\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 132, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/132\hfil A distributional solution]
{A distributional solution to a hyperbolic problem arising
in population dynamics}

\author[I. Kmit\hfil EJDE-2007/132\hfilneg]
{Irina Kmit}

\address{Irina Kmit \newline
Institute for Applied Problems of Mechanics and Mathematics \\
Ukrainian Academy of Sciences \\
Naukova St. 3b \\
79060 Lviv, Ukraine}
\email{kmit@informatik.hu-berlin.de}

\thanks{Submitted September 29, 2006. Published October 9, 2007.}
\thanks{This work was done while visiting the Institut
f\"ur Mathematik, Universit\"at
Wien. \hfill\break\indent Supported by an \"OAD grant.}
\subjclass[2000]{35L50, 35B65, 35Q80, 58J47}
\keywords{Population dynamics; hyperbolic equation; integral condition;
\hfill\break\indent singular data; distributional solution}

\begin{abstract}
 We consider a generalization of the Lotka-McKendrick problem
 describing the dynamics of an age-structured population with
 time-dependent vital rates.
 The generalization consists in allowing the initial and the
 boundary conditions to be derivatives of the Dirac measure.
 We construct a unique $\mathcal{D}'$-solution in the framework of intrinsic
 multiplication of distributions. We also investigate the regularity
 of this solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}\label{sec:intr}

We consider a non-classical  hyperbolic
problem with integral boundary condition
\begin{gather}
(\partial_t  + \partial_x) u = p(x,t)u + g(x,t), \quad (x,t)\in \Pi
  \label{eq:1}\\
u|_{t=0} = a(x), \quad x\in [0,L) \label{eq:2} \\
u|_{x=0} =  c(t)\int_{0}^{L}b(x)u\,dx,
  \quad  t\in[0,\infty) \label{eq:3} \,,
\end{gather}
where
$\Pi=\{(x,t)\in\mathbb{R}^2:0<x<L,t>0\}$.
 From the point of view of applications, (\ref{eq:1})--(\ref{eq:3})
describes the dynamics of an age-structured population (see i.e.
\cite{bu-ia,cushing,metz,Sanchez,webb}). There $u$ denotes the
distribution of individuals having age $x>0$ at time $t>0$, $a(x)$
is the initial distribution, $-p(x,t)$  denotes the mortality
rate, $b(x)$ denotes the age-dependent fertility rate, $c(t)$ is
the specific fertility rate of females, $g(x,t)$ is the
distribution of migrants, $L$ is the maximum age attained by
individuals. Furthermore, $b(x)=0$ on $[0,L]\setminus[L_1,L_2]$,
where $[L_1,L_2]\subset [0,L]$ is the fertility period of females.
The evolution of $u$ without diffusion is governed by
(\ref{eq:1})--(\ref{eq:3}). The system (\ref{eq:1})--(\ref{eq:3})
is a continuous model of a discrete structure. As in many problems
of such a kind, it is natural to consider singular initial and
boundary data.  We focus on the case when these  data have
singular support in finitely many points, i.e.
\begin{equation}\label{eq:0}
\begin{gathered}
a(x)=a_r(x)+\sum_{i=1}^md_{1i}\delta^{(m_i)}(x-x_i)\quad
\text{for some }d_{1i}\in\mathbb{R},\; m_i\in\mathbb{N}_0, x_i\in(0,L), \\
b(x)=b_r(x)+\sum_{k=1}^sd_{2i}\delta^{(n_k)}(x-x_k)\quad
\text{for some }d_{2i}\in\mathbb{R},\; n_k\in\mathbb{N}_0, x_k\in(0,L), \\
c(t)=c_r(t)+\sum_{j=1}^qd_{3i}
\delta^{(l_j)}(t-t_j)\quad \text{for some } d_{3i}\in\mathbb{R},\;
l_j\in\mathbb{N}_0, t_j\in(0,\infty).
\end{gathered}
\end{equation}
The data of the Dirac measure type enable us to model the
point-concentration of various demographic parameters.

The problem under consideration is of interest from both biological
and mathematical points of view.



