text:
\begin{document}

\title{Weak entropy solutions of nonlinear reaction-hyperbolic
systems for axonal transport}

\author{Hao YAN}
\address{Zhou Pei-Yuan Center for Appl. Math.\\
Tsinghua University\\
Beijing 100084, China} \email{yanhao06@mails.tsinghua.edu.cn}

\author{Wen-An YONG}
\address{Zhou Pei-Yuan Center for Appl. Math.\\
Tsinghua University\\
Beijing 100084, China} \email{wayong@tsinghua.edu.cn}

\keywords{}

\subjclass{}

\begin{abstract}
This paper is concerned with a class of nonlinear
reaction-hyperbolic systems as models for axonal transport in
neuroscience. We show the global existence of entropy-satisfying
BV-solutions to the initial-value problems by using hyperbolic-type
methods. Moreover, we rigorously justify the limit as the
biochemical processes are much faster than the transport ones.

\vskip 10 pt

\noindent {\bf Keywords:} axonal transport, relaxation limit,
difference scheme, BV-estimates, entropy.
\end{abstract}

\maketitle \markboth{Hao YAN and W.-A. YONG}{Weak entropy solutions
of nonlinear reaction-hyperbolic systems for axonal transport}

\tableofcontents

\section{Introduction}

The axonal transport is important for the maintenance and functions
of nerve cells. These cells are also called {\it neurons}.

A neuron consists of three parts mainly: cell body, dendrites and a
single axon. The axon is a long and thin pipe whose length can
exceed 10,000 times its diameter. It is this axon that distinguishes
neurons from other cells. The axon is responsible for signal
transmission in the nervous system. Its cytoplasm does not contain
rough endoplasmic reticulum and therefore its proteins can only be
transported from the cell body, where all proteins are synthesized.

The transport proceeds as follows. Proteins are stored in vesicles

as cargos. The vesicles are attached to kinesin (anterograde motors)
or dynein (retrograde motors) proteins. These motor proteins drive
the vesicles to walk along the cytoskeletal microtubules as track.
Here the kinesin proteins move the vesicles from the cell body to
synapse (anterograde transport), while the dynein proteins move the
vesicles in the opposite direction (retrograde transport). During
the transport, many biochemical processes are possible. For example,
the cargos can leave its track, can switch its motor proteins from
kinesin to dynein or vice verse, and can move back onto the track.
Thus, we can divide the cargos into a number of subpopulations, such
as free vesicles,

vesicle-kinesin compounds off track, moving vesicle-dynein compounds
on track, etc.

As the axon is long and thin, it is reasonable to assume the
transport only along the longitudinal direction of the axon. Denote
by $x>0$ the distance down the axon from the cell body

which is located at $x = 0$. Let $u_i = u_i(x, t)$ be the
concentration at space-time $(x, t)$ of the $i$-th subpopulations.
According to Reed and Blum \cite{RB}, the mathematical model for
axonal transport is partial differential equations of the form:
\begin{equation*}\label{pde0}
\partial_t u_i + \lambda_i\partial_x u_i = F_i (u_1, u_2, \cdots, u_r), \qquad i =
1, 2, \cdots, r .
\end{equation*}
Here the term $\lambda_i\partial_x u_i$ accounts for the transport
of the $i$-th subpopulation with constant velocity $\lambda_i$, and
$F_i (u_1, u_2, \cdots, u_r)$ describes the biochemical processes of
the constituents. It

is well recognized that the biochemical processes are much faster
than the transport in biosystems.

In \cite{CBF, FC, FH, RVB}, the authors studied the linear case,
where $F_i(u_1, u_2, \cdots, u_r)$ is linear with respect to the
$u_j$'s, in order to explain the approximate traveling waves
observed in experiments. Especially, in \cite{FH} Friedman and Hu
used parabolic-type estimates to analyse the diffusive limit of the
linear systems. However, it seems uneasy to deal with the nonlinear
problems with the parabolic-type techniques. On the other hand, in
\cite{Ca} Carr showed the existence of global classical solutions to
a class of nonlinear models with source terms of the form
$$
F_i(u_1, u_2, \cdots, u_r) = \begin{cases} f_1 , & i = 1\\
f_i - f_{i - 1} , & 1 < i < r \\
-f_{r-1} , & i = r
\end{cases},
$$
where $f_i = f_i(u_i, u_{i+1})$ is a continuously differentiable
function of two variables. We notice that in applications $f_i$ is a
polynomial of $u_i$ and $u_{i+1}$. In addition, the model in
\cite{CBF} is an example where $f_i$ depends only on $u_i$ and
$u_{i+1}$.

In this paper, we consider the same nonlinear systems, as in
\cite{Ca}, but with a small parameter $\epsilon > 0$:
 \begin{eqnarray}\label{pde1}
\left\{ \begin{array}{ll}
(\partial_t+\lambda_{1}\partial_x)u_1(x,t) & = \frac{1}{\epsilon}f_{1}(u_1,u_2), \\
\qquad\cdots  \\
(\partial_t+\lambda_{i}\partial_x)u_i(x,t) & =
-\frac{1}{\epsilon}f_{i-1}(u_{i-1},u_{i})+\frac{1}{\epsilon}f_{i}(u_i,u_{i+1}), \\
\qquad \cdots  \\
(\partial_t+\lambda_{r}\partial_x)u_r(x,t) & =
-\frac{1}{\epsilon}f_{r-1}(u_{r-1},u_{r}) .
\end{array} \right.
\end{eqnarray}
Here the small parameter $\epsilon$ characterizes the fact that the
biochemical processes are much faster than the transport. We assume,
throughout this paper, that each $f_i$ is strictly decreasing with
respect to the first argument and strictly increasing with respect
to the second. In addition, we assume that $f_i(0,0)=0$ and, for
fixed $v$, there exists $w$ such that $vf_i(v, w) \geq 0$. These
assumptions are consistent with those used in \cite{Ca, CBF, FC, FH,
RB, RVB}.

We will regard (\ref{pde1}) as a hyperbolic relaxation system
\cite{Y2} and use the corresponding techniques to study it. In
particular, we will use a difference scheme to show the global
existence of entropy-satisfying BV-solutions $(u_1^\epsilon,
u_2^\epsilon, \cdots, u_r^\epsilon)$ to the initial-value problems
of (\ref{pde1}) and investigate the limit as $\epsilon$ goes to
zero. For $r\leq 3$ and for linear problems, we prove that the limit
$(u_1^0, u_2^0, \cdots, u_r^0)$ is also an entropy-satisfying
BV-solution to the so-called equilibrium or reduced system of
$(\ref{pde1})$:
\begin{eqnarray}\label{pde2}
\left\{\begin{array}{ll}
\partial_t(u_1+u_2+\cdots+u_r) +
\partial_x(\lambda_1u_1+\lambda_2u_2+\cdots+\lambda_ru_r)=0,\\
f_1(u_1,u_2) = f_2(u_2,u_3) = \cdots = f_{r-1}(u_{r-1},u_r)=0 .
\end{array} \right.
\end{eqnarray}

Although the $BV$-framework is quite standard for nonlinear
hyperbolic problems (see, e.g.,  \cite{Da, RY, Y1}), some innovatory
ideas seem necessary to carry our the details. In particular, we use
the Brouwer fixed-point theorem, the $BV$-estimate and the special
structure of the source terms in (\ref{pde1}) to derive the
existence, boundedness and time-Lipschitz continuity of the
difference solutions. It seems not so easy to obtain the
time-Lipschitz continuity! Moreover, for an arbitrarily given convex
entropy function for the reduced system ({\ref{pde2}), we construct
a dissipative entropy function for the original system (\ref{pde1}).
In equilibrium, the constructed entropy function reduces to the
given one.

The paper is organized as follows. Section 2 is devoted to the
analysis of a difference scheme for the system $(\ref{pde1})$. The
time-Lipschitz continuity of the difference solutions is derived in
Section 3. In Section 4, we discuss entropy functions for the two
systems $(\ref{pde1})$ and $(\ref{pde2})$. The main results are
shown in Section 5.

\section{Difference Solutions}
\setcounter{equation}{0}

We begin with construction of approximation solutions to the
reaction-hyperbolic system $(\ref{pde1})$ by using difference
methods. For simplicity, we set
$$
\begin{array}{rl}
U = & (u_{1}, u_{2}, \cdots, u_{r})^{T}, \quad
\Lambda = \mbox{diag}(\lambda_1,\lambda_2,\cdots,\lambda_ r),
\\[4mm]
Q(U) = & (f_1, \cdots, -f_{i-1}+f_{i}, \cdots, - f_{r-1})^{T},
\end{array}
$$
where the superscript $T$ denotes the transpose of vectors or
matrices. Then the reaction-hyperbolic system $(\ref{pde1})$ can be
rewritten as
\begin{eqnarray}\label{pde3}
U_t + \Lambda U_x = \frac{1}{\epsilon}Q(U).
\end{eqnarray}
About this system, we make the following assumptions mentioned in
the introduction:\\

\begin{itemize}

\item[(1).] each $f_i$ is continuously differentiable, strictly decreasing
with respect to the first argument, and strictly increasing with
respect to the second;

\item[(2).] $f_i(0, 0) = 0$.\\

\end{itemize}
These assumptions are consistent with those used in \cite{Ca, CBF,
FC, FH, RB, RVB}.

Our difference approximation to $(\ref{pde3})$ is the following
semi-implicit upwind scheme
\begin{eqnarray}\label{dc1}
\frac{U_j^{n+1}-U_j^n}{\Delta
t}+\Lambda^+\frac{U_j^{n}-U_{j-1}^{n}}{\Delta
x}+\Lambda^-\frac{U_{j+1}^{n}-U_j^{n}}{\Delta x}=\frac{1}{\epsilon}
Q_j^{n+1}.
\end{eqnarray}
Here $\Delta x$ and $\Delta t$ denote the increments respectively in
$x$ and $t$; $U_{j}^n$ denotes the approximation of $U(x,t)$ over
the grid block $[x_j,x_{j+1})\times [t_n,t_{n+1})$ with $x_j=j\Delta
x$ and $t_n=n\Delta t$; $j=0,\pm1,\pm2,\cdots$ and $n=0,1,2,\cdots$;
$Q_j^n=Q(U_j^n);
\Lambda^+=diag(\lambda_1^+,\lambda_2^+,\cdots,\lambda_
r^+)\quad\mbox{and}\quad
\Lambda^-=diag(\lambda_1^-,\lambda_2^-,\cdots,\lambda_ r^-)$ with
\begin{eqnarray}\label{UPwind}
\lambda_{i}^+=\frac{\lambda_{i}+|\lambda_i|}{2} \ge 0
\quad\mbox{and}\quad \lambda_{i}^-=\frac{\lambda_{i}-|\lambda_i|}{2}
\le 0.
\end{eqnarray}
For $n=0$, we take
\begin{eqnarray}\label{inital}
U_j^0 = \begin{cases}
\frac{1}{\Delta x}\int_{x_j}^{x_{j+1}} U_0(x) dx, &
\mbox{if} \quad j \leq \frac{1}{\Delta x} \\[4mm]
0, & \mbox{if} \quad j > \frac{1}{\Delta x} \end{cases},
\end{eqnarray}
where $U_{0}(x):=(u_{10}(x),u_{20}(x),...,u_{r0}(x))^T$ is a bounded
measurable function of $x\in\mathbf{R}$. In addition, the 0 in
(\ref{inital}) can be replaced with any constant vector.

Throughout this paper, we assume that the grid sizes satisfy the
CFL-condition
\begin{eqnarray}\label{CFL}
\frac{\Delta t}{\Delta x}\max_i |\lambda_{i}|\le1.
\end{eqnarray}

To analyse the above scheme, we start with the following elementary
fact.

\begin{lem}\label{lem1}
Let $A=(a_{kl})_{n\times n}$ be a matrix satisfying $\sum_{k=1}^n
a_{kl}=1$ for $l=1,2,\cdots,n$ and $a_{kl}\leq 0 \ \textrm{for }
k\ne l$. Then $A$ is invertible, the spectral radius of $A^{-1}$ is
not bigger than 1, and the $1$-norm of $A^{-1}$ is 1.
\end{lem}

\begin{proof}
Let $\lambda$ be an eigenvalue of $A$. By the Gershgorin circle
theorem there is an integer $l$ such that
\begin{equation*}
|\lambda - a_{ll}|\leq\sum_{k\neq l}|a_{kl}| = a_{ll} - 1,
\end{equation*}
for $a_{ll}=1-\sum_{k\ne l} a_{kl}$ and $a_{kl}\leq 0 (k\ne l)$.
From this we easily deduce that $|\lambda|\geq1$. Thus $A$ is
invertible and the spectral radius of $A^{-1}$ is not bigger than 1.

