB(\xi)),\quad\xi\in{\bf R}_{\xi}^{n},$

where

$P:=\int_{{\bf R}^{n}}u_{1}(x)dx,$ $A(\xi):=\int_{{\bf
R}^{n}}(\cos(x\xi)-1)u_{1}(x)dx,\quad B(\xi):=\int_{{\bf
R}^{n}}\sin(x\xi)u_{1}(x)dx.$

It is known (see [8]) that with some constant $M>0$ one has

$|A(\xi)-iB(\xi)|\leq M|\xi|\|u_{1}\|_{L^{1,1}},\quad\xi\in{\bf R}_{\xi}^{n},$
(2.9)

in the case when $u_{1}\in L^{1,1}({\bf R}^{n})$. Then, it follows from (2.3)
that

$J_{1}(t)=\int_{|\xi|\leq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi$
$\geq\frac{P^{2}}{2}\int_{|\xi|\leq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}d\xi-\int_{|\xi|\leq\frac{\delta_{0}}{t}}|A(\xi)-iB(\xi)|^{2}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}d\xi$
$=\frac{P^{2}}{2}K_{1}(t)-K_{2}(t).$ (2.10)

$K_{2}(t)$ can be estimated from above as follows by using (2.9):

$K_{2}(t)\leq
M^{2}\|u_{1}\|_{L^{1,1}}^{2}\int_{|\xi|\leq\frac{\delta_{0}}{t}}\sin^{2}(t|\xi|)d\xi=M^{2}\|u_{1}\|_{L^{1,1}}^{2}\omega_{n}\int_{0}^{\frac{\delta_{0}}{t}}r^{n-1}dr$
$=\frac{M^{2}}{n}\omega_{n}\delta_{0}^{n}\|u_{1}\|_{L^{1,1}}^{2}t^{-n},\quad
t\gg 1.$ (2.11)

While, one can get the lower bound estimate for $K_{1}(t)$ because of (2.2):

$K_{1}(t)=t^{2}\int_{|\xi|\leq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{(t|\xi|)^{2}}d\xi\geq\frac{t^{2}}{4}\int_{|\xi|\leq\frac{\delta_{0}}{t}}d\xi$
$=\frac{\omega_{n}\delta_{0}^{n}}{4n}t^{2-n},\quad t\gg 1.$ (2.12)

Therefore from (2.10), (2.11) and (2.12) one can get the estimate from below
for $J_{1}(t)$:

$J_{1}(t)\geq\frac{P^{2}}{2}\frac{\omega_{n}\delta_{0}^{n}}{4n}t^{2-n}-\frac{M^{2}}{n}\omega_{n}\delta_{0}^{n}\|u_{1}\|_{L^{1,1}}^{2}t^{-n}\quad
t\gg 1.$ (2.13)

Since the upper bound estimate of $J_{2}(t)$ can be easily obtained as
follows:

$J_{2}(t)\leq\int_{|\xi|\leq\frac{\delta_{0}}{t}}|w_{0}(\xi)|^{2}d\xi\leq\|u_{0}\|_{L^{1}}^{2}\omega_{n}\int_{0}^{\frac{\delta_{0}}{t}}r^{n-1}dr=\frac{\omega_{n}\delta_{0}^{n}}{n}\|u_{0}\|_{L^{1}}^{2}t^{-n},$

by (2.13), (2.4) and (2.8) one has just arrived at the following crucial lower
bound estimate for $\|w(t,\cdot)\|$:

$\|w(t,\cdot)\|^{2}\geq
I_{low}^{n}(t)\geq\frac{P^{2}}{4}\frac{\omega_{n}\delta_{0}^{n}}{4n}t^{2-n}-\frac{M^{2}}{n}\omega_{n}\delta_{0}^{n}\|u_{1}\|_{L^{1,1}}^{2}t^{-n}-\frac{\omega_{n}\delta_{0}^{n}}{n}\|u_{0}\|_{L^{1}}^{2}t^{-n}\quad
t\gg 1.$ (2.14)

Thus, there exists a positive real number $t_{0}$ such that

$\|w(t,\cdot)\|^{2}\geq\frac{P^{2}}{32n}\omega_{n}\delta_{0}^{n}t^{2-n}$
(2.15)

for all $t\geq t_{0}$. It should be mentioned that $t_{0}>0$ depends on $n$
and the quantities $\|u_{1}\|_{L^{1.1}}$ and $\|u_{0}\|_{L^{1}}$.
By choosing $n=1$ in (2.15) one has the following blowup property of the
solution when $P\neq 0$.

###### Lemma 2.1

Let $n=1$, and $[u_{0},u_{1}]\in L^{1}({\bf R}^{n})\times L^{1,1}({\bf
R}^{n})$. Then, it holds that

$\|w(t,\cdot)\|^{2}\geq CP^{2}t,\quad t\gg 1.$

Let us prove the statement for $n=2$ at a stroke by using a trick with a
function $e^{-r^{2}}$. Indeed, it follows from (2.3) and a similar argument to
the one dimensional case that

$\|w(t,\cdot)\|^{2}\geq\frac{1}{2}\int_{{\bf
R}^{2}}\frac{\sin^{2}(tr)}{r^{2}}|w_{1}(\xi)|^{2}d\xi-\int_{{\bf
R}^{2}}\cos^{2}(tr)|w_{0}(\xi)|^{2}d\xi$ $\geq\frac{1}{2}\int_{{\bf
R}^{2}}e^{-r^{2}}\frac{\sin^{2}(tr)}{r^{2}}e^{r^{2}}|P+(A(\xi)-iB(\xi))|^{2}d\xi-\|u_{0}\|^{2}$
$\geq\frac{1}{4}P^{2}\int_{{\bf
R}^{2}}e^{-r^{2}}\frac{\sin^{2}(tr)}{r^{2}}d\xi-\frac{1}{2}\int_{{\bf
R}^{2}}e^{-r^{2}}\frac{\sin^{2}(tr)}{r^{2}}\left(M^{2}\|u_{1}\|_{L^{1,1}}^{2}r^{2}\right)d\xi-\|u_{0}\|^{2}$
$\geq\frac{1}{4}P^{2}\int_{{\bf
R}^{2}}e^{-r^{2}}\frac{\sin^{2}(tr)}{r^{2}}d\xi-\frac{1}{2}M^{2}\|u_{1}\|_{L^{1,1}}^{2}\int_{{\bf
R}^{2}}e^{-r^{2}}d\xi-\|u_{0}\|^{2}$
$=:\frac{1}{4}P^{2}T(t)-\frac{\omega_{2}}{4}M^{2}\|u_{1}\|_{L^{1,1}}^{2}-\|u_{0}\|^{2}.$
(2.16)