\subsection*{Biological motivation}
A basic model describing the evolution of an age-struc\-tured population
is given by the Lotka-McKendrick system
\begin{equation}\label{eq:M}
\begin{gathered}
(\partial_t  + \partial_x)u = -p(x)u   \\
u|_{t=0} = u_0(x)\\
u|_{x=0} =  \int_{0}^{L}b(x)u\,dx.
\end{gathered}
 \end{equation}
This differential equation describes the aging of the population.
While the integral $\int_{\alpha_1}^{\alpha_2}u(x,t)\,dx$ gives the number
of individuals, at time $t$, having age $x$ in the range
$\alpha_1\le x\le\alpha_2$. Thus, the third equation is responsible
for newborns, entering the population at age zero.


A biological generalization of (\ref{eq:M}) to
(\ref{eq:1})--(\ref{eq:3}) consists in  allowing the  fertility
and mortality rates to depend on $t$ (see e.g.
\cite{inaba1,kim,lopez}). In reality the vital rates are never
time-homogeneous and adapt to the changing social and
technological environment. Introducing  $\delta$-distributional
data in (\ref{eq:2}) and (\ref{eq:3}) also has a biological
meaning (see \cite{metz}).

In demography, $c(t)$ is the total fertility rate of the
population at time $t$, in other words, the average number of
childbirths per female during her reproductive period. On one
side, the results presented in the paper could shed a new light on
the so-called $c$-control problems when one wants to control the
population only through changing $c(t)$. Chinese scientists used
discrete models to provide mathematical background for the unicity
child policy ($c$-control problem) in the People's Republic of
China \cite{song,songyu,yu}. Continuous models in the context of
the $c$--control problem were considered in \cite{inaba}. In
contrast to the aforementioned papers, the presence of strongly
singular data in (\ref{eq:2}) and (\ref{eq:3}) allows one to
combine the continuity of the model with the discreteness of the
real evolutionary process. Occurrence of strong singularities in
$c(x)$ can be motivated by synchronized and concentrated
reproduction of the species. This also allows one  to introduce
statistical data in (\ref{eq:1})--(\ref{eq:3}) and perhaps makes
our model competitive with discrete-time and discrete-age models
\cite{caswell}.

Introducing strong singularities in the model could have another
interpretation: such singularities can be produced by a
linearization of nonlinear problems with discontinuous data. Thus
this opens a space for interesting nonlinear consequences.


\subsection*{Mathematical motivation}
We consider our paper as a further step in the study  of
generalized solutions to initial-boundary hyperbolic problems in
two variables.

Since the singularities given on $\partial\Pi$  expand inside
$\Pi$ along characteristic curves of the equation (\ref{eq:1}), a
solution preserves at least the same order of regularity as it has
on   $\partial\Pi$. This causes multiplication of distributions
under the integral sign in (\ref{eq:3}). In spite of this
complication, we find distributional solutions of
(\ref{eq:1})--(\ref{eq:3}). In parallel, we study propagation,
interaction and  creation of new singularities for the problem
(\ref{eq:1})--(\ref{eq:3}).


Semilinear hyperbolic initial-boundary value problems with
distributional data were studied, among others, in
\cite{13,Kmit,KmitHorm}.   There also appears a complication with
multiplication of distributions that is caused by nonlinear
right-hand sides of the differential equations and also  by
boundary conditions that are nonlinear (with bounded nonlinearity)
in \cite{13}, nonseparable in \cite{KmitHorm}, and integral in
\cite{Kmit}. To overcome this complication, the authors use the
framework of  {\it delta waves} (see \cite{RauchReed87}). In other
words, they find solutions by regularizing all singular data,
solving the regularized system and then passing to a weak limit in
the obtained sequential solution.


Boundary and initial-boundary value problems for a linear second
order hyperbolic equation \cite{89} and general strictly
hyperbolic systems in the Leray-Volevich sense \cite{91} are
studied in a complete scale of {\it Sobolev type spaces} depending
on parameters $s$ and $\tau$, where $s$ characterizes the
smoothness of a solution in all variables and $\tau$ characterizes
additional smoothness in the tangential variables. Sobolev-type a
priori estimates are obtained and, based on them, existence and
uniqueness results in Sobolev spaces are proved.


 In contrast to the aforementioned papers
we here treat {\it integral} boundary conditions and show that the
problem (\ref{eq:1})--(\ref{eq:3}) is solvable in the {\it
distributional} sense. We construct a unique distributional
solution by means of multiplication of distributions in the sense
of H\"ormander \cite{Horm}.