Furthermore, it follows from $\sum_{k=1}^n a_{kl} = 1$ that
\begin{equation*}
(1,1,\cdots,1)A=(1,1,\cdots,1)
\end{equation*}
and thereby
\begin{equation*}\label{matrix1}
(1,1,\cdots,1)A^{-1}=(1,1,\cdots,1).
\end{equation*}
This shows that the sum of each column of $A^{-1}$ is also 1. On the
other hand, we set $D = \mbox{diag}(a_{11}, a_{22}, \cdots, a_{nn})$
and $F = D - A$. It is obvious that the elements of $FD^{-1}$ are
nonnegative,
\begin{equation*}
A=D-F=(I-FD^{-1})D
\end{equation*}
and thereby
\begin{equation*}
A^{-1}=D^{-1}(I-FD^{-1})^{-1} .
\end{equation*}
Moreover, with the Gershgorin circle theorem it is not difficult to
see that the spectral radius of $FD^{-1}$ is less than 1. Thus we
can write
\begin{equation*}
A^{-1}=D^{-1}(I+FD^{-1}+(FD^{-1})^2+\cdots),
\end{equation*}
which shows that the elements of $A^{-1}$ are nonnegative. Hence the
$1$-norm of $A^{-1}$ is 1 for the sum of each column of $A^{-1}$ is
1. This completes the proof.
\end{proof}

Set $V=(v_1, v_2, \cdots, v_r)^T, W=(w_1, w_2, \cdots, w_r)^T, $
\begin{eqnarray}
A_i& = &A_i(V,W) = \int_0^1 \frac{\partial f_i}{\partial
u_i}(\theta v_i + (1-\theta )w_i, \theta v_{i+1} + (1-\theta)w_{i+1})d\theta,\label{int1}\\
B_i& = &B_i(V,W) = \int_0^1 \frac{\partial f_i}{\partial
u_{i+1}}(\theta v_i + (1-\theta)w_i, \theta v_{i+1} +
(1-\theta)w_{i+1})d\theta.\label{int2}
\end{eqnarray}
Assumption (1) implies that $A_i<0$ and $B_i>0.$ By the mean-value
theorem we have
\begin{equation}
Q(V)-Q(W)=\mathcal {Q}(V,W)(V-W).
\end{equation}
From the definition of $Q$, we have
\begin{eqnarray*}\label{Jocobi}
\mathcal {Q}(V,W)=\left( \begin{array}{cccccc}
A_{1} & B_{1}  &   &  &    \\
-A_{1} & -B_{1}+A_{2}& B_{2}  & & \\
{}&  -A_{2}& -B_{2}+A_{3}& B_{3}& \\
{}& &  \ldots& \ldots &\ldots\\
 {}& &  & -A_{r-1}& -B_{r-1}

\end{array} \right).
\end{eqnarray*}
It is easy to verify that $I-\frac{\Delta t}{\epsilon}\mathcal
{Q}(V,W)$ satisfies the conditions in Lemma $\ref{lem1}.$ Therefore
we have

\begin{cor}\label{cor}
$I-\frac{\Delta t}{\epsilon}\mathcal {Q}(V,W)$ is invertible, the
spectral radius of $(I-\frac{\Delta t}{\epsilon}\mathcal
{Q}(V,W))^{-1}$ is not bigger than 1, and the $1$-norm of
$(I-\frac{\Delta t}{\epsilon}\mathcal {Q}(V,W))^{-1}$ is 1 for any
$V,W \in \mathbb{R}^r$.
\end{cor}

With these preparations, we study the difference scheme
$(\ref{dc1})$. In what follows, we will denote by $|\cdot|$ the
$1$-norm of vectors and matrices.
\begin{lem}\label{lem2}
Given $U_j^n$, $U_j^{n+1}$ can be uniquely determined by solving the
nonlinear algebraic equation $(\ref{dc1})$.
\end{lem}

\begin{proof}
From $f_i(0,0)=0$ we see that $Q(0)=0$ and
\begin{eqnarray*}
Q_j^{n+1}=Q_j^{n+1}-Q(0)=\mathcal {Q}(U_j^{n+1},0)U_j^{n+1}.
\end{eqnarray*}
Thus, we deduce from the difference scheme $(\ref{dc1})$ that
\begin{eqnarray*}
(I-\frac{\Delta t}{\epsilon}\mathcal
{Q}(U_j^{n+1},0))U_j^{n+1}=\frac{\Delta t}{\Delta
x}\Lambda^+U_{j-1}^n+(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)U_{j}^n+(-\frac{\Delta t}{\Delta x}\Lambda^-)U_{j+1}^n.
\end{eqnarray*}

With this equation, we construct a mapping $F:V\rightarrow W$ as
follows
\begin{eqnarray*}
(I-\frac{\Delta t}{\epsilon}\mathcal {Q}(V,0))W=\frac{\Delta
t}{\Delta x}\Lambda^+U_{j-1}^n+(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)U_{j}^n+(-\frac{\Delta t}{\Delta x}\Lambda^-)U_{j+1}^n.
\end{eqnarray*}
Due to Corollary $\ref{cor}$, we can write
\begin{eqnarray*}
W=(I-\frac{\Delta t}{\epsilon}\mathcal {Q}(V,0))^{-1}[\frac{\Delta
t}{\Delta x}\Lambda^+U_{j-1}^n+(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)U_{j}^n+(-\frac{\Delta t}{\Delta x}\Lambda^-)U_{j+1}^n]
\end{eqnarray*}
and thereby
\begin{eqnarray*}
|W|&\leq& |(I-\frac{\Delta t}{\epsilon}\mathcal
{Q}(V,0))^{-1}||\frac{\Delta t}{\Delta
x}\Lambda^+U_{j-1}^n+(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)U_{j}^n+(-\frac{\Delta t}{\Delta
x}\Lambda^-)U_{j+1}^n|\\&=& |\frac{\Delta t}{\Delta
x}\Lambda^+U_{j-1}^n+(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)U_{j}^n+(-\frac{\Delta t}{\Delta x}\Lambda^-)U_{j+1}^n|.
\end{eqnarray*}
Therefore, $W$ is bounded for fixed $j$ and $n$. By the Brouwer
fixed-point theorem, the mapping $F$ has fixed points for $F$ is
continuous. This shows the existence of $U^{n+1}_j$.

To see the uniqueness, we assume that both $U_1$ and $U_2$ satisfy
the difference scheme $(\ref{dc1})$. With the mean-value theorem, we
see that
\begin{eqnarray*}
U_1-U_2=\frac{\Delta t}{\epsilon}(Q(U_1)-Q(U_2))=\frac{\Delta
t}{\epsilon}\mathcal {Q}(U_1,U_2)(U_1-U_2)
\end{eqnarray*}
and thereby
\begin{eqnarray*}
(I-\frac{\Delta t}{\epsilon}\mathcal {Q}(U_1,U_2))(U_1-U_2)=0.
\end{eqnarray*}
It follows from Corollary $\ref{cor}$ that $U_1=U_2$. This shows the
uniqueness and hence the proof is complete.
\end{proof}

\begin{rem}
We could use the contraction mapping principle to show the existence
and uniqueness of $U_j^{n+1}$. However, it requires that $\Delta t$
is much smaller than $\epsilon$. On the other hand, this lemma does
not tell the uniform boundedness of the difference solution $U_j^n$.
\end{rem}

Next we establish the $L^1$-stability of the difference scheme
$(\ref{dc1})$.
\begin{lem}\label{lem3}
Let $U_j^n$ and $V_j^n$ be two solutions of the difference scheme
$(\ref{dc1})$ with initial data $U_j^0$ and $V_j^0$, respectively.
Then it holds that
\begin{eqnarray*}
\sum_{j=-\infty}^{+\infty}|U_{j}^{n}-V_j^{n}|\leq\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-V_j^{0}|
\end{eqnarray*}
for all $n\ge 0$.
\end{lem}

\begin{proof}
It follows from the difference scheme $(\ref{dc1})$ that
\begin{eqnarray*}
U_j^{n+1}-V_j^{n+1}&=&\frac{\Delta
t}{\epsilon}(Q(U_j^{n+1})-Q(V_j^{n+1}))+\frac{\Delta t}{\Delta
x}\Lambda^+(U_{j-1}^n-V_{j-1}^n)\\&{}&+(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)+(-\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j+1}^n-V_{j+1}^n).
\end{eqnarray*}
Thus we deduce from the mean-value theorem and Corollary $\ref{cor}$
that
\begin{eqnarray*}
(I-\frac{\Delta t}{\epsilon}\mathcal
{Q}(U_j^{n+1},V_j^{n+1}))(U_j^{n+1}-V_j^{n+1})=\frac{\Delta
t}{\Delta x}\Lambda^+(U_{j-1}^n-V_{j-1}^n)\\
+(I-\frac{\Delta t}{\Delta x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)+(-\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j+1}^n-V_{j+1}^n) ,
\end{eqnarray*}
\begin{eqnarray*}
U_j^{n+1}-V_j^{n+1}&=&(I-\frac{\Delta t}{\epsilon}\mathcal
{Q}(U_j^{n+1},V_j^{n+1}))^{-1}[\frac{\Delta t}{\Delta
x}\Lambda^+(U_{j-1}^n-V_{j-1}^n)\\&{}&+(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)+(-\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j+1}^n-V_{j+1}^n)],
\end{eqnarray*}
\begin{eqnarray*}
|U_j^{n+1}-V_j^{n+1}|&\leq&|(I-\frac{\Delta t}{\epsilon}\mathcal
{Q}(U_j^{n+1},V_j^{n+1}))^{-1}||\frac{\Delta t}{\Delta
x}\Lambda^+(U_{j-1}^n-V_{j-1}^n)\\&{}&+(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)+(-\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j+1}^n-V_{j+1}^n)|\\
&\leq&|\frac{\Delta t}{\Delta
x}\Lambda^+(U_{j-1}^n-V_{j-1}^n)|\\&{}&+|(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)|+|(-\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j+1}^n-V_{j+1}^n)|,
\end{eqnarray*}
and thereby
\begin{eqnarray*}
&& \sum_{j=-\infty}^{+\infty}|U_j^{n+1}-V_j^{n+1}|
\leq\sum_{j=-\infty}^{+\infty}|\frac{\Delta t}{\Delta x}
\Lambda^+(U_{j-1}^n-V_{j-1}^n)|\\
&& \qquad + \sum_{j=-\infty}^{+\infty}|(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)|+\sum_{j=-\infty}^{+\infty}|(-\frac{\Delta
t}{\Delta x}\Lambda^-)(U_{j+1}^n-V_{j+1}^n)|\\
& = &\sum_{j=-\infty}^{+\infty}[|\frac{\Delta t}{\Delta
x}\Lambda^+(U_{j}^n-V_{j}^n)|+|(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)|\\
&& \qquad + |(-\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)|].
\end{eqnarray*}
In the last step we use the induction assumption that
$\sum_{j=-\infty}^{+\infty}|U_{j}^n-V_j^n|<\infty$, which is true if
$\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-V_j^{0}|<\infty$. On the other
hand, from the definition of $\lambda_i^\pm$ $(\ref{UPwind})$ and
the CFL-condition $(\ref{CFL})$ we deduce that
\begin{eqnarray*}
&{}&|\frac{\Delta t}{\Delta
x}\Lambda^+(U_{j}^n-V_{j}^n)|+|(I-\frac{\Delta t}{\Delta
x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)(U_{j}^n-V_{j}^n)|+|(-\frac{\Delta
t}{\Delta x}\Lambda^-)(U_{j}^n-V_{j}^n)|\\
&=&\sum_{i=1}^r\frac{\Delta t}{\Delta
x}\lambda_i^+|u_{ij}^n-v_{ij}^n|+\sum_{i=1}^r(1-\frac{\Delta
t}{\Delta x}\lambda_i^++\frac{\Delta t}{\Delta
x}\lambda_i^-)|u_{ij}^n-v_{ij}^n|- \sum_{i=1}^r\frac{\Delta
t}{\Delta x}\lambda_i^-|u_{ij}^n-v_{ij}^n|\\
&=&|U_j^n-V_j^n|.
\end{eqnarray*}
Hence we see that
\begin{eqnarray*}
\sum_{j=-\infty}^{+\infty}|U_j^{n+1} -
V_j^{n+1}|\leq\sum_{j=-\infty}^{+\infty}|U_j^n-V_j^n|\leq\cdots\leq\sum_{j=-\infty}^{+\infty}|U_j^{0}-V_j^{0}|.
\end{eqnarray*}
This completes the proof.
\end{proof}

By taking $V_j^n=U_{j-1}^n$ in Lemma $\ref{lem3}$, we get the
following corollary on BV-estimates of the difference solutions.
\begin{cor}\label{cor2}
Let $U_j^n$ be a solution to the difference scheme $(\ref{dc1})$
with initial data $U_j^0$. Then the $BV$-estimate
\begin{eqnarray*}
 \sum_{j=-\infty}^{+\infty}|U_{j}^{n}-U_{j-1}^{n}|\leq \sum_{j=-\infty}^{+\infty}|U_{j}^{0}-U_{j-1}^{0}|
\end{eqnarray*}
holds for all $n\ge 0$.
\end{cor}

Having Corollary $\ref{cor2}$, we show the uniform boundedness of
the difference solutions, which is not covered in Lemma \ref{lem2}.
\begin{lem}\label{boundedness}
\begin{eqnarray*}
\sup_j|U_j^n|\leq\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-U_{j-1}^{0}|
\end{eqnarray*}
holds for all $n\ge 0$.
\end{lem}

\begin{proof}
Thanks to the initial data $(\ref{inital})$, we have $U_{j_0}^n=0$
if $j_0$ is large enough. Thus, for any $j$ it follows that
\begin{eqnarray*}
|U_j^n|=|U_j^n - U_{j_0}^n|\leq \sum_{k=j + 1}^{j_0}|U_k^n -
U_{k-1}^n|\leq\sum_{k=-\infty}^{+\infty}|U_k^0 - U_{k-1}^0|.
\end{eqnarray*}
This completes the proof.
\end{proof}

\begin{rem}
This lemma requires that the initial data are of bounded variation.
Otherwise, we could not obtain the uniform boundedness of the
difference solutions.
\end{rem}

\section{Time-Lipschitz Continuity}
\setcounter{equation}{0}

In this section we show the time-Lipschitz continuity of the
difference solutions, which seems not so easy for the present
problem.

To begin with, we set
$G(U)=(f_1,f_2,\cdots,f_{r-1})^T\in\mathbb{R}^{r-1}$ and
$G_j^n=G(U_j^n)$. Note that
\begin{eqnarray}\label{QF}
Q(U)=KG(U), \qquad G(U)= K'Q(U),
\end{eqnarray}
where $K$ is the constant $r\times(r-1)$-matrix
\begin{eqnarray}\label{matrix}
K = \left( \begin{array}{cccccc}
1 &  &   &  &    \\
-1 & 1&   & & \\
{}&  -1& 1& & \\
{}& &  \ldots& \ldots &\ldots\\
 {}& &  & -1& 1\\
 {}& &  & & -1

\end{array} \right).
\end{eqnarray}
and $K'$ is the constant $(r-1)\times r$-matrix such that $K'K=I_{r
-1}$. Moreover, it follows from the mean-value theorem and the
definitions in $(\ref{int1})$ and $(\ref{int2})$ that
$$
G_j^{n+1}-G_j^n=\mathcal {M}(U_j^n, U_j^{n+1})(U_j^{n+1}-U_j^n)
$$
with $\mathcal {M}(U_j^n, U_j^{n+1})$ the following $(r-1)\times r$
matrix
\begin{eqnarray*}
\mathcal {M}(U_j^n, U_j^{n+1})=\left( \begin{array}{cccccc}
A_1 & B_1  &   &  &    \\
    &  A_2& B_2  & & \\
{}&  & A_3&B_3 & \\
{}& &  \ldots& \ldots &\ldots\\
 {}& &  & A_{r-1}&B_{r-1}

\end{array} \right).
\end{eqnarray*}
Set $M = M(U_j^n, U_j^{n+1})=\mathcal {M}(U_j^n,
U_j^{n+1})K\in\mathbb{R}^{(r-1)\times (r-1)}$. By calculation, we
have
\begin{eqnarray*}
M =\left( \begin{array}{cccccc}
A_1-B_1 & B_1  &   &  &    \\
  -A_2  & A_2-B_2 & B_2  & & \\
{}&  -A_3& A_3-B_3&B_3 & \\
{}& &  \ldots& \ldots &\ldots\\
 {}& &  & -A_{r-1}&A_{r-1}-B_{r-1}

\end{array} \right).
\end{eqnarray*}
Thanks to the strict monotonicity assumption (1) on the $f_i$'s, it
is an elementary fact that $I-\frac{\Delta t}{\epsilon}M$ is
invertible.