Now, we apply an idea which has its origin in [15]. For this purpose, we set

$\nu_{j}:=(\frac{1}{4}+j)\frac{\pi}{t},\quad\mu_{j}:=(\frac{3}{4}+j)\frac{\pi}{t}\quad(j=0,1,2,\cdots),$

and choose $t>1$ large enough such that $\nu_{1}=\frac{5\pi}{4t}<1$. Then,
since

$|\sin(tr)|\geq\frac{1}{\sqrt{2}}$

for $r\in[\nu_{j},\mu_{j}]$ and for each $j=0,1,2,\cdots$, one has the
estimate:

$T(t)=\int_{{\bf
R}^{2}}e^{-r^{2}}\frac{\sin^{2}(tr)}{r^{2}}d\xi\geq\frac{1}{2}\sum_{j=0}^{\infty}\int_{\nu_{j}\leq
r\leq\mu_{j}}\frac{e^{-r^{2}}}{r^{2}}d\xi=\frac{\omega_{2}}{2}\left(\sum_{j=0}^{\infty}\int_{\nu_{j}}^{\mu_{j}}\frac{e^{-r^{2}}}{r}dr\right)$
(2.17)
$\geq\frac{\omega_{2}}{2}\left(\frac{1}{2}\int_{\nu_{1}}^{\infty}\frac{e^{-r^{2}}}{r}dr\right)\geq\frac{\omega_{2}}{4}\int_{\nu_{1}}^{1}\frac{e^{-r^{2}}}{r}dr$
(2.18)
$\geq\omega_{2}\frac{e^{-1}}{4}\int_{\nu_{1}}^{1}\frac{1}{r}dr=\omega_{2}\frac{e^{-1}}{4}(\log
t+\log 4-\log 5\pi),$ (2.19)

where in the inequality from (2.17) to (2.18) one has just used the monotone
decreasing property of the function $r\mapsto\frac{e^{-r^{2}}}{r}$, and the
fact that $\frac{\pi}{2t}=\mu_{j}-\nu_{j}=\nu_{j+1}-\mu_{j}$ ($\forall j$).
Therefore, by (2.16) and (2.19) one has the following estimates for $n=2$ for
large $t>1$.

###### Lemma 2.2

Let $n=2$, and $[u_{0},u_{1}]\in L^{1}({\bf R}^{n})\times L^{1,1}({\bf
R}^{n})$. Then, it holds that

$\|w(t,\cdot)\|^{2}\geq CP^{2}\log t,\quad t\gg 1.$

##  3 $L^{2}$-upper bound estimates of the solution

In this section, one derives upper bound estimates of the quantity
$\|u(t,\cdot)\|$ as $t\to\infty$ by calculating the function $w(t,\xi)$ in
both high and low frequency region.

From (2.4) one first shows the upper bound estimate for $I_{low}^{(n)}(t)$ for
all $n\geq 1$. Indeed, by (2.8) one has

$I_{low}^{(n)}(t)\leq\int_{|\xi|\leq\frac{\delta_{0}}{t}}|\frac{\sin(t|\xi|)}{|\xi|}w_{1}(\xi)+\cos(t|\xi|)w_{0}(\xi)|^{2}d\xi$
$\leq
2\int_{|\xi|\leq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi+2\int_{|\xi|\leq\frac{\delta_{0}}{t}}\cos^{2}(t|\xi|)|w_{0}(\xi)|^{2}d\xi$
$=2L_{1}(t)+2L_{2}(t).$ (3.1)

To begin with, it is easy to get the estimate for $L_{2}(t)$ as in the
estimate for $J_{2}(t)$ in Section 2:

$L_{2}(t)\leq\int_{|\xi|\leq\frac{\delta_{0}}{t}}|w_{0}(\xi)|^{2}d\xi\leq\|u_{0}\|_{L^{1}}^{2}\omega_{n}\int_{0}^{\frac{\delta_{0}}{t}}r^{n-1}dr$
$=\frac{\omega_{n}\delta_{0}^{n}}{n}\|u_{0}\|_{L^{1}}^{2}t^{-n}.$ (3.2)

Let us estimate $L_{1}(t)$ as follows:

$L_{1}(t)=\int_{|\xi|\leq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi$
$\leq\|u_{1}\|_{L^{1}}^{2}\int_{|\xi|\leq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}d\xi.$
(3.3)

As in (3.2) one has

$\int_{|\xi|\leq\frac{\delta_{0}}{t}}d\xi\leq\frac{\omega_{n}\delta_{0}^{n}}{n}t^{-n},\quad
t\gg 1.$ (3.4)

Therefore, if one uses (2.1), (3.3) and (3.4) then it holds that

$L_{1}(t)\leq\|u_{1}\|_{L^{1}}^{2}L^{2}t^{2}\frac{\omega_{n}\delta_{0}^{n}}{n}t^{-n}=\|u_{1}\|_{L^{1}}^{2}L^{2}\frac{\omega_{n}\delta_{0}^{n}}{n}t^{2-n},\quad
t\gg 1.$ (3.5)

Thus, from (3.1), (3.2) and (3.5) one can obtain the low-frequency estimate

$I_{low}^{(n)}(t)\leq
C\left(\|u_{0}\|_{L^{1}}^{2}t^{-n}+\|u_{1}\|_{L^{1}}^{2}t^{2-n}\right),\quad
t\gg 1,$ (3.6)

with some constant $C>0$.

Next, one treats $I_{high}^{(n)}(t)$ to get the upper bound estimate. Indeed,
similar to the above estimate one stars with the following inequalities.

$I_{high}^{(n)}(t)\leq\int_{|\xi|\geq\frac{\delta_{0}}{t}}|\frac{\sin(t|\xi|)}{|\xi|}w_{1}(\xi)+\cos(t|\xi|)w_{0}(\xi)|^{2}d\xi$
$\leq
2\int_{|\xi|\geq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi+2\int_{|\xi|\geq\frac{\delta_{0}}{t}}\cos^{2}(t|\xi|)|w_{0}(\xi)|^{2}d\xi$
$=2N_{1}^{(n)}(t)+2N_{2}^{(n)}(t).$ (3.7)

It is easy to treat $N_{2}^{(n)}(t)$ as follows. This can hold for all $n\geq
1$.