We show that the boundary condition (\ref{eq:3}) causes anomalous
singularities at the time when singular characteristics and
vertical singular lines arising from the data of (\ref{eq:3})
intersect. In the case that the singular part of $b(x)$ is a sum
of derivatives of the Dirac measure, the solution becomes more
singular. In the case that the initial and the boundary data are
Dirac measures, the solution preserves the same order of
regularity. A similar phenomenon was shown in \cite{12}  for a
semilinear hyperbolic Cauchy   problem with strongly singular
initial data, where interaction of singularities was caused by
nonlinearity of the equations. Anomalous singularities were
considered also in \cite{RauchReed81} and \cite{Ober86}, where
propagation of singularities for, respectively, initial and
initial-boundary semilinear hyperbolic problems were studied.
There it was proved that, if the initial data have, at worst, jump
discontinuities, then the singularities at the common point of
singular characteristics of the differential equations  are
weaker. Furthermore, if the boundary data are regular enough, then
reflected singularities cannot be stronger than the corresponding
incoming singularities. It turns out \cite{Elt,LavLyu} that in
some cases of nonseparable boundary conditions the solution
becomes more regular in time, namely, for $C^1$-initial data it
becomes $k$-times continuously differentiable for any desired
$k\in \mathbb{N}_0$ in a finite time.

\subsection*{Organization of the paper} Section 2 contains some
basic facts from the theory of distributions. In Section 3 we
describe our problem in detail and state our results. Sections
4--9 present successive steps of construction of a distributional
solution to the problem. In particular, the integral boundary
condition is treated in Section 5. In parallel we analyze the
regularity of the solution. The uniqueness is proved in Section
10.

\section{Background}

For convenience of the reader we here recall the relevant material
from \cite{Joshi,Horm1,Horm,Shilow} without proofs. Throughout the
paper we will denote by $\langle\cdot,\cdot\rangle:
\mathcal{D}'\times\mathcal{D}\to\mathbb{R}$ the dual pairing on the space $\mathcal{D}$ of
$C^\infty$-functions having compact support.

\begin{definition}[{\cite[2.5 ]{Horm1}}] \label{def1} \rm
A distribution $u\in\mathcal{D}'(\mathbb{R}^2)$ is {\em microlocally smooth}
at $(x,t,\xi,\eta)$  $((\xi,\eta)\ne 0)$ if
the following condition holds: If  $u$ is localized
about $(x,t)$ by $\varphi\in \mathcal{D}(\mathbb{R}^2)$ with
$\varphi\equiv 1$ in a
neighborhood of $(x,t),$ then the Fourier transform of $\varphi u$
is rapidly decreasing in an open cone about $(\xi,\eta).$
The {\em wave front set} of $u$, $\mathop{\rm WF}(u)$, is the complement in $\mathbb{R}^4$
of the set of microlocally smooth points.
\end{definition}

\begin{proposition}[{\cite[8.1.5]{Horm}}] \label{prop2}
Let $u\in\mathcal{D}'(\mathbb{R}^2)$ and $P(x,D)$ be a linear differential operator with smooth
coefficients. Then
$$
\mathop{\rm WF}(Pu)\subset \mathop{\rm WF}(u).
$$
\end{proposition}

\begin{definition}[{\cite[6.1.2]{Horm}}] \label{def3} \rm
Let $X,Y\subset\mathbb{R}^2$ be open sets and $u\in\mathcal{D}'(Y)$. Let
$f:X\to Y$ be a smooth invertible map such that its derivative
is surjective. Then the {\em pullback of $u$ by $f$}, $f^*u$, is a unique
continuous linear map: $\mathcal{D}'(Y)\to\mathcal{D}'(X)$ such that for all
$\varphi\in\mathcal{D}(Y)$
$$
\langle f^*u,\varphi\rangle=\langle u,|J(f^{-1})|(\varphi\circ f^{-1})\rangle,
$$
where $J(f^{-1})$ is the Jacobian matrix of $f^{-1}$.
\end{definition}