We assume that there exists a $(U_j^n, U_j^{n+1})$-independent norm
$|\cdot|_\star$ on ${\bf R}^{r-1}$ and a positive constant $\lambda$
such that
\begin{eqnarray}\label{inverse}
|(I-\frac{\Delta t}{\epsilon}M)^{-1}|_\star \leq (1 + \frac{\Delta
t}{\epsilon}\lambda)^{-1}.
\end{eqnarray}
Although such a norm has not been found for general cases with $r
> 3$, this assumption is indeed true for $r\le 3$ and for
the case where $Q(U)$ is linear with respect to $U$. In fact, we
have

\begin{prop}\label{prop1}
For $r\le 3$, there exists a positive constant $\lambda$ such that
\begin{eqnarray*}
|(I-\frac{\Delta t}{\epsilon}M)^{-1}|_\infty\leq (1 + \frac{\Delta
t}{\epsilon}\lambda)^{-1}.
\end{eqnarray*}
\end{prop}

\begin{proof}
When $r=2$, $M$ is the negative number $(A_1 - B_1)$. Thus we have
$$
|(I-\frac{\Delta t}{\epsilon}M)^{-1}|_\infty = (1 + \frac{\Delta
t}{\epsilon}(B_1-A_1))^{-1}\leq (1 + \frac{\Delta
t}{\epsilon}\lambda)^{-1}
$$
with $\lambda = \min_{V, W}\{B_1(V, W) - A_1(V, W)\} > 0$. Here the
boundedness of the different solution established in Lemma
\ref{boundedness} has been used.

For $r=3$, $M$ is the $2\times2$ matrix
\begin{eqnarray*}
M =\left( \begin{array}{cccccc}
A_1-B_1 & B_1   \\
-A_{2}&A_{2}-B_{2}

\end{array} \right).
\end{eqnarray*}
Then we have
\begin{eqnarray*}
I-\frac{\Delta t}{\epsilon}M=\left( \begin{array}{cccccc}
1+\frac{\Delta t}{\epsilon}(B_1-A_1) & -\frac{\Delta t}{\epsilon}B_1  \\
\frac{\Delta t}{\epsilon}A_{2}&1+\frac{\Delta
 t}{\epsilon}(B_{2}-A_{2})

\end{array} \right)
\end{eqnarray*}
and
\begin{eqnarray*}
&{}&(I-\frac{\Delta t}{\epsilon}M)^{-1}\\
&=& \frac{1}{1+ A\frac{\Delta t}{\epsilon}+ B\frac{\Delta
 t^2}{\epsilon^2}}
\left( \begin{array}{cccccc} 1+\frac{\Delta t}{\epsilon}(B_2-A_2) &
\frac{\Delta t}{\epsilon}B_1   \\
-\frac{\Delta t}{\epsilon}A_{2}&1+\frac{\Delta t}{\epsilon}(B_1-A_1)

\end{array} \right)
\end{eqnarray*}
with $A=B_1-A_1+B_2-A_2$ and $B=B_1B_2 - A_1B_2 + A_1A_2$. Recall
that $A_i < 0$ and $B_i > 0$ for $i=1, 2$. Set
$$
a=\max\{B_1 + B_2-A_2, B_1-A_1 -A_2\}.
$$
It is obvious that $0 < a < A$ and $B>0$. Thus we deduce that
\begin{eqnarray*}
|(I-\frac{\Delta t}{\epsilon}M)^{-1}|_\infty = \frac{1 +
a\frac{\Delta t}{\epsilon}}{1+ A\frac{\Delta t}{\epsilon}+
B\frac{\Delta
 t^2}{\epsilon^2}}\leq (1 +
\frac{\Delta t}{\epsilon}\lambda)^{-1}
\end{eqnarray*}
with
$$
\lambda = \min\{A - a, B/a\} >0.
$$
This completes the proof.
\end{proof}

For $r>3$, $M$ is no longer strictly diagonally dominant and the
inequality in Proposition \ref{prop1} does not hold anymore.
However, we have

\begin{prop}\label{prop2}
If $Q(U)$ is linear with respect to $U$, then the above assumption
$(\ref{inverse})$ holds.

\end{prop}

\begin{proof}
Since $Q(U)$ is linear with respect to $U$, $M$ is a constant
matrix. Set
$\alpha_1=1,\alpha_{i+1}=-\alpha_{i}\frac{B_i}{A_{i+1}},$ and
$D=diag(\alpha_1,\alpha_2,\cdots,\alpha_{r-1})$. It is obvious that
the diagonal matrix $D$ is positive definite and $DM$ is symmetric.
Then $D^{\frac{1}{2}}MD^{-\frac{1}{2}}$ is also symmetric. Thus,
there exist an orthogonal matrix $T$ and a diagonal
negative-definite matrix $N$ such that
\begin{eqnarray*}
D^{\frac{1}{2}}MD^{-\frac{1}{2}}=T^{-1}N T,
\end{eqnarray*}
and thereby
\begin{eqnarray*}
(TD^{\frac{1}{2}})(I-\frac{\Delta
t}{\epsilon}M)^{-1}=(I-\frac{\Delta
t}{\epsilon}N)^{-1}(TD^{\frac{1}{2}}).
\end{eqnarray*}
Set $P=TD^{\frac{1}{2}}$. We define a norm on ${\bf R}^{r-1}$ as
$$
|\xi|_p = |P\xi|
$$
for $\xi\in {\bf R}^{r-1}$. Thus the corresponding matrix norm is
\begin{eqnarray*}
|(I-\frac{\Delta t}{\epsilon}M)^{-1}|_\star : = \sup_{|\xi|_p\leq
1}|(I-\frac{\Delta t}{\epsilon}M)^{-1}\xi|_p \\
= \sup_{|\xi|_p\leq 1}|P(I-\frac{\Delta
t}{\epsilon}M)^{-1}\xi|\\
= \sup_{|\xi|_p\leq 1}|(I-\frac{\Delta
t}{\epsilon}N)^{-1}P\xi|\\
\leq \sup_{|\xi|_p
\leq 1}|(I-\frac{\Delta t}{\epsilon}N)^{-1}||P\xi|\\
\leq|(I-\frac{\Delta t}{\epsilon}N)^{-1}| = (1 + \frac{\Delta
t}{\epsilon}\lambda)^{-1},
\end{eqnarray*}
where $\lambda$ is the smallest eigenvalue of $-N$.
\end{proof}

Now we turn to estimate $Q^n=Q(U_j^n)$.

\begin{lem}\label{lem4}
Assume (\ref{inverse}) holds. Then there is a positive constant $C$
such that
\begin{eqnarray*}
||Q^n||_{L^1}\le C((1+\lambda\frac{\Delta
t}{\epsilon})^{-n}||Q^0||_{L^1}+\epsilon).
\end{eqnarray*}
 for all $n$.
\end{lem}
\begin{proof}
From the difference scheme it follows that
\begin{eqnarray*}
G_j^{n+1}-G_j^n& = &\mathcal{M}(U_j^n, U_j^{n+1})(U_j^{n+1}-U_j^n)\\
& = & \mathcal{M}(U_j^n, U_j^{n+1})[\frac{\Delta
t}{\epsilon}Q_j^{n+1} - \frac{\Delta t}{\Delta
x}\Lambda^+(U_j^{n}-U_{j-1}^{n}) - \frac{\Delta t}{\Delta
x}\Lambda^-(U_{j+1}^{n}-U_j^{n})]\\
&=&\frac{\Delta t}{\epsilon}MG_j^{n+1}-\mathcal{M}(U_j^n,
U_j^{n+1})[\Lambda^+(U_j^{n}-U_{j-1}^{n})+\Lambda^-(U_{j+1}^{n}-U_j^{n})]\frac{\Delta
t}{\Delta x}
\end{eqnarray*}
and thereby
\begin{eqnarray*}
(I-\frac{\Delta t}{\epsilon}M)G_j^{n+1}=G_j^n-\mathcal {M}(U_j^n,
U_j^{n+1})[\Lambda^+(U_j^{n}-U_{j-1}^{n})\\+
\Lambda^-(U_{j+1}^{n}-U_j^{n})]\frac{\Delta t}{\Delta x}.\nonumber
\end{eqnarray*}
Thus we deduce from the inequality (\ref{inverse}) that
\begin{eqnarray*}
&&(1+\lambda\frac{\Delta t}{\epsilon})|G_j^{n+1}|_\star \leq
|(I-\frac{\Delta t}{\epsilon}M)G_j^{n+1}|_\star \\
& = & |G_j^n-\mathcal {M}(U_j^n,
U_j^{n+1})[\Lambda^+(U_j^{n}-U_{j-1}^{n})+\Lambda^-(U_{j+1}^{n}-U_j^{n})]\frac{\Delta
t}{\Delta x}|_\star\\
& \leq & |G_j^n|_\star + |\mathcal {M}(U_j^n,
U_j^{n+1})[\Lambda^+(U_j^{n}-U_{j-1}^{n}) +
\Lambda^-(U_{j+1}^{n}-U_j^{n})]|_\star\frac{\Delta t}{\Delta x} .
\end{eqnarray*}
Since $|\cdot|_\star$ is equivalent to the $1$-norm $|\cdot|$ on
${\bf R}^{r-1}$ and since the difference solution is bounded, there
is a positive constant $C$ such that
\begin{eqnarray*}
(1+\lambda\frac{\Delta t}{\epsilon})|G_j^{n+1}|_\star \leq
|G_j^n|_\star + C\frac{\Delta t}{\Delta x}(|U_j^{n}-U_{j-1}^{n}| +
|U_{j+1}^{n}-U_j^{n}|).
\end{eqnarray*}
Consequently, we use Corollary $\ref{cor2}$ to obtain
\begin{eqnarray*}
(1+\lambda\frac{\Delta t}{\epsilon})\sum_j|G_j^{n+1}|_\star\Delta x
&\le& \sum_j|G_j^n|_\star\Delta x+C\Delta
t\sum_j|U_j^{n}-U_{j-1}^{n}|\\
&\le& \sum_j|G_j^n|_\star\Delta x + C\Delta t.
\end{eqnarray*}

From the last inequality, it is easy to verify that
\begin{eqnarray*}
\sum_j|G_j^{n}|_\star\Delta x\le (1+\lambda\frac{\Delta
t}{\epsilon})^{-n}\sum_j|G_j^{0}|_\star\Delta x+C\epsilon\le
C((1+\lambda\frac{\Delta t}{\epsilon})^{-n}\sum_j|G_j^{0}|\Delta
x+\epsilon).
\end{eqnarray*}
Thus we get
\begin{eqnarray*}
||G^n||_{L^1} \leq C\sum_j|G_j^{n}|_\star\Delta x \le
C((1+\lambda\frac{\Delta t}{\epsilon})^{-n}\sum_j|G_j^{0}|\Delta
x+\epsilon).
\end{eqnarray*}
Finally, from $(\ref{QF})$ it follows that
\begin{eqnarray*}
||Q^n||_{L^1}\le C||G^n||_{L^1} \qquad \mbox{and} \qquad
||G^0||_{L^1}\le C||Q^0||_{L^1}.
\end{eqnarray*}
Hence
\begin{eqnarray*}
||Q^n||_{L^1}\le C((1+\lambda\frac{\Delta
t}{\epsilon})^{-n}||Q^0||_{L^1}+\epsilon)
\end{eqnarray*}
and the proof is complete.
\end{proof}

Now we can easily show the time-Lipschitz continuity of $U^n$.
\begin{lem}\label{lem5}
Assume (\ref{inverse}) holds. Then
\begin{eqnarray*}
||U_j^{n+1}-U_j^n||_{L^1}\le C(\frac{\Delta
t}{\epsilon}(1+\lambda\frac{\Delta
t}{\epsilon})^{-(n+1)}||Q^0||_{L^1}+\Delta t)
\end{eqnarray*}
for $n=0,1,2,\cdots.$
\end{lem}

\begin{proof}
From the difference scheme $(\ref{dc1})$ we have
\begin{eqnarray*}
U_j^{n+1}-U_j^n=\frac{\Delta t}{\epsilon}Q_j^{n+1}-\frac{\Delta
t}{\Delta x}\Lambda^+(U_j^{n}-U_{j-1}^{n})-\frac{\Delta t}{\Delta
x}\Lambda^-(U_{j+1}^{n}-U_j^{n})
\end{eqnarray*}
and therefore
\begin{eqnarray*}
|U_j^{n+1}-U_j^n|\le\frac{\Delta
t}{\epsilon}|Q_j^{n+1}|+C\frac{\Delta t}{\Delta
x}(|U_j^{n}-U_{j-1}^{n}|+|U_{j+1}^{n}-U_j^{n}|).
\end{eqnarray*}
Thus, from Corollary $\ref{cor2}$ and Lemma $\ref{lem4}$ we deduce
that
\begin{eqnarray*}
||U_j^{n+1}-U_j^n||_{L^1}\le\frac{\Delta
t}{\epsilon}||Q_j^{n+1}||_{L^1}+C\Delta t\leq C(\frac{\Delta
t}{\epsilon}(1+\lambda\frac{\Delta
t}{\epsilon})^{-(n+1)}||Q^0||_{L^1}+\Delta t).
\end{eqnarray*}
This completes the proof.
\end{proof}

\section{Entropies}
\setcounter{equation}{0}

In this section, we discuss entropy functions for the
reaction-hyperbolic system $(\ref{pde1})$ and its equilibrium system
$(\ref{pde2})$. For this purpose, we make the following additional assumption\\

\begin{itemize}

\item[(3).] For any fixed $v$, there exists $w$ such that $vf_i(v, w) \geq
0$ for $i=1, 2, \cdots, r-1$.\\
\end{itemize}

\noindent This assumption, together with those made in the previous
section, ensures that there exists a unique and globally-defined
function $h_i$ of one variable such that
\begin{equation}\label{31}
f_i(u_i, u_{i+1})= 0 \qquad \mbox{iff} \qquad u_{i+1} = h_i(u_i)
\end{equation}
for $i=1, 2, \cdots, r-1$. Obviously, $h_i$ is strictly increasing
and $h_i(0)=0$. By the implicit function theorem, $h_i$ is
continuously differentiable. Thus the equilibrium system
$(\ref{pde2})$ can be written as
\begin{equation}\label{32}
\begin{split}
\partial_t(u_1+u_2+\cdots+u_r) +
\partial_x(\lambda_1u_1+\lambda_2u_2+\cdots+\lambda_ru_r)=0,\\
u_2=h_1(u_1), \ u_3 = h_2(u_2), \ \cdots, \ u_r = h_{r-1}(u_{r-1}).
\end{split}
\end{equation}

Set

\begin{eqnarray*}
v = u_1 + h_1(u_1) + h_2\circ h_1(u_1) + \cdots + h_{r-1}\circ
h_{r-2}\circ \cdots \circ h_1(u_1).
\end{eqnarray*}
Since the right-hand side is strictly increasing with respect to
$u_1$, $u_1$ can be expressed as a function of $v$, say $u_1=
u_1(v)$. By the inverse function theorem, $u_1(v)$ is continuously
differentiable. Set
$$
u_{i+1}(v) = h_i(u_i(v))
$$
for $i=1, 2, \cdots, r-1$. Consequently, each $u_i(v)$ is strictly
increasing, continuously differentiable and $u_i(0)=0$. Set
\begin{eqnarray*}
v=u_1(v)+u_2(v)+\cdots+u_r(v) \quad \mbox{and} \quad h(v) =
\lambda_1u_1(v) + \lambda_2u_2(v) + \cdots + \lambda_r u_r(v).
\end{eqnarray*}
Then the equilibrium system $(\ref{32})$ can be rewritten as
\begin{eqnarray}\label{pde2*}
\partial_tv + \partial_xh(v)=0.
\end{eqnarray}
Recall that any convex function is a convex entropy function for
scalar conservation laws like (\ref{pde2*}) (see, e.g., \cite{Da}).