$N_{2}^{(n)}(t)\leq\int_{|\xi|\geq\frac{\delta_{0}}{t}}|w_{0}(\xi)|^{2}d\xi\leq\|u_{0}\|^{2}.$
(3.8)

Let us estimate $N_{1}^{(n)}(t)$ when $n=1,2$. First, for all $n\geq 1$ one
has

$N_{1}^{(n)}(t)=\int_{|\xi|\geq\frac{\delta_{0}}{\sqrt{t}}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi+\int_{\frac{\delta_{0}}{t}\leq|\xi|\leq\frac{\delta_{0}}{\sqrt{t}}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi$
$=:O_{1}^{(n)}(t)+O_{2}^{(n)}(t).$ (3.9)

To begin with, we consider the case $n=1$. In this case, for $O_{1}^{(1)}(t)$
one can get the estimate

$O_{1}^{(1)}(t)\leq\frac{t}{\delta_{0}^{2}}\int_{|\xi|\geq\frac{\delta_{0}}{\sqrt{t}}}\sin^{2}(t|\xi|)|w_{1}(\xi)|^{2}d\xi$
$\leq\frac{t}{\delta_{0}^{2}}\int_{|\xi|\geq\frac{\delta_{0}}{\sqrt{t}}}|w_{1}(\xi)|^{2}d\xi\leq\frac{t}{\delta_{0}^{2}}\|u_{1}\|^{2},\quad
t\gg 1.$ (3.10)

While, $O_{2}^{(1)}(t)$ can be treated as follows:

$O_{2}^{(1)}(t)\leq\|u_{1}\|_{1}^{2}\int_{\frac{\delta_{0}}{t}\leq|\xi|\leq\frac{\delta_{0}}{\sqrt{t}}}\frac{1}{|\xi|^{2}}d\xi$
$=\|u_{1}\|_{1}^{2}\omega_{1}\int_{\frac{\delta_{0}}{t}}^{\frac{\delta_{0}}{\sqrt{t}}}\frac{1}{r^{2}}dr=\frac{\omega_{1}}{\delta_{0}}\|u_{1}\|_{1}^{2}(t-\sqrt{t}),\quad
t\gg 1.$ (3.11)

Thus, from (3.7), (3.8), (3.9), (3.10) and (3.11) one has the estimate for
$I_{high}^{(n)}(t)$ in the case of $n=1$:

$I_{high}^{(1)}(t)=C\left(\|u_{0}\|^{2}+\|u_{1}\|^{2}t+\|u_{1}\|_{L^{1}}^{2}(t-\sqrt{t})\right),\quad
t\gg 1.$ (3.12)

Next, let us give sharp estimates for $N_{1}^{(2)}(t)$ in the case $n=2$ at a
stroke by using a more precise task. For this purpose, by choosing $t>1$ large
enough to realize the relation $1\leq\log t\leq t\leq t^{2}$ one has a
decomposition of the desired integrand:

$N_{1}^{(2)}(t)=\int_{|\xi|\geq\frac{\delta_{0}}{\sqrt{\log
t}}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi$
$+\int_{\frac{\delta_{0}}{\sqrt{\log
t}}\geq|\xi|\geq\frac{\delta_{0}}{\sqrt{t}}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi+\int_{\frac{\delta_{0}}{\sqrt{t}}\geq|\xi|\geq\frac{\delta_{0}}{t}}\frac{\sin^{2}(t|\xi|)}{|\xi|^{2}}|w_{1}(\xi)|^{2}d\xi$
$=:O_{1}(t)+O_{2}(t)+O_{3}(t).$ (3.13)

Let us estimate them in order.
First, one has

$O_{1}(t)\leq\frac{\log
t}{\delta_{0}^{2}}\int_{|\xi|\geq\frac{\delta_{0}}{\sqrt{\log
t}}}|w_{1}(\xi)|^{2}d\xi$ $\leq\frac{\log
t}{\delta_{0}^{2}}\|u_{1}\|^{2},\quad t\gg 1.$ (3.14)

For $O_{2}(t)$ one can proceed as follows:

$O_{2}(t)\leq\|u_{1}\|_{L^{1}}^{2}\omega_{2}\int_{\frac{\delta_{0}}{\sqrt{t}}}^{\frac{\delta_{0}}{\sqrt{\log
t}}}\frac{1}{r}dr$ $\leq\omega_{2}\|u_{1}\|_{L^{1}}^{2}2^{-1}\left(\log
t-\log(\log t)\right),\quad t\gg 1.$ (3.15)

Lastly, we treat $O_{3}(t)$ to give the following estimate such that

$O_{3}(t)\leq\|u_{1}\|_{L^{1}}^{2}\int_{\frac{\delta_{0}}{t}\leq|\xi|\leq\frac{\delta_{0}}{\sqrt{t}}}\frac{1}{r^{2}}d\xi$
$\leq\omega_{2}\|u_{1}\|_{L^{1}}^{2}2^{-1}\log t,\quad t\gg 1.$ (3.16)

Thus, it follows from (3.13), (3.14), (3.15) and (3.16) that for large $t\gg
1$

$N_{1}^{(2)}(t)\leq C\left(\|u_{1}\|_{L^{1}}^{2}\log t+\|u_{1}\|^{2}\log
t+\|u_{1}\|_{L^{1}}^{2}(\log t-\log(\log t))\right),\quad t\gg 1$ (3.17)

with some constant $C>0$. Therefore, by combining (3.7), (3.8) and (3.17) one
can get the high-frequency estimate for $n=2$:

$I_{high}^{(2)}(t)\leq
C\left(\|u_{0}\|^{2}+\|u_{1}\|^{2}+\|u_{1}\|_{L^{1}}^{2}\right)\log t,\quad
t\gg 1.$ (3.18)

Finally, it follows from (2.4), (3.6), (3.12) and (3.18) one can get the
crucial upper bound estimates for each $n=1$ and $n=2$.

###### Lemma 3.1

Let $n=1$, and $[u_{0},u_{1}]\in(L^{1}({\bf R}^{n})\cap L^{2}({\bf
R}^{n}))\times(L^{1}({\bf R}^{n})\cap L^{2}({\bf R}^{n}))$. Then, it holds
that

$\|w(t,\cdot)\|^{2}\leq
C(\|u_{0}\|^{2}+\|u_{0}\|_{L^{1}}^{2}+\|u_{1}\|^{2}+\|u_{1}\|_{L^{1}}^{2})t,\quad
t\gg 1.$

###### Lemma 3.2

Let $n=2$, and $[u_{0},u_{1}]\in(L^{1}({\bf R}^{n})\cap L^{2}({\bf
R}^{n}))\times(L^{1}({\bf R}^{n})\cap L^{2}({\bf R}^{n}))$. Then, it holds
that

$\|w(t,\cdot)\|^{2}\leq
C(\|u_{0}\|^{2}+\|u_{0}\|_{L^{1}}^{2}+\|u_{1}\|^{2}+\|u_{1}\|_{L^{1}}^{2})\log
t,\quad t\gg 1.$

Finally, the proofs of Theorems 1.1 and 1.2 are direct consequence of Lemmas
2.1, 2.2, 3.1 and 3.2, and the Plancherel Theorem.