\begin{theorem}[{\cite[8.2.7]{Horm}}] \label{thm4}
Let $X$ be a manifold and $Y$ a submanifold with normal bundle denoted
by $N(Y)$. For every distribution $u$ in $X$ with $\mathop{\rm WF}(u)$ disjoint from
$N(Y)$, the restriction $u|_Y$ of $u$ to $Y$ is a well-defined distribution on
$Y$ that is the pullback by the inclusion $Y\hookrightarrow X.$
\end{theorem}

\begin{theorem}[{\cite[5.1.1]{Horm}}] \label{thm5}
For any distributions $u\in\mathcal{D}'(X_1)$ and $v\in\mathcal{D}'(X_2)$
there exists  a unique
distribution $w\in\mathcal{D}'(X_1\times X_2)$ such that
\begin{gather*}
\langle w,\varphi_1\otimes\varphi_2\rangle
=\langle u,\varphi_1\rangle\langle v,\varphi_2\rangle,\quad
\varphi_i\in \mathcal{D}(X_i), \\
\langle w,\varphi\rangle=\langle u,\langle v,\varphi(x_1,x_2)\rangle\rangle
=\langle v,\langle u,\varphi(x_1,x_2)\rangle\rangle,\quad
\varphi\in \mathcal{D}(X_1\times X_2).
\end{gather*}
Here $u$ acts on $\varphi(x_1,x_2)$
as on a function of $x_1$ and  $v$ acts on $\varphi(x_1,x_2)$
as on a function of $x_2$.
\end{theorem}

The distribution $w$ as in the above theorem is called the {\em tensor product}
of $u$ and $v$, and denoted by $w=u\otimes v$.

\begin{theorem}[{\cite[11.2.2]{Joshi}}] \label{thm6}
Let $X,Y$ be open sets in $\mathbb{R}^2$ and let $f:X\to Y$ be a diffeomorphism.
If $u\in\mathcal{D}'(Y)$, then $f^*u$, the pull-back of $u$, is well defined,
and we have
$$
\mathop{\rm WF}(f^*(u))=\{(x,df_x^t\eta):(f(x),\eta)\in\mathop{\rm WF}(u)\}.
$$
\end{theorem}

\begin{theorem}[{\cite[8.2.10]{Horm}}] \label{thm:times}
If $v,w\in\mathcal{D}'(X)$, then the product $v\cdot w$
is well defined as the pullback of the tensor product $v\otimes w$ by the
diagonal map   $\delta:\mathbb{R}\to \mathbb{R}\times \mathbb{R}$ unless
$(x,t,\xi,\eta)\in \mathop{\rm WF}(v)$ and $(x,t,-\xi,-\eta)\in \mathop{\rm WF}(w)$
for some $(x,t,\xi,\eta)$.
\end{theorem}

\begin{theorem}[{\cite[8.6]{Shilow}}] \label{thm:shilow}
If a distribution $u$ is identically equal to 0 on each of the
domains $G_i$, $i\ge 1$, then
$u$ is identically equal to 0 on $G=\bigcup_{i\ge 1}G_i$.
\end{theorem}

\section{Statement of the results}\label{sec:multipl}

For simplicity of technicalities we assume that both
the initial and the boundary data have singular
supports  at a single point and are Dirac measures or
 derivatives of the Dirac measure.
This causes no loss of generality for the problem if the singular
parts of the initial and the boundary data are finite sums of the
Dirac  measures and derivatives thereof, i.e. they are of the form
(\ref{eq:0}). Specifically, we consider the following system
\begin{gather}
(\partial_t  + \partial_x) u = p(x,t)u+g(x,t),\quad (x,t)\in\Pi
  \label{eq:51}\\
u|_{t=0}  = a_r(x)+\delta^{(m)}(x-x_1^*), \quad x\in [0,L) \label{eq:52} \\
u|_{x=0}  = (c_r(t)+\delta^{(j)}(t-t_1))\int_{0}^{L}
(b_r(x)+\delta^{(n)}(x-x_1))u\,dx,
    \, t\in[0,\infty),\label{eq:53}
\end{gather}
where $x_1>0,x_1^*>0,t_1>0$, and $m,j,n\in\mathbb{N}_0$.
Without loss of generality we can assume that  $x_1^*<x_1$.
We introduce the following assumptions:
\begin{itemize}

\item[(A1)] $a_r^{(i)}(0)=0,  c_r^{(i)}(0)=0$ for all $i\in\mathbb{N}_0$.