Next, we turn to discuss the entropy functions for system
$(\ref{pde2*})$ and the reaction-hyperbolic system $(\ref{pde1})$.

\begin{lem}\label{entropy1}
Given a strictly convex smooth function $\tilde{\eta}(v)$, there is
a dissipative entropy function $\eta(U)$, in the sense of \cite{Y3},
for the reaction-hyperbolic system $(\ref{pde1})$ such that
\begin{eqnarray*}
\tilde{\eta}(v)=\eta(u_1(v), u_2(v), \cdots,
u_r(v))\equiv\eta(U(v)).
\end{eqnarray*}
\end{lem}

\begin{proof}
We inductively define
\begin{eqnarray*}
\eta_r(u) &=& \int^u_0\tilde{\eta}'(u_r^{-1}(w))dw, \\
\eta_{i-1}(u) &=& \int^u_0\eta_i'(h_{i-1}(w))dw
\end{eqnarray*}
for $i= r, r-1, \cdots, 2$. Then we have
\begin{eqnarray}\label{35}
\eta_{i-1}'(u_{i-1}) = \eta_i'(h_{i-1}(u_{i-1}))  \qquad \mbox{and}
\qquad \eta_r'(u) = \tilde{\eta}'(u_r^{-1}(u)) .
\end{eqnarray}

Since $\tilde{\eta}(v)$ is strictly convex and $u_r^{-1}$ and $h_i$
are all strictly increasing, it is clear that $\eta_1', \eta_2',
\cdots, \eta_r'$ are strictly increasing. Therefore,
\begin{eqnarray}\label{36}
\eta(U): =\sum_{i=1}^r\eta_i(u_i) + \tilde{\eta}(0)
\end{eqnarray}
is a strictly convex function of $U$. Recall the matrix $K$ defined
in (\ref{matrix}). We see that
$$
(\eta'_1 - \eta'_2, \eta'_2 - \eta'_3, \cdots, \eta'_{r-1} -
\eta'_r)^T = K^T\eta_U(U)
$$
and, moreover, from (\ref{QF}) that
\begin{equation}\label{37}
\begin{split}
Q(U) & = KG(U) \\
&=K\mbox{diag}(\frac{f_1(u_1, u_2)}{\eta'_1 - \eta'_2},
\frac{f_2(u_2, u_3)}{\eta'_2 - \eta'_3}, \cdots,
\frac{f_{r-1}(u_{r-1}, u_r)}{\eta'_{r-1} - \eta'_r})K^T\eta_U(U)\\
&\equiv S(U)\eta_U(U).
\end{split}
\end{equation}
Thanks to the relations in (\ref{35}), we deduce from (\ref{31}),
the convexity of $\eta_i$ and the monotonicity assumption (1) that
\begin{eqnarray*}
\frac{f_i(u_i, u_{i+1})}{\eta'_i - \eta'_{i+1}} &=& \frac{f_i(u_i,
u_{i+1})}{\eta'_{i+1}(h_i(u_i)) -
\eta'_{i+1}(u_{i+1})}\\
& = & -\frac{\int_0^1 f_{iu_{i+1}}(u_i, h_i(u_i) + \sigma(u_{i+1} -
h_i(u_i)))d\sigma}{\int_0^1 \eta''_{i+1}(h_i(u_i) + \sigma(u_{i+1} -
h_i(u_i)))d\sigma} < 0.
\end{eqnarray*}
Thus, $S(U)$ is a symmetric and non-positive definite matrix. Its
null space is obviously that of $K^T$, which is independent of $U$.
Consequently, $\eta(U)$ is a dissipative entropy function, in the
sense of \cite{Y3}, for the reaction-hyperbolic system
$(\ref{pde1})$.

Furthermore, we deduce from the relations in (\ref{35}) that
\begin{eqnarray*}
\eta_1'(u_1(v))&=&\eta_2'(h_1(u_1(v)))\\
&=&\eta_3'(h_2\circ h_1(u_1))\\
&=&\cdots\\
&=&\eta_r'(h_{r-1}\circ h_{r-2}\circ \cdots \circ h_1(u_1))\\
&=&\eta_r'(u_r(v))\\
&=&\tilde{\eta}'(v).
\end{eqnarray*}
Similarly, we have
\begin{eqnarray*}
\eta_2'(u_2(v))=\cdots=\eta_r'(u_r(v))=\tilde{\eta}'(v).\\
\end{eqnarray*}
Therefore, we deduce from $u_1(v)+u_2(v)+\cdots+u_r(v)\equiv v$ that
\begin{eqnarray*}
(\eta(U(v)))'&=&(\sum_{i=1}^r\eta_i(u_i(v)))'\\
&=&\eta_1'(u_1(v))u_1'(v)+\eta_2'(u_2(v))u_2'(v)+\cdots+\eta_r'(u_r(v))u_r'(v)\\
&=&\tilde{\eta}'(v).
\end{eqnarray*}
In view of $\eta_i(0)=0=u_i(0)$, we see from (\ref{36}) that
$\eta(U(0))=\tilde\eta(0)$ and hence
\begin{eqnarray*}
\tilde{\eta}(v)=\eta(U(v)).
\end{eqnarray*}
This completes the proof.
\end{proof}

We conclude this section with a discrete entropy inequality for the
difference solutions.
\begin{lem}\label{entropy}
Let $U_j^n$ be a solution to the difference scheme $(\ref{dc1})$.
Then, for any smooth convex function
$\eta(U)=\sum_{i=1}^r\eta_i(u_i)$, there exists a Lipschitz
continuous function $\Psi$ of two variables such that for all $j\in
\mathbb{Z}$ and $n\geq 0$, the following cell entropy inequalities
hold:
\begin{eqnarray*}
\eta(U_j^{n+1})&\leq& \eta(U_j^n)-\frac{\Delta t}{\Delta
x}(\Psi(U_{j}^n,U_{j+1}^n)-\Psi(U_{j-1}^n,U_{j}^n))+\frac{\Delta
t}{\epsilon}\eta_U(U_j^{n+1})Q(U_j^{n+1}).
\end{eqnarray*}
Moreover, the Lipschitz continuous function satisfies the
consistency relation
\begin{eqnarray*}
\Psi(U,U)=\sum_{i=1}^r\lambda_i\eta_i(u_i).
\end{eqnarray*}
\end{lem}

\begin{proof}
For any smooth convex function $\eta_i$ and any two real numbers
$a,b\in \mathbb{R}$, it is standard that
\begin{eqnarray*}
\eta_i(b)-\eta_i(a)\leq \eta_i'(b)(b-a).
\end{eqnarray*}
Thus, for the given convex function
$\eta(U)=\sum_{i=1}^r\eta_i(u_i)$ and for any $V,W\in \mathbb{R}^r$
we have
\begin{eqnarray*}
\eta(V)-\eta(W)\leq \eta_U(V)(V-W).
\end{eqnarray*}
Thus, it follows from the original difference scheme that
\begin{eqnarray*}
\eta(U_{j}^{n+1})\leq\eta(U_j^n-\frac{\Delta t}{\Delta
x}\Lambda^+(U_j^{n}-U_{j-1}^{n})-\frac{\Delta t}{\Delta
x}\Lambda^-(U_{j+1}^{n}-U_j^{n}))+\eta_U(U_{j}^{n+1})\frac{\Delta
t}{\epsilon}Q_j^{n+1},
\end{eqnarray*}
since
\begin{eqnarray*}
U_j^{n+1}=\frac{\Delta t}{\epsilon}Q_j^{n+1}+U_j^n-\frac{\Delta
t}{\Delta x}\Lambda^+(U_j^{n}-U_{j-1}^{n})-\frac{\Delta t}{\Delta
x}\Lambda^-(U_{j+1}^{n}-U_j^{n}).
\end{eqnarray*}
On the other hand, we deduce from $(\ref{UPwind}), (\ref{CFL})$ and
the convexity of $\eta_i$ that
\begin{eqnarray*}
&{}&\eta(U_j^n-\frac{\Delta t}{\Delta
x}\Lambda^+(U_j^{n}-U_{j-1}^{n})-\frac{\Delta t}{\Delta
x}\Lambda^-(U_{j+1}^{n}-U_j^{n}))\\
&=&\eta(\frac{\Delta t}{\Delta x}\Lambda^+U_{j-1}^n+(I-\frac{\Delta
t}{\Delta x}\Lambda^++\frac{\Delta t}{\Delta
x}\Lambda^-)U_j^n-\frac{\Delta t}{\Delta
x}\Lambda^-U_{j+1}^{n})\\
&=&\sum_{i=1}^r\eta_i(\lambda_i^+\frac{\Delta t}{\Delta
x}u_{i,j-1}^n+(1-\lambda_i^+\frac{\Delta t}{\Delta
x}+\lambda_i^-\frac{\Delta t}{\Delta x})u_{i,j}^n
-\lambda_i^-\frac{\Delta t}{\Delta
x}u_{i,j+1}^n)\\
&\leq&\sum_{i=1}^r[\lambda_i^+\frac{\Delta t}{\Delta
x}\eta_i(u_{i,j-1}^n)+(1-\lambda_i^+\frac{\Delta t}{\Delta
x}+\lambda_i^-\frac{\Delta t}{\Delta
x})\eta_i(u_{i,j}^n)-\lambda_i^-\frac{\Delta t}{\Delta
x}\eta_i(u_{i,j+1}^n)]\\
&=&\sum_{i=1}^r[\eta_i(u_{i,j}^n)-\frac{\Delta t}{\Delta
x}(|\lambda_i|\eta_i(u_{i,j}^n)-\lambda_i^+\eta_i(u_{i,j-1}^n)+\lambda_i^-\eta_i(u_{i,j+1}^n))].
\end{eqnarray*}
Define
\begin{eqnarray*}
\Psi(U,V)=\sum_{i=1}^r[\frac{\lambda_i}{2}(\eta_i(u_i)+\eta_i(v_i))
+\frac{|\lambda_i|}{2}(\eta_i(u_i)-\eta_i(v_i))].
\end{eqnarray*}
This $\Psi$ is obviously Lipschitz continuous and satisfies the
consistency relation. Moreover, the above inequalities lead directly
to
\begin{eqnarray*}
\eta(U_j^{n+1})&\leq& \eta(U_j^n)-\frac{\Delta t}{\Delta
x}(\Psi(U_{j}^n,U_{j+1}^n)-\Psi(U_{j-1}^n,U_{j}^n))+\frac{\Delta
t}{\epsilon}\eta_U(U_j^{n+1})Q(U_j^{n+1}).
\end{eqnarray*}
This completes the proof.
\end{proof}

\section{Main Results}
\setcounter{equation}{0}

In this section we prove the main results of this paper. To this
end, we define
$$
U^\Delta(x,t)=(u_1^{\Delta}(x,t), u_2^{\Delta}(x,t),\cdots,
u_r^{\Delta}(x,t))^T:=(u_{1,j}^{n}, u_{2,j}^{n},\cdots,
u_{r,j}^{n})^T
$$
for $(x,t)\in [j\Delta x,(j+1)\Delta x)\times[n\Delta t,(n+1)\Delta
t)$. With this definition, it simply follows from Corollary
$\ref{cor2}$, Lemma \ref{boundedness} and Lemma \ref{lem5} that
\begin{lem}\label{41}
The piecewise constant function $U^{\Delta}(x,t)$ satisfies the
following estimates
\begin{eqnarray}
&{}&|U^{\Delta}(x,t)|\leq\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-U_{j-1}^{0}| \quad
\textrm{ for all } (x,t),\label{1}\\
&{}&TV(U^{\Delta}(\cdot,t))\leq\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-U_{j-1}^{0}|,\label{estimate1}\\
&{}&||U^\Delta(\cdot,t)-U^\Delta(\cdot,t_1)||_{L^1}\leq
C(\frac{1}{\epsilon}(1+\lambda\frac{\Delta
t}{\epsilon})^{{-\frac{\min\{t,t_1\}}{\Delta t}}}||Q^0||_{L^1}+1)
(|t-t_1|+\Delta t)\label{2}
\end{eqnarray}
for all $t, t_1>0$.
\end{lem}

\begin{proof}
We only need to show the last inequality. Let $k,k_1$ be two
integers such that $t\in [k\Delta t,(k+1)\Delta t),t_1\in [k_1\Delta
t,(k_1+1)\Delta t)$. Without loss of generality, we assume $k_1\leq
k.$ Then we deduce from the definition of $U^\Delta$ and Lemma
$\ref{lem5}$ that
\begin{eqnarray*}
||U^\Delta(\cdot,t)-U^\Delta(\cdot,t_1)||_{L^1}&=&\sum_{j}|U_j^k-U_j^{k_1}|\Delta
x\\&\leq&\sum_{n=k_1}^{k-1}\sum_{j}|U_j^{n+1}-U_j^{n}|\Delta
x\\&\leq&\sum_{n=k_1}^{k-1}C(\frac{1}{\epsilon}(1+\lambda\frac{\Delta
t}{\epsilon})^{-(n+1)}||Q^0||_{L^1}+1)\Delta t\\
&\leq& C(\frac{1}{\epsilon}(1+\lambda\frac{\Delta
t}{\epsilon})^{-(k_1+1)}||Q^0||_{L^1}+1)(k-k_1)\Delta t\\
&\le&C(\frac{1}{\epsilon}(1+\lambda\frac{\Delta
t}{\epsilon})^{-\frac{\min\{t,t_1\}}{\Delta
t}}||Q^0||_{L^1}+1)(|t-t_1| + \Delta t).
\end{eqnarray*}
This completes the proof.
\end{proof}