##  4 Application to local energy decay problems

In this section we apply Theorem 1.2 to the local energy decay problems. This
problem is very important problem in the wave equation field, and one can
observe many important previous papers (see [1, 2, 3, 4, 5, 6, 9, 10, 13, 16,
18, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 35, 36, 37, 39, 40]), and a
recent interesting application to inverse scattering problem of the local
energy decay can be found in [19]. In particular, Morawetz [22] studied such
problems by constructing the so-called Morawetz identity to develop the
multiplier method (see [7] for its text book). The author in [22] considered
the following exterior mixed problems:

$\displaystyle u_{tt}-\Delta u=0,\ \ \ (t,x)\in(0,\infty)\times\Omega,$ (4.1)
$\displaystyle u(0,x)=u_{0}(x),\quad u_{t}(0,x)=u_{1}(x),\ \ \ x\in\Omega,$
(4.2) $\displaystyle u(t,\sigma)=0,\ \ \ \sigma\in\partial\Omega,$ (4.3)

where $\Omega\subset{\bf R}^{n}$ is a smooth exterior domain with bounded
boundary $\partial\Omega$ and star-shaped compliment $\Omega^{c}$. In [22] the
author derived the crucial estimate:

$\int_{B_{R}\cap\Omega}\left(|u_{t}(t,x)|^{2}+|\nabla u(t,x)|^{2}\right)dx\leq
C_{R}t^{-1},\quad t\gg 1,$

where $C_{R}>0$ is a constant depending on $R>0$ and initial data
$[u_{0},u_{1}]\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)$, and
$B_{R}:=\\{x\in{\bf R}^{n}\,:\,|x|\leq R\\}$ ($R>0$). To study it, the author
assumed the compactness of the support of the initial data (say CSI in short).
The most difficult point to derive the result is to show the
$L^{2}$-boundedness of the solution, that is to say, we have to show that
$\|u(t,\cdot)\|\leq\exists M$ for all $t\geq 0$. Such CSI was essentially used
to derive the $L^{2}$-boundedness of the solution. So, a natural question
arises: can one remove the CSI assumption? This question can be solved by
Zachmanoglou [40], Muraveĭ[24], and the Ikehata’s series of previous papers
[1, 5, 9, 10, 13, 16]. In particular, in [13] the authors developed a refined
multiplier method based on [14] combined with the weighted energy method
originally developed by Todorova-Yordanov [34] in order to accomplish the
removal of CSI. In particular, the method introduced in [14] is useful to get
such $L^{2}$-boundedness of the solution, and the essential idea in [14] is
like this: if one has the so-called Hardy-type inequality, one can get the
$L^{2}$-boundedness of the solution. In this connection, in [22] the author
never used the Hardy inequality. In stead of using the Hardy type one, in [22]
the author solved a corresponding Poisson equation to get $L^{2}$-boundedness
of the solution using CSI. These methods can be also applied to the Cauchy
problem (1.1)-(1.2) in ${\bf R}^{n}$, and in fact, without CSI condition
Charaõ-Ikehata [5] have derived the local energy decay results by using the
method due to [13] to problem (1.1)-(1.2). However, it should be remarked that
in [5] only the two dimensional case is not studied yet. This is because in
the two dimensional case, one can not obtain the Hardy type inequality as was
pointed out by [25, (1.14) of Theorem 5]. One of useful ideas to treat the two
dimensional case is to impose the extra assumption $\int_{{\bf
R}^{2}}u_{1}(x)dx=0$ to get $L^{2}$-boundedness of the solution, and this is
tried in [11] recently. The method in [11] is independent from establishing
the Hardy-type inequality, and the Fourier transform is suitably used like in
this paper. From this observation, in the whole space case one can apply the
Fourier transform, and it suffices to analyze the $L^{2}$-behavior of the
Fourier transformed solution $\hat{u}(t,\xi)$ through the Plancherel Theorem.
At present, in the two dimensional case, under the non-vanishing condition of
the moment $\int_{{\bf R}^{2}}u_{1}(x)dx$ it seems open to get the local
energy decay because in general, the $L^{2}$-boundedness of the solution is
not trivial.

Now, from Theorem 1.2 one can get the upper bound estimate with $log$-order
under the loose condition on the initial data (not necessarily $\int_{{\bf
R}^{2}}u_{1}(x)dx=0$). Although this is not the boundedness, it may work
nicely to get the local energy decay in the two dimensional whole space case.
It should be mentioned once more that one does not rely on any Hardy type
inequalities.

Let us mention our story. First the Morawetz identity in ${\bf R}^{n}$-version
can be stated as follows. Let $u\in C([0,\infty);H^{1}({\bf R}^{n}))\cap
C^{1}([0,\infty);L^{2}({\bf R}^{n}))$ be the weak solution to problem
(1.1)-(1.2). Then, one has the following identity.

$tE(t)=\frac{n-1}{2}\int_{{\bf R}^{n}}u_{1}(x)u_{0}(x)dx+\int_{{\bf
R}^{n}}u_{1}(x)(x\cdot\nabla u_{0}(x))dx-\frac{n-1}{2}\int_{{\bf
R}^{n}}u_{t}(t,x)u(t,x)dx$ $-\int_{{\bf R}^{n}}u_{t}(t,x)(x\cdot\nabla
u(t,x))dx,\quad t\geq 0.$ (4.4)

$L^{2}$-boundedness directly affects the part $\int_{{\bf
R}^{n}}u_{t}(t,x)u(t,x)dx$ because of the Schwarz inequality and energy
identity (1.3):

$|\int_{{\bf R}^{n}}u_{t}(t,x)u(t,x)dx|\leq\left(\int_{{\bf
R}^{n}}|u(t,x)|^{2}dx\right)^{1/2}\left(\int_{{\bf
R}^{n}}|u_{t}(t,x)|^{2}dx\right)^{1/2}=\|u(t,\cdot)\|\|u_{t}(t,\cdot)\|$
$\leq\|u(t,\cdot)\|\sqrt{2E(0)},\quad t\geq 0.$ (4.5)

So, it suffices to control the quantity $\|u(t,\cdot)\|$ in order to get some
good estimate of the term

$F(t):=\int_{{\bf R}^{n}}u_{t}(t,x)u(t,x)dx.$

###### Remark 4.1

In the case when $n=1$, the term $F(t)$ vanishes in (4.4), so in the one
dimensional case we only control the term

$G(t):=\int_{{\bf R}^{n}}u_{t}(t,x)(x\cdot\nabla u(t,x))dx.$

The analysis on $G(t)$ can be done in [5, 13, 16] by the (modified) weighted
energy method due to [34] in order to absorb this term to a part of the total
energy under the non-compact support assumption of the initial data. So, this
topic on $G(t)$ is out of scope of this paper.

Now, one introduces the midway formula derived in [5, 13] to get the local
energy decay in the case when the CSI condition is not assumed.
For each $R>0$, we define the local energy in the region $B_{R}\subset{\bf
R}^{2}$:

$E_{R}(t):=\int_{B_{R}}\left(|u_{t}(t,x)|^{2}+|\nabla u(t,x)|^{2}\right)dx.$

Then, the result in [13, (2.13) at p. 272] and/or [5, (3.5) with $C(x)\equiv
1$ and $\eta=0$] tell us the following fact.

###### Proposition 4.1

Let $n=2$, $R>0$, and assume that $[u_{0},u_{1}]\in H^{1}({\bf R}^{2})\times
L^{2}({\bf R}^{2})$ further satisfies

$\int_{{\bf R}^{2}}|x|\left(|u_{1}(x)|^{2}+|\nabla
u_{0}(x)|^{2}\right)dx<+\infty.$

Then, the weak solution $u(t,x)$ to problem (1.1)-(1.2) satisfies

$(t-R)E_{R}(t)\leq K_{0}+\frac{1}{2}\left|\int_{{\bf
R}^{2}}u_{t}(t,x)u(t,x)dx\right|,\quad\forall t>R,$

where

$K_{0}:=\int_{{\bf R}^{2}}u_{1}(x)(x\cdot\nabla
u_{0}(x))dx+\frac{1}{2}\int_{{\bf R}^{2}}u_{1}(x)u_{0}(x)dx+E(0).$

Therefore, from this Proposition 4.1 one can observe that if one can control
the function $F(t)$ then one has a possibility to get the local energy decay.
In [5] the authors have imposed rather stronger condition such that

$\int_{{\bf R}^{2}}u_{1}(x)dx=0$

to get the ”boundedness” of the function $F(t)$. This is their weak point in
[5] in the two dimensional case.
Let us apply Theorem 2 to control (4.5). Indeed, by applying Theorem 2 to
(4.5) one can get

$\left|F(t)\right|\leq C\sqrt{2E(0)}I_{0,2}\sqrt{\log t},\quad t\gg 1,$ (4.6)

where the constant $C>0$ does not depend on any $R>0$. By combining (4.6) with
Proposition 4.1 one has arrived at the following crucial result.

###### Theorem 4.1

Let $n=2$, $R>0$, and assume that $[u_{0},u_{1}]\in(H^{1}({\bf R}^{2})\cap
L^{1}({\bf R}^{2}))\times(L^{2}({\bf R}^{2})\cap L^{1}({\bf R}^{2}))$ further
satisfies

$\int_{{\bf R}^{2}}|x|\left(|u_{1}(x)|^{2}+|\nabla
u_{0}(x)|^{2}\right)dx<+\infty.$

Then, the weak solution $u(t,x)$ to problem (1.1)-(1.2) satisfies

$E_{R}(t)\leq\frac{K_{0}}{t-R}+\frac{C}{2}\sqrt{2E(0)}I_{0,2}\frac{\sqrt{\log
t}}{t-R},\quad\forall t>R.$

###### Remark 4.2

It should be strongly mentioned that Theorem 4.1 can be derived without
assuming CSI. For the one dimensional case, the effect of the function $F(t)$
is nothing, so one has no any good applications of Theorem 1.1 yet.

###### Remark 4.3

Shapiro [30] has also obtained the local energy decay estimates for $n=2$ with
$(\log t)^{-2}$-order for more general wave equations with variable
coefficients including (1.1), however, the work of [30] has been done under
stronger assumptions such as CSI and high regularity on the initial data. On
the other hand, by [27, Theorem 2.1] applied to the case for $n=2$ one can get
the local energy decay with rate $t^{-1}$ under high regularity assumptions on
the initial data (see also [29, (22)]).

###### Remark 4.4

By combining our method with the Morawetz identity, in the forthcoming project
we will study similar local energy decay estimates for the linear Klein-Gordon
equation in the framework of non-compactly supported initial data.

Acknowledgement. The author would like to thank Masaru Ikehata (Hiroshima
University) for his fruitful discussion at the preparation stage of this
paper. The work of the author was supported in part by Grant-in-Aid for
Scientific Research (C)20K03682 of JSPS.

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# Mapping Temporary Slums from Satellite Imagery using a Semi-Supervised
Approach

M. Fasi ur Rehman, Izza Ali, Waqas Sultani, Mohsen Ali M. Fasi ur Rehman and
Izza Ali are with the School of Humanities and Social Sciences and Waqas
Sultani and Mohsen Ali are with the department of computer science,
Information Technology University, Lahore. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

One billion people worldwide are estimated to be living in slums, and
documenting and analyzing these regions is a challenging task. As compared to
regular slums; the small, scattered and temporary nature of temporary slums
makes data collection and labeling tedious and time-consuming. To tackle this
challenging problem of temporary slums detection, we present a semi-supervised
deep learning segmentation-based approach; with the strategy to detect initial
seed images in the zero-labeled data settings. A small set of seed samples (32
in our case) are automatically discovered by analyzing the temporal changes,
which are manually labeled to train a segmentation and representation learning
module. The segmentation module gathers high dimensional image
representations, and the representation learning module transforms image
representations into embedding vectors. After that, a scoring module uses the
embedding vectors to sample images from a large pool of unlabeled images and
generates pseudo-labels for the sampled images. These sampled images with
their pseudo-labels are added to the training set to update the segmentation
and representation learning modules iteratively. To analyze the effectiveness
of our technique, we construct a large geographically marked dataset of
temporary slums. This dataset constitutes more than 200 potential temporary
slum locations (2.28 square kilometers) found by sieving sixty-eight thousand
images from 12 metropolitan cities of Pakistan covering 8000 square
kilometers. Furthermore, our proposed method outperforms several competitive
semi-supervised semantic segmentation baselines on a similar setting. The code
and the dataset will be made publicly available.