Having the estimates in Lemma \ref{41} and the discrete entropy
inequality in Lemma $\ref{entropy}$, we follow the standard argument
in \cite{Da} to obtain ($\epsilon$ is fixed)

\begin{thm}\label{th1}
Suppose the initial data
$U_0(x)=(u_{10}(x),u_{20}(x),\cdots,u_{r0}(x))$ have bounded
variations, the grid sizes $\Delta t$ and $\Delta x$ satisfy the
CFL-condition $(\ref{CFL})$, the $f_i's$ satisfy the assumptions
(1)--(3), and the inequality (\ref{inverse}) holds. Then, as the
grid sizes $\Delta t,\Delta x$ tend to zero, there is a subsequence
of the function family $U^{\Delta}(x,t)=(u_1^{\Delta}(x,t),
u_2^{\Delta}(x,t),\cdots, u_r^{\Delta}(x,t))$ converging in
$(L_{loc}^1(\mathbb{R}^1 \times \mathbb{R}^+) )^r$ to an entropy
solution $U^{\epsilon}(x,t)=(u_1^{\epsilon}, u_2^{\epsilon},\cdots,
u_r^{\epsilon})$ of the reaction-hyperbolic system $(\ref{pde1})$
with initial data $U_0(x)$. Furthermore, the solution fulfills the
following estimates
\begin{eqnarray}\label{estimate2}
&{}&|U^{\epsilon}(x,t)|\leq\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-U_{j-1}^{0}|
\quad \textrm{ for almost all } (x,t),\label{3}\\
&{}&TV(U^{\epsilon}(\cdot,t))\leq\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-U_{j-1}^{0}|,\\
&{}&||U^\epsilon(\cdot,t)-U^\epsilon(\cdot,t_1)||_{L^1}\leq
C(\frac{1}{\epsilon}\exp(-\frac{\lambda\min\{t,t_1\}}{\epsilon})||Q^0||_{L^1}+1)|t-t_1|.
\label{4}
\end{eqnarray}
for all $t,t_1>0$.
\end{thm}

In the framework of BV-solutions, the zero-relaxation limit can be
very easily discussed. In fact, the standard argument in \cite{Da}
proves that the embedding of
$L^\infty(\mathbb{R}_+\times\mathbb{R})\cap L^\infty(\mathbb{R}_+,
BV(\mathbb{R}))\cap Lip(\mathbb{R}_+, L^1(\mathbb{R}))$ into
$L^1_{loc}(\mathbb{R}_+\times\mathbb{R})$ is compact. On the other
hand, the estimates in $(\ref{3})-(\ref{4})$ show that
$\{U^\epsilon\}_{\epsilon>0}$ lies in a bounded subset of
$L^\infty(\mathbb{R}_+\times\mathbb{R})\cap L^\infty(\mathbb{R}_+,
BV(\mathbb{R}))\cap Lip(\mathbb{R}_+, L^1(\mathbb{R}))$ by assuming
\begin{equation}\label{47}
\|Q^0\|_{L^1}=0 .
\end{equation}
Namely, the initial data are assumed to be in equilibrium. Thus, we
have
\begin{thm} Under the conditions of
Theorem $\ref{th1}$ and the equilibrium assumption (\ref{47}), there
exist a bounded measurable function
$U^*(x,t)=(u_1^*(x,t),u_2^*(x,t),\cdots,u_r^*(x,t))$ and a
subsequence(denoted in the same way) of set
$\{U^\epsilon(x,t)=(u_1^{\epsilon}, u_2^{\epsilon},\cdots,
u_r^{\epsilon})\}$ such that as $\epsilon\rightarrow0$,
\begin{eqnarray*}
U^\epsilon(x,t) \rightarrow U^*(x,t) &in& (L_{loc}^1(\mathbb{R}^1
\times \mathbb{R}^+) )^r.
\end{eqnarray*}
Moreover, the function $U^*(x,t)$ is a weak entropy solution to the
Cauchy problem $(\ref{pde2})$ with initial data $U_0(x)$ and
satisfies
\begin{eqnarray*}
&{}&|U^{*}(x,t)|\leq\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-U_{j-1}^{0}| \quad \textrm{ for almost all } (x,t),\label{5}\\
&{}&TV(U^{*}(\cdot,t))\leq\sum_{j=-\infty}^{+\infty}|U_{j}^{0}-U_{j-1}^{0}|,\\
&{}&||U^*(\cdot,t)-U^*(\cdot,t_1)||_{L^1}\leq C|t-t_1|.\label{6}
\end{eqnarray*}
\end{thm}

\begin{rem}
The fact that $U^*(x,t)$ satisfies the entropy conditions for the
equilibrium system $(\ref{pde2})$ follows from Lemmas \ref{31} and
\ref{32}. Indeed, because $S(U)$ in (\ref{37}) is symmetric and
non-positive, the term $\eta_U(U^{n+1})Q(U^{n+1})$ in Lemma \ref{32}
is
$$
\eta_U(U^{n+1})Q(U^{n+1})=\eta_U(U^{n+1})S(U^{n+1})\eta_U(U^{n+1})\leq
0
$$
if $\eta$ is chosen to be that constructed in Lemma \ref{31}.
\end{rem}

\begin{rem}
Without the equilibrium assumption in (\ref{47}), $U^\epsilon$ can
only converge to $U^*$ for $t>0$ but not up to $t=0$, because
$$
\exp(-\lambda\min\{t, t_1\}/\epsilon)\leq
\frac{\epsilon}{e\lambda\min\{t, t_1\}}
$$
in $(\ref{4})$. Indeed, without the equilibrium assumption, initial
boundary-layers occur. See also \cite{Y1}.
\end{rem}

config: arxiv


text:
\chapter{Future directions}
\label{sec:future-directions}

\headerquote{The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design.}{---~-F.~Dyson}

\headerquote{An expert is a man who has made all the mistakes, which can be made, in a very narrow field.}{---~N.~Bohr}

\begin{remark}\label{rem:fut:boundary}
  We have done some groundwork in \S\ref{sec:the-boundary} for the formal construction of the boundary of an infinite resistance network, however there is much more to be done. The development of this boundary theory is currently underway in \cite{bdG}, where we make explicit the connections between our boundary, Martin boundary, and the theory of graph ends. As in this book, the notions of dipoles, monopoles, and harmonic functions play key roles.
\end{remark}

\begin{remark}\label{rem:fut:fractals}
  In \cite{JoPea10b}, we attempt to apply some results of the present investigation to the theory of fractal analysis.\footnote{Finally! If you remember from the introduction, this was our initial aim!} For now, we just show that the resistance distance as defined by \eqref{eqn:def:R(x,y)-Lap} extends to the context of analysis on \emph{PCF self-similar fractals}. The reader is referred to the definitive text \cite{Kig01} and the excellent tutorial \cite{Str06} for motivation and definitions.

  Suppose that \sF is a post-critically finite (PCF) self-similar set with an approximating sequence of graphs $\Graph_1, \Graph_m, \dots \Graph_m, \dots$ with $\Graph = \bigcup_m \verts_m$ and \sF is the closure of \Graph in resistance metric (which is equivalent to closure in Euclidean metric; see \cite[(1.6.10)]{Str06}). The definition of PCF can be found in \cite[Def.~1.3.4 and Def.~1.3.13]{Kig01}. In the following proof, the subscript $m$ indicates that the relevant quantity is computed on the corresponding \ERN $(\Graph_m,R_m)$. For example, $\Pot_m(x,y)$ is the set of dipoles on $\Graph_m$ (cf.~Definition~\ref{eqn:def:dipole}) and $\energy_m(u)$ is the appropriately renormalized energy of a function $u:\verts_m \to \bR$ (cf.~\cite[(1.3.20)]{Str06}).

  \begin{theorem}\label{thm:R(x,y)-Lap-fractal}
    For $x,y \in \sF$, the resistance distance is given by
    \linenopax
    \begin{equation}\label{eqn:R(x,y)-Lap-fractal}
      \min \{v(x)-v(y) \suth v \in \dom \energy, \Lap v = \gd_x-\gd_y\}.
    \end{equation}
    \begin{proof}
      By the definition of \sF, it suffices to consider the case when $x,y$ are junction points, that is, $x,y \in \Graph_m$ for some $m$. Then the proof follows for general $x,y \in \sF$ by taking limits.

      For $x,y \in \Graph_m$, let $v=v_m$ denote the element of $\Pot_m(x,y)$ of minimal energy; the existence and uniqueness of $v_m$ is justified by the results of \S\ref{sec:potential-functions}. From Theorem~\ref{thm:effective-resistance-metric} we have $R_m(x,y) = \energy(v_m)$.
      Next, apply the harmonic extension algorithm to $v_m$ to obtain $v_{m+1}$ on $\Graph_{m+1}$. By \cite[Lem.~1.3.1]{Str06},
      \linenopax
      \begin{align*}
        v_m(x)-v_m(y) = R_m(x,y) = \energy(v_m) = \energy(v_{m+1}) = \dots = \energy(\tilde v),
      \end{align*}
      where $\tilde v$ is the harmonic extension of $v$ to all of \Graph. It is clear by construction and the cited results that $\tilde v$ minimizes \eqref{eqn:R(x,y)-Lap-fractal}. Note that we do not need to worry about the possible appearance of nontrivial harmonic functions, as $\tilde v$ is constructed as a limit of functions with finite support.
    \end{proof}
  \end{theorem}

  This theorem offers a practical improvement over the formulation of resistance metric as found in the literature on fractals in a couple of respects:
  \begin{enumerate}
    \item \eqref{eqn:R(x,y)-Lap-fractal} provides a formula (or at least, an equation to solve) for the explicit function which gives the minimum in \cite[(1.6.1) or (1.6.2)]{Str06}.
    \item One can compute $v=v_m$ on $\Graph_m$ by basic methods, i.e., Kirchhoff's law and the cycle condition. To find $R(x,y)$, one need only evaluate $v(x)-v(y)$, and this may be done without even fully computing $v$ on all of $\Graph_m$.
  \end{enumerate}
\end{remark}

\version{
\begin{remark}\label{rem:spectral-reciprocity}
  The authors have uncovered a form of spectral reciprocity relating the Laplacian to the matrix $[\la v_x,v_y\ra_\energy]$. This topic is currently under investigation in \cite{SRAMO}.
\end{remark}
}{}

\begin{remark}\label{rem:quantum-graphs}
  The metric graphs and their analysis presented in this volume are ubiquitous, and we can not do justice to the vast literature. However, the application to quantum communication appears especially intriguing, and we refer to the following papers for detail: \cite{NeBr08,DuBr07,Fab06,GHB05,HCDB07}.
  \begin{quote}
    ``$\dots$ the computational power of an important class of quantum states called graph states, representing resources for measurement-based quantum computation, is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs.'' \q from \cite{NeBr08}.
  \end{quote}
  \emph{Quantum graphs} (also called \emph{cable systems} in \cite{Kig03} and \emph{graph refinements} in \cite{Telcs06a}) are essentially a refinement of \ERNs where the edges are replaced by intervals and functions are allowed to vary continuously for different values of $x$ in a single edge.
\end{remark}

\begin{remark}\label{rem:fut:rank-Pperp-is-numcycles}
  As noted in Remark~\ref{rem:rank-Pperp-is-numcycles}, the rank of $\Pdrp^\perp$ is an invariant related to the space of cycles in \Graph. However, this object is rather a blunt tool, and it would interesting to see if one can obtain a more refined analysis by applying extensions of the techniques Terras and Stark, as in  \cite{GIL06b}, for example.
\end{remark}


text:
\begin{document}

\title{\LARGE  Identifying the Fake Base Station: A Location Based Approach
\thanks{
The work was supported in part by the
National Natural Science Foundation of China under Grant 61671364 and in part
by the Outstanding Young Research Fund of Shaanxi Province.

}
\thanks{
K.-W. Huang and H.-M. Wang are with the Ministry
of Education Key Lab for Intelligent Networks and Network Security, Xi'an Jiaotong
University, Xi'an, 710049, Shaanxi, P. R. China. Email: {\tt
xjtu-huangkw@outlook.com,
xjbswhm@gmail.com}.
}
\author{Ke-Wen Huang, \hspace{0.05in} Hui-Ming Wang,~\IEEEmembership{Senior Member,~IEEE}

}
}
\maketitle

\begin{abstract}
    Fake base station (FBS) attack is a great security challenge to wireless user equipment (UE). During the cell selection stage, the UE receives multiple synchronization signals (SSs) from multiple nearby base stations (BSs), and then synchronizes itself with the strongest SS. A FBS also can transmit a SS with sufficient power to confuse the UE, which makes the UE connect to the FBS, and may lead to the leakage of private information.
    In this letter, countermeasure to the FBS attack by utilizing the location information is investigated.  Two location awareness based FBS-resistance schemes are proposed by checking the received signal strength according to the position of the UE and a legitimate BS map.
    The successful cheating rate (SCR) definded as the probability that the UE will connect to the FBS is investigated.
    Numeric results show that with the two proposed schemes, the SCR can be greatly reduced especially when the transmit power of the FBS is large. Beyond that,
    a cooperation aided method is further proposed to improve the performance, and
    we show that the cooperation aided method can further suppress the SCR when the signal strength from the FBS is similar to that from the legitimate BS.
\end{abstract}
\begin{IEEEkeywords}
Fake base station, location awareness, physical layer authentication, detection.
\end{IEEEkeywords}

\section{Introduction}
\label{Sec:Introduction}
The initial cell selection (CS) stage during which an user equipment (UE) searches for a suitable base station (BS) to camp on is vulnerable to a fake base station (FBS) attack.
In the CS stage, the UE keeps listening to the wireless broadcast channel and searches for the synchronizing signal (SS) from surrounding base stations (BSs).
After that, the UE selects a suitable BS based on the received SSs, and then begins to establish a wireless connection with the BS \cite{Book:S.Sesia}.
If a FBS transmits a spoofing SS during the CS stage with sufficiently high power ({referred to as the SS spoofing attack \cite{M.LichtmanCM2016}}), the UE may be attracted by and attempt to camp on the FBS rather than any legitimate BS (LBS) \cite{M.Labib2015,M.Labib2016}.
Currently, the widely-used authentication method for the UE to distinguish the FBS from the LBSs is based on encryption.
However, though the FBS will fail during the key-based authentication procedure, it still significantly degrades the system performance, for example, the FBS can lead to the significant increase of the delay before the UE can successfully connect to the legitimate network and more seriously, if the FBS further spoof the control signal, it can even cause the access denial to the network \cite{M.Labib2015,M.Labib2016}.

Recently, using the physical layer parameters to authenticate the signal source, namely physical layer authentication, has gained considerable attention which does not depends on any private key, see \cite{X.Wang2016} and references therein \footnote{We note that physical layer based approach has also been extensive investigated to secure data transmission in wireless channel, namely physical layer security, as a complementary scheme to the conventional cryptography-based scheme, see \cite{YLiu2017CST} and references therein.}.
For example, channel impulse response and device fingerprint based schemes were proposed in  \cite{L.Xiao2008} and \cite{A.C.Polak2011}, respectively, to verify the transmitter's identity.
However, these methods are almost impossible to be exploited to combat with the SS spoofing attack because they require the pre-recorded estimations of the physical layer parameters which is impossible to be obtained  during the CS stage, as in this stage, there is exactly no wireless connection between the UE and any LBS.
To combat with the SS spoofing attack, in this letter, we propose a location-based physical layer approach.
To the best of our knowledge, no existing literature is specifically focused on this issue.

Our proposed schemes require that the UE knows its own position and the locations of the LBSs. Note that this requirement is not hard to satisfy in view of the facts that nowadays, the UE (such as a smartphone) is usually equipped with a positioning module (GPS) so that the UE can know its own position, and
the locations of the LBSs are generally fixed and invariant which can be also known by the UE via an off-line map (GoogleMap in a smartphone).

Our main idea is that with the location information of the UE and LBSs, taking the path loss, the shadowing effect and the small-scale fading into consideration, the average received synchronization signal strength (ARSSS) from LBSs should be within a proper range. On the other hand, a FBS usually transmits with a high power level to attract the UE \cite{M.LichtmanCM2016}. The UE can check the ARSSS and
once the ARSSS exceed its normal range, it is reasonable to suspect that the corresponding SS is transmitted from a FBS. Note that location awareness has been proposed to improve the performance of future networks, e.g., in \cite{M.Koivisto}, but has not been utilized for the UEs to identify the FBS.
The major benefit of the proposed methods in this letter is that once the ARSSSs are obtained, they can make a decision immediately, while the conventional key-based methods require the UE to synchronize itself with the SS possibly transmitted by the FBS.
The contributions of this letter are summarized as follows:
1) based on the location information, we provide two different ARSSS checking criterions, which are referred as suspicious ARSSS region (SAR) criterion  and maximum-likelihood (ML) criterion, respectively;
2) by taking the shadow fading and small-scale fading effects into consideration, the successful cheating rates (SCR), defined as the probability that a given UE will connect to the FBS, are derived for our proposed methods;
3) a cooperative ARSSS checking scheme is proposed to improve the performance when the ARSSS from the FBS approaches to that from the closest LBS.

\section{System Model and Problem Statement}
In this section, we first introduce our system model. Then we evaluate the SCR of a given UE when there is no security mechanism during the CS stage.

A comprehensive system model is given in Fig. \ref{Model}.
We consider that there are $M$ LBSs, referred to as LBS $1$, LBS $2$, $\cdots$, LBS $M$, and a FBS around the considered UE broadcasting mutually orthgonal SSs, denoted by $\mathcal{Z}\triangleq\{\bm{z}_1,\bm{z}_2,\cdots,\bm{z}_{M+1}\}$, to enable the nearby UEs to connect to them. Assume $\bm{z}_{m}\in\mathcal{C}^{\tau}$, $m\in\{1,2,\cdots,M\}$, and $\bm{z}_{M+1}\in\mathcal{C}^{\tau}$ are the SS transmitted by the $m$-th LBS and the FBS, where $\tau$ is the length of the SSs satisfying $\tau\geq M+1$.
For $m=1,2,\cdots,M+1$, we assume that $\|\bm{z}_m\|^2=\tau$.
During the CS stage, the UE first searches the existence of the SSs and obtains the set $\mathcal{Z}$ \footnote{This can be simply realized by matching all the possible SSs with the received signal and check whether the output power exceeds a pre-designed threshold. For simplicity, we assume all the SSs in $\mathcal{Z}$ can be successfully detected in this letter.}.
Then it keeps listening to the SSs for a total of $L$ observing time slots (TSs) and record the ARSSS of each SS. Finally, we assume the UE synchronize itself to the SS with largest ARSSS
\footnote{The 3GPP specification requires the UE to search for the strongest cell except for the some special cases such as when the strongest cell is in ``barred'' status \cite{3GPP}. For simplicity, in this letter, we only consider the situation where the UE is able to connect to the strongest cell to provide a basic understanding of the effect of the FBS attack. Other special cases are more complicated and are left for future research.}.

{
The received signal at the UE during the $l$-th TS is
\begin{align}
\bm{y}\left[l\right]&=\sum_{m=1}^{M+1}\sqrt{P_m\Psi_{m}}d_{m}^{-\alpha/2}h_m\left[l\right]\bm{z}_m +\bm{n}\left[l\right],\label{Simultaneousarrive}
\end{align}
where $\{d_m,h_m[l],\Psi_m, P_m\}$, for $m\in\{1,2,\cdots,M\}$, and $\{d_{M+1},h_{M+1}[l],\Psi_{M+1},P_{M+1}\}$ are the distance, the Rayleigh small scale fading factor during the $l$-th TS, the log-normal shadow fading, and the transmit power of the $m$-th LBS and the FBS,  respectively, $\bm{n}\left[l\right]\sim \mathcal{CN}\left(\bm{0},\sigma^2\bm{I}\right)$ is the received noise, and $\alpha$ is the exponential factor of the path loss.
We assume the log-normal shadow fading factors remain unchanged during the whole $L$ TSs, and they are identically and independently distributed as $\mathcal{N}\left(0, \sigma_\Psi^2\right)$ in decibels, and all the Rayleigh small scale fading factors are identically and independently distributed as $\mathcal{CN}\left(0, \sigma_h^2\right)$.
For simplicity, we assume that the LBSs transmit with the same power $P$, i.e., $P=P_1=P_2=\cdots=P_M$.

Matching $\bm{y}\left[l\right]$ with $\bm{z}_m$, for $m\in\{1,2,\cdots,M+1\}$, the UE obtains the output power, which can be written as
\begin{align}
p_{m}\left[l\right]&\triangleq |\bm{y}\left[l\right]\bm{z}_m^H|^2/\tau= \left|\sqrt{P_m\Psi_{m}d_{m}^{-\alpha}}h_m\left[l\right]+n_m\left[l\right]\right|^2,\nonumber
\end{align}
where $n_m\left[l\right]\triangleq\bm{n}\left[l\right]\bm{z}_m^H/\tau^2\sim\mathcal{CN}\left(0,\sigma^2/\tau\right)$.
{Note that in general, the SSs are designed to be detectable at a low SNR \cite{M.LichtmanCM2016},} and therefore for simplicity, we assume that the noise term $n_m\left[l\right]$ can be neglected.
In fact, by increasing $\tau$, the impact of the noise gets smaller.
Therefore, we have $p_{m}\left[l\right]\approx P_md_{m}^{-\alpha}\Psi_{m}\left|h_m\left[l\right]\right|^2$, for $m\in\{1,2,\cdots,M+1\}$.

The ARSSS of the $m$-th SS, i.e., $\bm{z}_m$, is defined as
\begin{align}
\mathrm{S}_m &\triangleq \frac{1}{L}\sum_{l=1}^{L}\mathrm{lt}\left(p_{m}\left[l\right]\right)=
U_m + \frac{1}{L}\sum_{l=1}^{L}X_m\left[l\right],\label{ARSSSExpression}
\end{align}
where we have $\mathrm{lt}\left(x\right)\triangleq10\log_{10}\left(x\right)$,
$X_m\left[l\right]\triangleq 2\mathrm{lt}\left(\left|h_m\left[l\right]\right|\right)$, and $U_m\triangleq\mathrm{lt}\left(P_m\right) - \alpha\mathrm{lt}\left( d_m\right)+\mathrm{lt}\left( \Psi_m\right)$.
}

We have to emphasize here that at this stage, the UE only obtain the set of SSs from surrounding BSs (both LBSs and FBS), i.e., $\mathcal{Z}$, and the corresponding set of the ARSSSs, denoted by $\mathcal{S}\triangleq\left\{\mathrm{S}_1,\mathrm{S}_2,\cdots,\mathrm{S}_{M+1}\right\}$, but for each received SS, the UE does not know which BS is the signal source.

Based on \eqref{ARSSSExpression}, we can evaluate the SCR when there is no authentication mechanism for the UE to distinguish the FBS from the LBSs during the CS stage.
We denote the SCR as $\mathcal{P}_{S}$, then we have
\begin{align}
\nonumber
\mathcal{P}_{S}&\triangleq\mathcal{P}\left\{\mathrm{S}_{M+1}>\hat{S}\right\} =\int_{-\infty}^{+\infty}f_{M+1}\left(x\right)\prod_{m=1}^MF_m\left(x\right)\mathrm{d}x,
\end{align}
where $\hat{S} \triangleq \max\limits_{1\leq m\leq M}\mathrm{S}_m$, and for $1\leq m\leq M+1$, $f_{m}(x)$ and $F_{m}(x)$ are the probability density function (PDF) and cumulative distribution function (CDF) of $S_m$, respectively.

According to the central-limit theorem, as $L$ becomes large, $S_m$ is asymptotically distributed as $\mathcal{N}\left(u_m,\sigma_S^2\right)$, where $u_m\triangleq \mathrm{lt} (\sigma_h^2P_m) - \alpha\mathrm{lt} \left(d_m\right) - \gamma$ and $\sigma_S^2\triangleq \sigma_\Psi^2 + \frac{1}{L}\sigma_X^2$ with $\gamma$ being the Eular's constant and $\sigma_X^2\triangleq \left(\gamma + \mathrm{lt}\left(\sigma_h^2\right)\right)^2 + \frac{\pi^2}{6}$. Therefore, we have
\begin{align}
\mathcal{P}_{S}\doteq
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{-\frac{t^2}{2}}\prod_{m=1}^M
\Phi\left(t+\frac{u_{M+1}-u_m}{\sigma_S}\right)\mathrm{d}x,\nonumber
\end{align}
where $\Phi(x)$ is the CDF of a standard Gaussian random variable, and ``$\doteq$'' means ``asymptotically equals to'' under the condition that $L\rightarrow+\infty$.
It should be pointed out that, if $u_{M+1}-u_{m}$ is sufficiently large for $m=1,2,\cdots,M$, then $\mathcal{P}_{S}\rightarrow 1$, which means that the UE almost always choose to connect to the FBS.
\begin{remark}
{ In practice, the different SSs from the different BSs usually arrive at the UE in an asynchronous manner.
However, for simplicity of mathematical description, we assumed that the different SSs simultaneously arrive the UE as in \eqref{Simultaneousarrive}.
Note that this simplification does not change the basic process during the CS stage.}
\qed
\end{remark}

\begin{figure}
    \centering
    \includegraphics[width=1.7 in, height = 0.8 in]{FakeBaseStationModel.eps}
    \caption{\small System model with three LBSs and one FBS.}
    \label{Model}
    \vspace{-6mm}
\end{figure}

\section{Identify the FBS Through ARSSS}
In this section, based on the UE's own position and the prior locations information of the LBSs according to an off-line map, we provide two practical methods to combat with the SS spoofing attack at the UE-side during the CS stage.
With the location information, we assume the UE knows $u_m$ for $m=1,2,\cdots,M$ and $\sigma_S^2$.

Once the UE obtains $\mathcal{Z}$ and $\mathcal{S}$, we provide the following two criterions which allow the UE to choose a SS from the observed SS set $\mathcal{Z}$ in a more secure manner. The basic idea is that by checking the relative locations of the UE itself and LBSs in the map, the ARSSS from  LBSs should within a proper range. A significant large ARSSS could be suspected to be from the FBS.
For notational simplicity, in this section, we denote the final SS chosen by the UE as $\bm{z}_{*}$.

\subsection{SAR based ARSSS checking}
As in current cellular networks, the UE always searches the strongest cell, the greedy FBS may transmit with a higher power level than the LBSs to attract the UE \cite{M.LichtmanCM2016}.
Therefore, we define a SAR for the ARSSSs in $\mathcal{S}$, which is denoted by $\mathcal{I}\triangleq\left(\bar{S},+\infty\right)$.
More specifically,
all the elements in $\mathcal{S}$ that are within $\mathcal{I}$ will be suspected to be from the FBS,
and the UE should synchronize itself to the strongest SS in $\mathcal{S}\setminus\mathcal{I}$.
Here, $\bar{S}$ is chosen such that
$\mathcal{P}\left\{ \hat{S} > \bar{S} \right\}\leq\delta$ with $0<\delta<1$ being a small pre-designed value, which is similar to the false alarm rate in hypothesis test theory \cite{M.Barkat}.
Note that we have
$\mathcal{P}\left\{\hat{S}>\bar{S}\right\}\doteq1 - \prod_{m=1}^M \left(1- Q\left(\frac{\bar{S}-u_m}{\sigma_S}\right)\right)$, and the value of $\bar{S}$ can be searched through a bisection method. As a result, in the SAR based method, we have $\bm{z}_{*} = \bm{z}_{\hat{m}}$ with $S_{\hat{m}} =
\mathrm{argmax}_{s\in\mathcal{S}\setminus\mathcal{I}}~s$.

As we have assumed that $\bm{z}_{M+1}$ is transmitted by the FBS, under the proposed strategy, the SCR satisfies
\begin{align}
&\mathcal{P}_{S}=\mathcal{P}\left\{\bm{z}_{*}=\bm{z}_{M+1}\right\}\overset{(a)}{\leq}
\mathcal{P}\left\{\hat{S}<S_{M+1}< \bar{S}\right\} +\delta\nonumber \\
&\doteq
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\frac{\bar{S}-u_{M+1}}{\sigma_S}}e^{-\frac{t^2}{2}}
\prod_{m=1}^M\Phi\left(t+\frac{u_{M+1}-u_m}{\sigma_S}\right)\mathrm{d}t  +\delta,\nonumber
\end{align}
where the inequality in $(a)$ is obtained by assuming that the UE will always connect to the FBS if $\hat{S}>\bar{S}$.
In the following two reasonable scenarios, $\bar{S}$ can be approximated by some simple and computationally efficient expressions:
\subsubsection{when the UE is much closer to one of the LBS}
For example, when the UE is much closer to LBS 1, then we have $u_1\gg u_m$ for $m=2,3,\cdots,M$, and $\hat{S}\approx S_1$. Therefore, we have $\mathcal{P}\left\{\hat{S}>\bar{S}\right\}\approx \mathcal{P}\left\{S_1>\bar{S}\right\}\doteq Q\left(\frac{\bar{S}-u_1}{\sigma_S}\right)$. Accordingly, we can obtain $\bar{S}\approx \sigma_SQ^{-1}\left(\delta\right)+u_1$.

\subsubsection{when the UE is at the coverage edge of several LBSs}
For example, when the UE is at the coverage edge of LBS 1, LBS 2,$\cdots$, and LBS K, then
we have $u_1\approx u_2\approx\cdots\approx u_K\gg u_{j}$, for $K+1\leq j\leq M$. Therefore, we have $\mathcal{P}\left\{\hat{S}>\bar{S}\right\}\approx \mathcal{P}\left\{\max\limits_{1\leq k\leq K} S_k>\bar{S}\right\}\approx 1 - \left(1 - Q\left(\frac{\bar{S}-u_1}{\sigma_S}\right)\right)^K$. Accordingly, we can obtain $\bar{S}\approx \sigma_SQ^{-1}\left(1 - \sqrt[k]{1-\delta}\right)+u_1$.