###### Index Terms:

Temporary Slums, semi-supervised approach, new temporary slums dataset.

##  I Introduction

Slums are home to about a billion humans worldwide [1]. The lack of
opportunities in the rural areas pushes people towards urban areas; thus, two-
thirds of the global population is expected to be living in urban areas by
2050 [2].

Figure 1:  (a) We have used U-Net architecture as a segmentation model. Two
one-dimensional fully connected layers of 256 and 64 units are appended at the
end of the centermost layer of U-Net. The arrows from the centermost layer of
U-Net to the dense layers indicate the flow of feature maps. (b) Converting
pixel-level labels to image-level labels: y=1 means the image is labeled as a
slum. In contrast, y=0 means the image is labeled as non-slum, and $\mu_{y}$
is the mean value of the segmentation mask or the ratio of area covered by
pixels labeled as a slum. (c) represent 64-dimensional embedding for labeled
and unlabeled data. (d) The scoring module is based on Cosine similarity
between labeled and unlabeled embeddings. (e) The scoring module returns two
lists of the sorted list, one for each class i.e, slum and non-slum. Top K
images are selected from each list.  Figure 2: Examples of samples from
Lahore, Pakistan.

. Slums can be detected manually [3] and automatically using object-based
image analysis (OBIA) [4, 5] and by employing a grey-level-co-occurrence-
matrix and support vector machines [6, 7]. Recently deep learning-based slums
detection approaches have been proposed, including [8, 9]. Wurm et al., [10]
used fully convolutional network architecture to segment out slum areas from
non-slum. Similarly, authors in [9], and [11] showed that the slum mapping-
related information could be transferred between models trained on different
image resolutions, thus improving the segmentation quality across the board.

UN-Habitat [12] defines a slum by the lack of one or more of the following:
durable housing, sufficient living space, easy access to safe water, access to
adequate sanitation, and security of tenure. Normal/formal slum areas,
although unplanned, due to their long-time existence, take the shape of semi-
formal structure with a population stretching to thousands. Many of these have
some partial local economic activity and formal sources of basic amenities,
electricity, and water [13]. The temporary slums, as the name suggests, are
temporary both in the context of existence (time) and the building structures
present there. Due to their temporary nature, the area lacks much of the
formal structure, access to amenities and mainly consists of buildings that
could be moved easily (make-shift houses, tents). Therefore, temporary slum
dwellers are even more marginalized and economically and socially more
divested. The unplanned nature of the development in these dwellings makes
human-centered information gathering an expansive exercise. The distinction
between temporary (informal) and formal slums is often overlooked. We could
not find any reliable source of geographical locations of temporary slums and
the number of people living in temporary slums around the world. Through this
work, we intend to provide geographical locations of temporary slums in
Pakistan and a framework that can be used in other parts of the world so we
can be made aware of the scale of temporary slums.

Our work is focused specifically on identifying temporary slums from satellite
imagery (examples of temporary slums are shown in Fig. 2). One of the
challenging aspects of detecting these structures is the lack of data in terms
of visual datasets and location information. Temporary slums are very small
and scattered around the cities; deciding which images to label or, more
simply put, ”where to look for temporary slums” is a vital issue.

To overcome the limitation of the availability of the geo-located datasets, we
have designed a technique to collect seed images exploiting the temporary
nature of these slums and design a semi-supervised learning approach to
segment out the slums in the region. Our methodology allows it to be
applicable to any region since it learns from the structure in the local
images. Specifically, our work addresses the challenge of finding temporary
slums in an automated way from visible spectrum (RGB) satellite imagery by
learning from a minimal dataset and then iteratively extending the dataset. In
summary, our work has the following contributions: (1) A semi-supervised
semantic-segmentation method is presented that learns to identify and segment
out the temporary slum from very few positive seed samples, (2) We devise a
strategy to start from a zero-dataset scenario. The make-shift nature of the
temporary slums is exploited to discover the small set of seed images that
might contain the temporary slum areas, (3) We put forward a new large and
diverse satellite imagery-based dataset of temporary slums location. The total
footprint of the collected dataset is 2.28 square kilometers and covers all 12
metropolitan cities of Pakistan, (4) The proposed approach outperforms several
competitive baselines. Specifically, it outperforms supervised (trained on
seed images) baseline by 1.8 times and semi-supervised baselines by 1.9 times.

##  II Methodology

Our proposed approach starts with collecting seed satellite images
representing temporary slums for the semi-supervised learning task. To
overcome the limitation of information regarding where temporary slums are
located, we design an automatic strategy to collect seed images using temporal
changes in the status of buildings. The semi-supervised framework of extending
the seed dataset (labeled pool) is illustrated in (Fig. 1). Using the seed
dataset, we train a segmentation model and an encoding model. The encoding
model is then used to extract encoding from both labeled and unlabeled images.
Encoding of slum labeled images are compared will all of the embeddings of the
unlabeled pool and are sorted based on cosine similarity, and the same process
is repeated for non-slum labeled images. As a result, for each class (slum and
non-slum), we have a sorted list of images that are similar to the relative
class. We select the top few (top $K$) images from each list and generate
segmentation masks. If the segmented region is not of significant size, it is
removed from the top list. The remaining images in the top $K$ list along with
their segmentation masks are made a part of the seed dataset.

###  II-A Preliminaries

We start with two set of images $\mathcal{D}_{pool}$ and $\mathcal{D}_{lab}$.
$\mathcal{D}_{pool}$ consist of unlabeled images $X_{pool}$, whereas
$\mathcal{D}_{lab}$ consist of labeled images $X_{lab}$ with segmentation
labels $Y_{lab}$. Let $X_{pseudo}$ be the images taken from the pool
$\mathcal{D}_{pool}$ and added to the $\mathcal{D}_{lab}$ over multiple
iterations of the self-supervised learning algorithm. Each image in
$X_{pool}$, $X_{lab}$ and $X_{pseudo}$ are such that $x\in$IRW×H where
$W\times H$ are the spatial dimensions of the images. Our segmentation model
$f_{s}$ outputs, $p=f_{s}(x)$, probability of each pixel belonging to the
temporary slum. We design an embedding model $f_{emb}$ which consists of the
encoder part of $f_{s}$ extended with fully connected layers. The output of
this $f_{emb}$ model is $\mathcal{E}(x)\in$IR64×1. Unlike many previous works
in semi-supervised learning, we start with empty $\mathcal{D}_{lab}$ and,
using our temporal consistency strategy we select a few images which are hand-
tagged to construct initial $\mathcal{D}_{lab}$. Over time, both the $f_{s}$
and $f_{emb}$ are trained jointly over the labeled and pseudo-labeled data.