\subsection{ML based ARSSS checking}
In this subsection, we introduce a ML based method for the UE to choose a proper SS.
Different from the SAR based method where the ARSSSs are compared with a pre-designed threshold, the
ML based method determines a SS by directly checking the likelihood function \cite{M.Barkat}.
More specifically, $\bm{z}_{*}$ satisfies
$\bm{z}_{*} = \bm{z}_{\check{m}}$ with $\check{m}=\mathrm{argmax}_{m}~f_{\mathrm{max}}\left(S_{m}\right)$ and $f_{\mathrm{max}}\left(x\right)=\sum_{k=1}^M f_k\left(x\right) \prod_{m=1,m\neq k}^M F_m\left(x\right)$ being the PDF of $\hat{S}$.
Obviously, the main idea behind the ML based ARSSS checking method is that the UE chooses the SS which is most likely to be a realization of $\hat{S}$.

Under the ML based ARSSS checking strategy, the SCR can be written as
\begin{align}
\mathcal{P}_S &= \mathcal{P}\left\{  f_{\mathrm{max}}\left(S_{M+1}\right) > \max_{1\leq m\leq M} f_{\mathrm{max}}\left(S_m\right)  \right\}
\nonumber \\
&= \int_{-\infty}^{+\infty}f_{M+1}(x)\prod_{m=1}^M \left\{\int_{\Omega\left(x\right)} f_{m}\left(t\right) \mathrm{d}t\right\}
\mathrm{d}x \label{MLCARSSSc}
\end{align}
where $\Omega\left(x\right)\triangleq\left\{t|f_{\mathrm{max}}(t)<f_{\mathrm{max}}(x)\right\}$ .

In general, numerical calculation of \eqref{MLCARSSSc} is involved due to the fact that $\Omega\left(x\right)$ does not have a closed form expression. For a special case when the UE is much closer to one of the LBS, for example LBS 1, we have $f_{\mathrm{max}}(x) \approx f_1(x)\doteq \frac{1}{\sqrt{2\pi}\sigma_S}\exp\left(-\frac{(x-u_1)^2}{2\sigma_S^2}\right)$, and thus
\begin{align}
\Omega\left(x\right)\approx\left\{
\begin{aligned}
\left\{t|t<x~\mathrm{or}~t>2u_1-x\right\},\quad x<u_1\\
\left\{t|t<2u_1-x~\mathrm{or}~t>x\right\},\quad x\geq u_1
\end{aligned}
\right..
\end{align}
\subsection{Numerical example \& Discussion}
In Fig. \ref{PAC}, we plot $\mathcal{P}_S$ versus $u_{M+1}$ when there is no ARSSS checking and when the SAR or ML based schemes are utilized.
As we can see, with the increase of $u_{M+1}$, the UE will connect to the FBS with probability approaching one.
However, with the proposed two ARSSS checking methods, for large $u_{M+1}$, the FBS will be easily distinguished by the UE, and in this cases, $\mathcal{P}_S$ becomes very small.
We note that each of the proposed schemes has its own advantages and disadvantages.
We observe that within a vast range value of $u_{M+1}$, the ML scheme outperforms the SAR scheme in term of suppressing the SCR $\mathcal{P}_S$. This is because the ML based scheme tends to find the SS that is most likely to be from the LBSs.
However, the complexity of the ML scheme is much higher than the SAR scheme. The SAR scheme only needs to compare the largest ARSSS with a pre-given number while the ML requires to calculate $f_{\max}(x)$, which involves exponential and Q functions, for $M+1$ times.
As for the SAR scheme, it associates the UE with the BS who provides the strongest signal strength out of the suspicious power region. As a result, conditioning on the UE will connect to a LBS, the BS selected by the SAR scheme may provide a higher link capacity than that by the ML scheme.

\begin{figure}[t]
  \centering
  \includegraphics[width=3 in]{F1.eps}\\
  \caption{\small $\mathcal{P}_S$ versus $u_{M+1}$, where we set $M=3$, $d_1=80$ (m), $d_2=250$ (m), $d_3=250$ (m), $P=40~$(dBm), $\sigma_h^2=1$, $\sigma_\Psi^2=3$, $L = 10$, $\alpha = 3$, and $\delta=0.01$.}\label{PAC}
  \vspace{-5mm}
\end{figure}

\section{Cooperation aided ARSSS checking}
\label{Sec:PLAScheme}

In this section, we introduce a method to further improve the performance of identifying the FBS by utilizing the cooperative nodes (CNs).
We assume there are several geometrically distributed and friendly cooperative nodes (CNs) around the UE \footnote{For example, the CNs can be other legitimate but idle UEs}. For simplicity, we only consider the cases where the UE is close to one LBS, and the ARSSSs from other LBSs is much lower than the closest one, and thus those LBSs  are neglected. More general scenarios are left for future works.

To facilitate the proposed method, reliable communication links between the UE and the CNs are required. This can be realized by the techniques such as device-to-device communication.

The basic idea here is that except for the UE, the CNs also record the ARSSSs of the their received SSs.
Then, the CNs feed their ARSSSs back to the UE. With these extra information provided by the CNs, the UE can make a more secure decision.

Denote  the PDF of the ARSSS from the LBS to the $i$-th CN as $f_{C,i}(x)$.
According to the central-limit theorem, we have $f_{C,i}(x)\doteq \frac{1}{\sqrt{2\pi}\sigma_S}\exp\left({-\frac{\left(x-u_{C,i}\right)^2}{2\sigma_S^2}}\right)$, where $u_{C,i}$ is the mean value of the ARSSS from the LBS to the $i$-th CN which can be obtained from the location information of the LBS and the $i$-th CN.
During the CS stage, both the UE and the CNs receives two strong SSs. Without loss of generality, the two SSs are denoted by $\bm{z}_1$ and $\bm{z}_2$. For $j\in\{1,2\}$, the ARSSSs of $\bm{z}_j$ at the UE and the $i$-th CN are denoted by $S_{j}$ and $S_{j,i}$, respectively.

With the help of the CNs, we summarize a detailed steps for the UE to distinguish the SS of the LBS as follows:
1) when the UE is much closer to one LBS but observes two strong SSs, i.e., $\bm{z}_1$ and $\bm{z}_2$, the UE broadcasts the index of the two indistinguishable SSs to nearby CNs;
2) The $i$-th CN observes the ARSSSs of these two SSs, and feeds back the values of $f_{C,i}\left(S_{1,i}\right)$ and $f_{C,i}\left(S_{2,i}\right)$;
3) The UE obtains the feedback from the CNs and determines that $\bm{z}_{j^*}$ is the SS from the LBS, where $j^*=\mathrm{argmax}_{j\in\{1,2\}}~f_1\left(Z_{j}\right)\prod_{i}f_{C,i}\left(S_{j,i}\right)$.

In Fig. \ref{PACR}, we evaluate the performance of the proposed CN aided ARSSS checking method in terms of SCR. In our simulation, the UE and the LBS are located at $(0,0)$ and $(0,R_L)$, respectively. The simulation is carried out for $10000$ realizations. For each realization, two CNs are uniformly and randomly generated within $\mathcal{R}\left(r_C\right)$, and a FBS is uniformly and randomly generated within $\mathcal{R}\left(r_O\right)\setminus\mathcal{R}\left(r_I\right)$, where $\mathcal{R}\left(x\right)$ denotes the ball region whose center and radius are $(0,0)$ and $x$, respectively. As we can see from Fig. \ref{PACR}, with the extra information provided by the CNs, the SCR is greatly reduced.
This is mainly due to the fact that with more distributed nodes recording the ARSSS, we obtain a higher resolution when distinguishing the location of the source of the received SS, and with the prior location information of the LBSs, we can identify the FBS in a more reliable manner.
From Fig. \ref{PACR}, we can also observe that different from the cases without ARSSS checking, when the proposed methods are adopted, the SCR reaches the maximum when the FBS uses a moderate transmit power level. This is because in this case, the ARSSSs from the FBS and the LBS are less different, which makes it harder to distinguish between them.

\begin{figure}[t]
  \centering
  \includegraphics[width=3 in]{F2New.eps}\\
  \caption{\small  $\mathcal{P}_S$ versus $P_{M+1}$, $P=40~$(dBm), $\sigma_h^2=1$, $\sigma_\Psi^2=3$, $L = 10$, $\alpha = 3$, $R_L=100~$m, $r_C=50~$m, $r_O=150~$m, $r_I=90~$m and $\delta=0.01$.}\label{PACR}
  \vspace{-5mm}
\end{figure}

\section{Conclusion}
\label{Sec:Conclusion}
In this letter, we proposed two location based schemes to combat the SS spoofing attack, i.e., the SAR and the ML based method.
We showed via numerical results that both the SAR and the ML based method deal well with the SS spoofing attack when the malicious FBS is greedy who adopts a large transmit power.
Besides, we also proposed a cooperation based method to further enhance the performance of combating the SS spoofing attack. It is shown that with the cooperation based method, the SDR can be greatly reduced in a vast range of spoofing power.


TITLE: Find an equation for the largest sphere that is centered at $(2,1,-2)$ and intersects the sphere $x^2 +y^2+z^2=1$
QUESTION [1 upvotes]: So, basically my process so far has been to find the distance between the two centers; that is, the distance between $(0,0,0)$ and $(2,1,-2)$. Using the distance formula, I get that the distance between the two centers is $3$. Then, since the the "unit" sphere has a radius of $1$, I add $1$ to $3$ so that the radius of the circle centered at $(2,1,-2)$ is $4$. I'm not sure if this is the correct way to go about this, since the problem specifically asks for intersection.

REPLY [2 votes]: The equation of the largest circle centered at $(2,1,-2)$ and intersecting $$x^2+y^2+z^2=1$$ is $$(x-2)^2+(y-1)^2+ (z+2)^2= 16.$$
Note that the radius of such a circle is the distance between the two cneters plus the radius of the smaller circle which is  $3+1=4.$


TITLE: Is this an action?
QUESTION [5 upvotes]: Suppose that you have some group $G$ acting on a set $S$ under the action $\cdot: G \curvearrowright S$, and then you have a set of all functions $T = \{ f: S \to \mathbb{C} \}$. I want to define an action $*: G \curvearrowright T$.
Let $\omega_g: s \in S \mapsto g \cdot s$, then I want to define $g * f(x) = f(g \cdot x)$. This seems to be associative:
$a * ( b * f(x)) = a *(f(b \cdot x)) = f(a\cdot (b \cdot x)) = f((ab) \cdot x) = ab * f(x)$
However, when I try to write this action out in terms of $\omega_g$ it gets problematic (I'll denote the action here by $\times$ for clarity)
$a \times ( b \times f(x)) = a \times ( f \circ \omega_b) = (f \circ \omega_b) \circ \omega_a = f \circ \omega_{ba} = ba \times f(x)$
These two actions are obviously not the same, but I can't seem to write my first action in the $\omega_g$ notation.
I am wondering whether the first action is really an action, or whether the notation is hiding something which makes it not an action. Moreover, how can it be written in the $\omega_g$ notation? I can fix the omega notation by mapping to $\omega_{g^{-1}}$, but paradoxically it doesn't seem to be associative if I use my first notation.

REPLY [4 votes]: Your "associative property" is incorrect. In fact, if $(g\ast f)(x):=f(g\cdot x)$ then
$$ (a\ast(b\ast f))(x)=(b\ast f)(a\cdot x)=f(b\cdot (a\cdot x)),$$
which is "backwards." Indeed if you define $h(x)=f(b\cdot x)=(b\ast f)(x)$ then
$$ (a\ast (b\ast f))(x)=(a\ast h)(x)=h(a\cdot x)=f(b\cdot (a\cdot x)). $$
The reason things look backwards is that $(g\ast f)(x):=f(g\cdot x)$ does not define a left action of $G$ on the function space $T$, it defines a right action. It would be better to write $(f\ast g)(x)=f(gx)$.
More often, sources will define a left action of $G$ on $T$ via $(g\ast f)(x):=f(g^{-1}\cdot x)$. This is called the contragredient action. This left-vs-right issue is recurring in some other places too...
Example 1. Consider transformations of graphs of functions. If we begin with a graph of $y=f(x)$ then to move it up or down respectively we graph $y=f(x)+c$ and $y=f(x)-c$. And yet, contrary to intuition, to move the graph to the right by $c$ units (where $c>0$), we get the graph
$$\{(x+c,f(x)):x\in\mathbb{R}\}=\{(x,f(x-c)):x\in\mathbb{R}\}. $$
That is, to move the graph to the right, we graph $y=f(x\color{Red}{\mathbf{-}}c)$ even though on the number line we know that $x\mapsto x+c$ moves right and $x\mapsto x-c$ moves left. This issue confounds many a student of college algebra.
Example 2. Consider the set $[n]=\{1,\cdots,n\}$. Then the function space $X^{[n]}$ may be identified with the set of $n$-tuples with entries in $X$. (Any function $f:[n]\to X$ can be thought of as the tuple $(f(1),\cdots,f(n))$ and vice-versa.) There is an action of the symmetric group of $S_n$ on $[n]$ using permutations, and hence on $n$-tuples. Naturally, we want $\sigma\in S_n$ to permute the coordinates of a tuple the same way it permutes $\{1,\cdots,n\}$. So for instance, we would want $\sigma=(123)$ to send shift the entries of $(x_1,x_2,x_3)$ rightwards, yielding $(x_3,x_1,x_2)$. The problem is that this is not $(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)})$, instead it's $(x_{\sigma^{-1}(1)},x_{\sigma^{-1}(2)},x_{\sigma^{-1}(3)})$.
This example also occurs with $S_n$ acting on the tensor power $V^{\otimes n}$. In my experience I've seen numerous authors of textbooks and lecture notes mistake left and right actions in this context.


text:
\section{Scaled Real Function that Decreases Without Bound}
Tags: Unbounded Mappings

\begin{theorem}
Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let $\lambda \in \R_{\ne 0}$ be a nonzero [[Definition:Constant|constant]].
Then:
For $\lambda > 0$:
:$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$
:$\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = -\infty$
For $\lambda < 0$:
:$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$
:$\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = +\infty$
\end{theorem}

\begin{proof}
Let $\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty$.
From the definition of [[Definition:Infinite Limit at Infinity|infinite limits at infinity]], this means that:
:$\forall M < 0: \exists N > 0: x > N \implies \map f x < M$
Suppose $\lambda > 0$.
Then $M < 0 \iff \lambda^{-1}M < 0$.
Also, $\map f x < \lambda^{-1}M \iff \lambda \map f x < M$
So:
:$\forall M < 0: \exists N > 0: x > N \implies \lambda \map f x < M$
From the definition of [[Definition:Infinite Limit at Infinity|infinite limits at infinity]]:
:$\displaystyle \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$
The proof for $\displaystyle  \lim_{x \mathop \to -\infty} \map f x = -\infty$ is analogous.
Now, suppose $\lambda < 0$.
Then $-\lambda f < 0$, and so $-\lambda f \to -\infty$, from above.
Write $\lambda f = -\paren {-\lambda f}$ and the result follows from [[Negative of Real Function that Decreases Without Bound]].
{{qed}}
\end{proof}

config: wiki


TITLE: How do I figure out how many N-m it takes to move a man on a skateboard?
QUESTION [0 upvotes]: Given that the skateboard is on a flat surface, and let's say that the man weights 150lbs. How would I figure out how many N-m it takes to move that man? Does the wheel size matter?
Ignore friction of wheels and other environmental variables for a second, I am trying to figure out how one calculates how much N-m it takes to move a person? Are there other variables I need to know?
I am trying to ask this question without sounding ignorant, but it is true that I lack knowledge in physics. Please excuse me if I do sound ignorant.