###  II-B Discovering Seed Images

We design an automatic strategy to overcome the limitation of the non-
existence of geo-location information about the temporary slums. Specifically,
we use the non-permanent presence of temporary slums as a means of identifying
them with the help of a built-up region segmentation model. A U-Net-based
segmentation model is trained over the Village Finder [14] dataset for the
build-up segmentation. We use the change in segmentation results (intersection
over union) to classify images where the change in build-up regions occurred;
either new settlements were created, or settlements were removed. Any regions
with non-nucleated settlement detection or having detected settlements lower
than a certain threshold were removed. Working conservatively, only a few
images with small IoU across the time are selected as possible temporary slums
and pixel-level tagging is performed (Fig. 3).

Change detection might fail to capture all the temporary slums, since they
might not change in the time window we choose for this purpose, and might
capture images that are not temporary slums. Therefore, this method results in
only a few seed images 32 in our case which were curated and annotated
manually, this process was done for just one city (Lahore).

###  II-C Dataset

The satellite imagery is collected in patches of 256 $\times$ 256 pixels using
a sliding window with an overlap of 56 pixels. The unlabeled pool consists of
68,051 images, while the labeled pool contains 32 temporary slum and 4831 non-
temporary slum images, which are then split into training (20 slums and 3938
non-slum) and validation (12 slums and 861 slums). The seed-image dataset is
highly imbalanced (against temporary slums) and extremely small to train a
robust semantic-segmentation network. To cater to the data imbalance, we use
data augmentations where slum images are augmented more times than non-slum
images. To enrich our negative class, we make sure that we add examples from
‘formal slums’ or permanent slums as non-temporary slum examples (Fig. 2). The
source of satellite imagery is Google Earth at zoom level 19, which translates
to approximately 1m per pixel resolution. Rawalpindi, Islamabad, Multan,
Hyderabad, Faisalabad, Gujranwala, Sialkot, Peshawar, Quetta, Sargodha are
part of the unlabeled set (whole city), Lahore is part of the training set
(train and Val), whereas Karachi is a part of the test set.

Figure 3: An example of an area where the built-up area changed over time. The
relative difference between the two Satellite Images taken at two different
Timestamps is visible in the zoomed-in views on the right. The area zoomed in
blue had no built-up, while some dwellings appear in the later view and
opposite for the area zoomed in red.

###  II-D Temporary Slum segmentation Module

To achieve the pixel-wise location of temporary slums regions, we employ a
semantic segmentation module. Our segmentation model $f_{s}$ is based on U-Net
[15] architecture with the ResNet-32 [16] backbone pre-trained on ImageNet
[17]. It is trained on $D_{lab}$ which is our initial labeled dataset in a
fully supervised fashion. The U-Net is trained through binary cross-entropy
loss.

###  II-E Embedding Learning Module

To learn a more robust and discriminative embedding, we have employed deep
metric learning. The goal is to push the model into squeezing discriminative
semantic information in a one-dimensional vector, commonly referred to as
embedding vector, by penalizing the model if the representation of the
instances of two different classes is similar. To generate embeddings, we add
three fully connected layers of size $512$, $256$, and $64$ neurons,
respectively, over the features extracted from the encoder. ReLu activation is
used after the first two layers, whereas the third one (output layer) is
without any activation layer.

The output of this module is a $\mathcal{E}(I)\in$IR64×1 dimensional
embedding-vector of the image $I$. We use triplet loss as an objective
function.

$\mathcal{L}_{emb}=max(\lVert(\mathcal{E}(I_{a}),\mathcal{E}(I_{p})\lVert)-\lVert(\mathcal{E}(I_{a}),\mathcal{E}(I_{n}))\lVert+m,0),$
(1)

where $\mathcal{E}(I_{a})$, $\mathcal{E}(I_{p})$, $\mathcal{E}(I_{n})$
represents embedding vector for the anchor, positive and negative images, and
$m$ is a margin. Positive and anchor images belong to the same class
(temporary slum), whereas the negative images are from the non-temporary slum
areas. Minimizing the triplet loss forces the embedding of images from the
temporary slum areas to be similar while maximizing the distance between
embedding from temporary slum and non-temporary slum areas. The triplets are
created on a fly from the batch.

###  II-F Discovering Temporary Slum from Unlabelled Pool

We start with $\mathcal{D}_{pool}$ which consist of unlabeled images
$X_{pool}$ and $\mathcal{D}_{lab}$ which consist of labeled images $X_{lab}$
with segmentation labels $Y_{lab}$. At each iteration of our algorithm,
selected samples from $\mathcal{D}_{pool}$ are moved to the
$\mathcal{D}_{lab}$ along with their pseudo-labels. Let $X_{pseudo}$ be such
images. Each image in $X_{pool}$,$X_{lab}$ and $X_{pseudo}$ are such that
$x_{s}\in$IRW×H where $W$ and $H$ are the spatial dimensions of the images.

####  II-F1 Scoring Module

The trained semantic segmentation $f_{s}$ network with embedding head is used
to generate the image embedding for all the images in the unlabeled pool as
well as the labeled pool, $X_{pool}\cup X_{lab}$. We compute the cosine
similarity between embeddings across the pool to identify unlabeled pool
images similar to images in the labeled pool. Specifically, embedding of each
unlabeled image $x_{p}^{i}\in\mathcal{X}_{pool}$ is compared with the
embedding of all slum and non-slum images $x_{lab}^{i}\in\mathcal{X}_{lab}$
from the labeled pool. A few examples are shown in Fig. 4. Instead of
calculating pixel-level similarity, calculating similarity over the image
embedding allows us to compare the encoded information in the latent space.
Let $K_{p}^{t}$ and $K_{l}^{t}$ be the total number of images in
$\mathcal{X}_{pool}$ and $\mathcal{X}_{lab}$ at iteration $t$. Let
$d(i_{p},i_{l})=\mathcal{C}(\mathcal{E}(x_{p}^{i}),\mathcal{E}(x_{l}^{i}))$ be
the cosine similarity between embeddings of $x_{p}^{i}\in\mathcal{X}_{pool}$
and $x_{l}^{i}\in\mathcal{X}_{lab}$. Let $\mu_{s}^{i}$ be the average
similarity of $x_{p}^{i}$ with all the temporary slum images in the
$\mathcal{X}_{lab}$ and $\mu_{n}^{i}$ be the average similarity with all the
non-slum images in the $\mathcal{X}_{lab}$. $\mu_{s}^{i}$ and $\mu_{n}^{i}$
are sorted in descending order and top $K$ images selected to make sets
$S_{s}^{t}$ and $S_{ns}^{t}$. Images that get selected in both sets are
removed to make exclusive sets and are added back to the unlabeled pool.

Figure 4: Image retrieval using cosine similarity. Image retrieval is done by
comparing the labeled dataset’s embedding vectors with the unlabeled dataset’s
embedding vectors. Top 9 Query results for temporary slum examples are shown
on the left, and for non-slum, examples are shown on the right.

####  II-F2 Pixel level Pseudo Labeling

After $S_{s}^{t}$ and $S_{ns}^{t}$ have been selected based on image-level
information, the second step of filtering is performed, this time over the
pixel-level information. We compute the segmentation mask using the model
$f_{s}$ updated in the last iteration for all the images in the selected sets.
Images in $S_{s}^{t}$, images with predicted temporary slum segmentation area
less than 5 percent of the total image size are removed from the set. While
for images from non-temporary slum set $S_{ns}^{t}$ if the slum class
prediction covers more than 5 percent area, they are removed from the set. The
remaining images from each set are added to the labeled data pool along with
their segmentation pseudo labels, which are computed by applying thresholds to
prediction probabilities at 0.5.

Figure 5: Iterations taken to reach desired set of images by algorithm on
different K settings. Stopping point shown as dashed line in black. The solid
lines show the cumulative count of images added to $S_{s}^{t}$ over the course
of the experiment.

##  III Experiments and Results

###  III-A Experimental Setup

Segmentation model $f_{s}$ and embedding model $f_{e}$ are trained for at most
50 epochs with batch size 8 and learning rate of $10^{-3}$ and $10^{-5}$
respectively. Since the proposed method is an iterative process, the stopping
criterion is defined as follows; when the ratio of the size of $S_{s}^{t}$ to
the initial size of the unlabeled pool $\mathcal{D}_{pool}$ becomes equal to
(or greater than) ratio of slums to non-slums as present in the initial
training set, no further iterations are performed. This criterion is based on
the simple intuition that the slum to the non-slum ratio in the initial
training set is indicative of the actual presence of slums. Our initial
training set consists of $3938$ images in which $32$ ($20$ used for training
and $10$ used for validation) are labeled as slums while the rest are labeled
non-slum. In other words, only $0.507\%$ of the training pool is labeled as a
slum. In this case, to reach the same ratio, we have to identify at least 345
slum images from the unlabeled pool. Our test set 486 images, of which $196$
are labeled as temporary slums.

TABLE I:  Quantitative Comparison of a few semi supervised learning base semantic segmentation methods  Method | mIoU | Precision | Recall | F1
---|---|---|---|---
Supervised baseline (U-Net) | 0.17 | 0.27 | 0.27 | 0.22
Cutmix[18] | 0.11 | 0.35 | 0.13 | 0.16
Cutout[18] | 0.14 | 0.39 | 0.18 | 0.19
ICT[19] | 0.18 | 0.38 | 0.24 | 0.25
VAT[20] | 0.18 | 0.40 | 0.23 | 0.24
Ours | 0.33 | 0.49 | 0.49 | 0.43
TABLE II:  Ablation studies our approach for different values of ‘K’  Method | mIoU | Precision | Recall | F1
---|---|---|---|---
Ours K=100 | 0.25 | 0.50 | 0.35 | 0.34
Ours K=50 | 0.32 | 0.48 | 0.51 | 0.42
Ours K=30 | 0.33 | 0.49 | 0.49 | 0.43

###  III-B Experimental Results

We have compared the proposed approach with several competitive baselines
settings proposed in [18]. The authors in [18] used different regularization
techniques such as Cutout [21], Cutmix [18], Interpolation Consistency
Training (ICT) [19], Virtual Adversarial Training (VAT) [20]. For all the
comparisons, we followed the experimental setup of [18] and used the same
training, testing, and validation data as used in our experiment. The
quantitative comparison of our approach in Table I demonstrates that the
proposed approach significantly outperforms the baselines (Fig. 6). Evaluation
metrics: mIoU: IOU is the ratio of the area of intersection and area of Union
between prediction and ground truth. The mean value of IOU over all test
images is denoted as mIOU. Precision is the ratio of true positives and all
positives (true and false Positive). Recall: is the ratio of true positives
and true positives plus false negatives. Finally, F1-Score combines the
Precision and recall of a classifier into a single metric by taking their
harmonic mean.

Figure 6: A comparison of the segmentation results of all the methods
mentioned in Table I. The segmentation masks generated by the respective
methods are overlayed over the satellite imagery.

Ablation Studies: We evaluate our approach for different values of $K$ and
demonstrate the results in Fig. 5 and Table II. The results demonstrate that
the proposed approach outperforms the baselines for different values of $K$.
Fig. 5 shows that for ${K}$ set to 30, our algorithm takes most iterations to
reach the required threshold. However, this also results in the best results
(mIoU), as indicated in Table I.

Discussion: While our proposed method performs significantly better against
the other methods on segmentation metrics, the overall mIOU of is low below
ideal. We consider the main contribution of our work is a methodology that
automatically discovers temporary slum locations starting from minimal seed
data. In our methodology the segmentation masks are just intermediate outputs,
the problem we want to tackle in this work is ‘where to look for temporary
slums?’. The only alternative to our work is to scan the whole city’s
satellite imagery manually. In the future, our approach can be improved in
several ways. For instance, seed Selection can be improved if satellite
imagery is available at more than two timestamps. Our methodology can be used
in different geographical settings to identify the scale of temporary slums
globally.

Figure 7: This figure shows the identified temporary slum locations in
Faisalabad and Sargodha, two of the 12 metropolitan cities of Pakistan that we
covered. The green dots represent the visually verified locations, and the
blue dots represent the locations where we are uncertain about our
predictions.

##  IV Conclusion

Finding temporary slums is a challenging task, as these areas exist mainly on
the fringes of the cities, and the usual surveying methods would require
significant resources both in terms of time and money.In our work, we have
provided a semi-supervised learning-based solution that can be initiated from
a small seed dataset. We used the temporal consistency of built-up structures
to find those small seed images. We have generated temporary slums dataset for
12 metropolitan cities in Pakistan that can be updated and extended to other
regions very easily. Our extensive experimental results and discussion
validate the proposed ideas and framework. We intend to use our works as a
methodology that answers the question of ”where to look for temporary slums?”
and enables us to identify and visualize the scale of temporary slums.

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