REPLY [1 votes]: A moving skateboarder on a flat slope has only kinetic energy. The kinetic energy of you and your skateboard is (1/2)mv^2 where m is rhe total mass of you and your skateboard. Both you and your skateboard start out with 0J (or 0 Nm) of kinetic energy. This means you have to do work on the skateboard to give it kinetic energy. How much work? Exactly (1/2)mv^2 Joules. Also, everyone who knows things once learnt them for the first time; it's fine if it's your first encounter with this. Anyway, I hope this helped answer your question, and have a nice day.


text:
\section{Integer Divisor Results/One Divides all Integers}
Tags: Divisors

\begin{theorem}
{{begin-eqn}}
{{eqn | l = 1
      | o = \divides
      | r = n
}}
{{eqn | l = -1
      | o = \divides
      | r = n
}}
{{end-eqn}}
\end{theorem}

\begin{proof}
From [[Integers form Integral Domain]], the concept '''[[Definition:Divisor of Integer|divisibility]]''' is fully applicable to the [[Definition:Integer|integers]].
Therefore this result follows directly from [[Unity Divides All Elements]].
{{qed}}
\end{proof}


TITLE: Limit of $\frac{1-2\cos(x)+\cos^2(2x)}{x^2}$
QUESTION [3 upvotes]: I tried to find the value of this limit without L'Hopital , but no luck
$$\lim_{x\to 0}\frac{1-2\cos(x)+\cos^2(2x)}{x^2}$$

REPLY [1 votes]: Why do you say "L'Hopital does not work"?  Differentiating the numerator and denominator separately, we get $\frac{2 sin(x)- 4 sin(2x)cos(2x)}{2x}$.  Since both numerator and denominator still go to 0, do it again:  $\frac{2cos(x)- 8cos^2(2x)+ 8sin^2(2x)}{2}$.  Now we can take the limit, as x goes to 0 by evaluating at x= 0:  $\frac{2(1)- 8+ 0}{2}= -\frac{6}{2}= -3$.


TITLE: Angular momentum of a black hole
QUESTION [2 upvotes]: I recently read this Phys.SE post and, since I didn't know that black holes had a spin, a question came to my mind: how can I calculate the spin velocity of a black hole? Does mass or radius affects it? I googled it but I couldn't understand much, all I found was about Orbital Velocity of a planet...

REPLY [2 votes]: Black holes don't have a "spin velocity", since there is no surface at which to measure the rotation speed. Instead, they are characterised by an angular momentum $J$ and a specific angular momentum $J/M$ (in units where $G=1$, $c=1$).
A non-zero specific angular momentum changes the space-time metric around the black hole - one uses the Kerr metric, rather than the Schwarzschild metric. This in turn changes the dynamics of material orbiting within a few Schwarzschild radii of the event horizon. In particular it alters the radius of the innermost stable circular orbit (ISCO). The dynamics of orbiting material can be measured if it is sufficiently luminous. This is often the case in accreting black holes, where accreting matter is compressed, becomes very hot and therefore emits lots of electromagnetic radiation. Detailed fitting of the profiles of spectral features can be used to estimate $J/M$. An example of this approach can be found in Patrick et al. (2011), where X-ray emission line profiles are used to constrain $J/M$ in the accreting supermassive black holes at the centres of active galactic nuclei. A similar approach has been employed for accreting black holes in stellar binary systems (e.g. Chen et al. 2016).
A second approach, that has become apparent since this question was posted is to look at the gravitational wave signature of merging black hole binaries. The spin of the black hole components imprints a subtle signature on the gravitational wave signals. At present it appears that the data are most sensitive to the relative orientation of the spin axes of the black holes, rather than the magnitude of the individual spins (e.g. Bailes et al. 2017).


TITLE: How do the fundamental relation of thermodynamics and the 2nd law jive with each other?
QUESTION [0 upvotes]: I have never been able to figure out how these work together. The fundamental relation in terms of entropy is $$dS = \frac{1}{T}dU + \frac{P}{T}dV$$ And for an isolated system the second law gives $$dS\ge0$$ But for an isolated system $dU$ and $dV$ are both zero. It therefore seems that $dS$ must always equal zero for an isolated system? But this is of course nonsense.
Example. An isolated box is our system. In the box is an elevated platform and a ball sets on top of the platform. Eventually the ball rolls off the platform and comes to rest in a small recess in the bottom of the box. Clearly the entropy has increased in the box. However, from the external surroundings POV, the box has neither changed its volume nor transferred any heat to the surroundings; it's energy is therefore unchanged and its volume is unchanged. How can $dS\gt0$ with $dU=dV=0$?
The same question applies to the fundamental equation in terms of any of the thermodynamic potentials. The condition for a process to be spontaneous at constant $T$ and $P$ is that $\Delta G\lt0$, however, the fundamental equation is $$dG=-SdT+VdP$$How can $\Delta G=\int dG\lt0$ with $dP=dT=0$?
My only rationalization is that perhaps the fundamental relation only applies to a system that is already in internal equilibrium and concerns changes between two possible system macrostates that are already in internal equilibrium via external influence and does not apply to the change within a system on the approach to internal equilibrium? Is this correct?

REPLY [0 votes]: In the fundamental thermodynamic relation, you need to include all the macroscopic parameters and the associated generalized forces. The general relation is:
$$dU = T dS - \sum_{j}X_j dx_j + \sum_{r}\mu_r dN_r$$
where $X_j$ is the generalized force associated with external parameter $x_j$, and the $\mu_r$ are the chemical potentials associated with the particle numbers $N_r$. The pressure is the generalized force conjugate to the volume of the system. If you only include this term in the fundamental thermodynamic relation, then you are declaring that the system doesn't have any other external parameters that are needed to give an adequate macroscopic description of the system, which is false in case of ball inside the box.
If there is a ball inside the box, then it's position is a parameter that must be included. But then the next question is how to proceed when all the relevant parameters are included. While the thermodynamic variables in it refer to equilibrium conditions, the changes do not need to be the result of slow quasi-static changes during which internal equilibrium is maintained.
What matters is that the system was in equilibrium before the change and after the change of the parameters. Thermodynamic state variables will then have well defined values, the change from going from one to the other is therefore also well defined. So, whether or not the change happened due to  a slow quasi-static process or a rapid non-equilibrium process, doesn't matter.
In case of changes in the Gibbs energy, exactly the same logic applies. Take e.g. mixing salt with water. You then need to include the particles numbers for the number of salt ions that are dissolved in water than the numbers for the numbers in the solid state. If you describe the system with only the temperature and the pressure, you are leaving out the relevant macroscopic parameter that describes the change in the system.


TITLE: Continuity of fractional part function for negative integers
QUESTION [0 upvotes]: Given a function $f(x)=frac(x^2)-(frac(x))^2$
We have to check continuity at $x=-2$
I know that $frac(x)= x-[x]$ where $[x]$ is the greatest integer function less than or equal to x.
Checking continuity -
$f(-2)=0$
$f(-2^+)=frac(4^+)-(frac(-2^+))^2=0$
But according to the textbook, right-hand limit should be 1. And hence the function should be discontinuous. Where am I wrong? I used the same way to check continuity at $x=2$ and it came out to be continuous which is correct according to the textbook.

REPLY [1 votes]: To check continuity you want to evaluate the function close to (say, just above) $-2$. If $x$ is just above $-2$ then $x^2$ is just below $4$. So $\mathrm{frac}(x^2)$ is a number just below $1$.


TITLE: To prove neccesary and sufficient condition for set W to be a subspace
QUESTION [1 upvotes]: The necessary and sufficient condition for a non - empty subset W of a vector space V(F) to be a subspace of V is
$a$,$b$ in F and $\alpha$ , $\beta$ in W  implies a$\alpha$ + b$\beta$ in W
I need to prove the postulates of vector space with this condition .Hints ?
Thanks

REPLY [1 votes]: Hint
The first axiom is to show that if $u,v \in W$, then $u+v \in W$. Since $F$ is a field, therefore $ 1 \in F$. From the property given to you, you can choose $a=b=1$ to get this axiom.
Now try to go with the remaining axioms by making appropriate choices.


TITLE: Are all asymptotic symmetries and their meaning known?
QUESTION [1 upvotes]: Beyond the Standard Model and the General relativity invariant groups, recently we have met (again) the BSM groups of asymptotic symmetries given by the Bondi-Metzner-Sachs (BMS) or the extended BSM groups. As far as I understand, there are at least 3 BSM symmetries:

Supertranslations. Meaning: angle-dependent directional time translations (angle with respect what?I suppose the whole flat spacetime but not sure) so I could understand it as the energy entering into a given direction, I think.
Superrotations. Meaning: angle-dependent directional space translations. I believe it can be understood as the angular momentum entering from a given direction.
SuperLorentz transformations. Meaning: angle-dependent directional boost. Interpretation: centre-of-mass-energy conservation.

Is there any other asymptotic symmetry out there? What is the physical interpretation of these new possible symmetries?
BONUS: are the 3 symmetries above contained in the Diff(S²) group? Can they be generalized to higher dimensions?

REPLY [0 votes]: I believe the answer to this question demands one general definition of what an asymptotic symmetry is. In general, this can be answered employing the Covariant Phase Space formalism. References are https://arxiv.org/abs/1801.07064 and https://arxiv.org/abs/2009.14334. I'll provide a summary here. Suppose we have a theory in which we have fields $\Phi_i$ possibly including the metric tensor. Let $\mathbf{L}$ be its Lagrangian form. The Covariant Phase Space prescription tells us that starting from a generic variation of $\mathbf{L}$ it can be put in the form $$\delta{\mathbf{L}}=E[\Phi_i]\delta \Phi_i+d{\pmb\Theta}[\Phi_i,\delta\Phi_i]\tag{1}$$
which defines a pre-sympletic potential density ${\pmb \Theta}$. Then, taking one anti-symmetric variation we define the pre-sympletic density $${\pmb \omega}[\Phi_i,\delta_1\Phi_i,\delta_2\Phi_i]=\delta_1{\pmb\Theta}[\Phi_i,\delta_2\Phi_i]-\delta_2{\pmb\Theta}[\Phi_i,\delta_1\Phi_i].\tag{2}$$
From ${\pmb \omega}$ we may construct the pre-sympletic form ${\pmb \Omega}$
$${\pmb\Omega}[\Phi_i,\delta_1\Phi_i,\delta_2\Phi_i]=\int_\Sigma {\pmb \omega}[\Phi_i,\delta_1\Phi_i,\delta_2\Phi_i]\tag{3}.$$
Now suppose the theory under consideration has gauge/local symmetries. Let $\delta_1\Phi_i=\delta_\varepsilon\Phi_i$ be such a gauge transformation and $\delta_2\Phi_i=\delta\Phi_i$ a generic variation. From standard Hamiltonian mechanics, we would expect $${\pmb\Omega}[\Phi_i,\delta_\varepsilon\Phi_i,\delta\Phi_i]=-\delta Q_\varepsilon[\Phi_i]\tag{4}$$
where $Q_\varepsilon[\Phi_i]$ is the charge associated to such symmetry, so one is motivated to study (4) for such local symmetries. What one finds is that as a consequence of Noether's second theorem, valid for local symmetries, (4) is actually one codimension two integral over $\partial \Sigma$. One then finds the following possibilities:

For some set of $\varepsilon$ that integral vanishes. This means from (4) that $\delta_\varepsilon$ is a degenerate vector field for ${\pmb \Omega}$ thereby making the phase space ill-defined. These are therefore not true symmetries, these are the gauge transformations which are mere redundancies and must be gauge-fixed.

For some set of $\varepsilon$ one finds $Q_\varepsilon\neq 0$ as a codimension two integral over $\partial\Sigma$. These are the asymptotic symmetries.

For some set of $\varepsilon$ one does not actually find (4). Instead of finding $-\delta Q_\varepsilon[\Phi_i]$ one finds and additional term which is not $\delta$ of something. This gives one non-integrable charge.

The subject of your question is (2). This is the definition of an asymptotic symmetry: a gauge transformation with non-trivial action at the boundary of spacetime, so that the charge computed from (4) is non-zero and therefore generates the symmetry in a classical phase space through the Poisson brackets defined by ${\pmb \Omega}$.
So it is important to understand that your question "is there any other asymptotic symmetry out there?" on the outset has the issue that this question is theory dependent. What is the Lagrangian? What are the boundary conditions? What is the background? All these things have impact on what asymptotic symmetries you will find.
Now some examples, in Einstein gravity in asymptotically flat spacetimes one finds the extended BMS group with supertranslations and superrotations. On the other hand, for Einstein gravity in asymptotically AdS$_4$ spacetimes with Dirichlet boundary conditions one finds instead the AdS isometry group while with Neumann boundary conditions the asymptotic symmetry group is empty. There is even the possibility of adopting a certain mixed boundary condition which gives you $\mathbb{R}\times{\cal A}$ where $\mathbb{R}$ are time translations and ${\cal A}$ is a group of two-dimensional area preserving diffeomorphisms, see https://arxiv.org/abs/1905.00971 for details. In asymptotically AdS$_3$ spacetimes one finds a Virasoro algebra. These are just Einstein gravity examples. One also may study asymptotic symmetries in gauge theories with generic gauge group, which you can found in https://arxiv.org/abs/2009.14334. There one finds for example Large Gauge Transformations.
Finally, about ${\rm Diff}(S^2)$ on asymptotically flat spacetimes and its relation to the extended BMS group. Lorentz transformations and their Virasoro extension to superrotations indeed comprise a subset of ${\rm Diff}(S^2)$. Supertranslations have a different nature because they are actually shifts along the generators of ${\cal I}^\pm$ shifting $u\to u+f(z,\bar z)$ and $v\to v+f(z,\bar z)$.
Regarding the full ${\rm Diff}(S^2)$ as an asymptotic symmetry, there are some authors who have argued in the past that the superrotations should be extended to the whole ${\rm Diff}(S^2)$, but there has been one recent argument against that constructed from the celestial holography point of view. See https://arxiv.org/abs/2208.13304 for the details of that.


TITLE: Prove a set is countable
QUESTION [0 upvotes]: Prove the set $A = \{n+m\sqrt{2} | n,m\in\mathbb{Z}\}$ is countable

So I guess I need to find an injection for $A$ into a countable set, but I don't seem to get anywhere. Any hints?

REPLY [1 votes]: Define $f\colon A\to\mathbb{Z}\times\mathbb{Z}$ by $f(n+m\sqrt{2})=(n,m)$. This is one possible injection, and, since $\mathbb{Z}\times\mathbb{Z}$ is countable, it follows that $A$ is countable.