\item[(A2)] $b_r^{(i)}(L)=0$ for all $i\in\mathbb{N}_0$ and there exists
$\varepsilon>0$ such that  $b_r(x)=0$ for
$x\in[0,\varepsilon]$.

\item[(A3)] The functions  $p$ and $g$ are  smooth in
$\mathbb{R}^2$, $a_r$ is smooth on $[0,L)$,  $b_r$ is  smooth on $[0,L]$,
and $c_r$ is  smooth on $[0,\infty)$.

\end{itemize}
Note that (A1) ensures an arbitrary order compatibility
between (\ref{eq:52}) and \eqref{eq:53}. (A2) is
 not particularly restrictive from the practical point of
view, since $[0,L]$ covers the fertility period of females.

All characteristics of the differential equation (\ref{eq:1}) are
solutions to the following initial value problem for ordinary
differential equation
$$
\frac{dx}{dt}=1,\quad x(t_0)=x_0,\quad\text{where } (x_0,t_0)\in\mathbb{R}^2,
$$
and therefore are given by the formula $x=t+x_0-t_0$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1}
\end{center}
\caption{Singular characteristics $\chi_n$ and $\xi_n$ in
the case $t_1=t_2^*$.}
\end{figure}

\begin{definition} \label{def9} \rm
Define $\chi_n$ and $\xi_n$, subsets
of $\mathbb{R}^2$, inductively:
\begin{itemize}
\item
$\chi_0$ is the characteristic passing through the point $(x_1^*,0)$ and
$\xi_0$ is the characteristic passing through the point $(0,t_1)$.

\item Let $n\ge 1$. Then
$\chi_n$ is the characteristic passing through the point $(0,t)$ such that $(x_1,t)\in\chi_{n-1}$.
Furthermore,
$\xi_n$ is the characteristic passing through the point $(0,t)$ such that $(x_1,t)\in\xi_{n-1}$.

\end{itemize}
Also, we set $I=\bigcup_{n\ge 0}(\chi_n\cup\xi_n)$.
\end{definition}

For characteristics contributing into $I$ denote their
intersection points   with the positive semiaxis $x=0$ by
$t_1^*,t_2^*,\dots$. We assume that $t_j^*<t_{j+1}^*$ for $j\ge 1$
(see Figure 1, where we chose $t_1=t_2^*$).
 The union of all singular characteristics
of the initial problem, as it will be shown,
is included in the set $I$. In fact, we will show that
$\mathop{\rm sign}\mathop{\rm supp} u\subset I$.

\begin{itemize}

\item[(A4)] $(x_1,t_1)\notin\chi_n$ for all $n\ge 0$.

\end{itemize}
This assumption excludes the
situation when three different singularities
intersect at the same point. Without this assumption
the distributional solution does not exist, because there appears a
 multiplication of two Dirac measures at the same point.


Our goal is, using distributional multiplication, to obtain a
distributional solution to \eqref{eq:51}--\eqref{eq:53}. We use
the notion of the so-called ''WF favorable'' product which is due
to L. H\"ormander \cite{Horm} and is in the second level
 of M. Ober\-gug\-genberger's
hierarchy of intrinsic distributional products \cite[p. 69]{11}.

We will actually obtain a distributional solution  in the domain
 $$
 \Omega=\{(x,t)\in\mathbb{R}^2: x<t+L\}.
 $$
This is the  domain of influence of the data
 on the part of the boundary of $\Pi$ where the conditions
 (\ref{eq:52}) and \eqref{eq:53} are given.


 \begin{definition} \label{defn:Omega} \rm
 A distribution $u$ is called a $\mathcal{D}'(\Omega)$-solution to
 the problem \eqref{eq:51}--\eqref{eq:53} if the following conditions are
met.
\begin{enumerate}
\item The equation \eqref{eq:51} is satisfied in $\mathcal{D}'(\Omega)$: for every
 $\varphi\in\mathcal{D}(\Omega)$
 $$
\langle(\partial_t+\partial_x-p(x,t))u,\varphi\rangle=\langle  g(x,t),\varphi\rangle.
 $$
\item $u$ is restrictable to $[0,L)\times\{0\}$
 in the sense of H\"ormander (see Theorem \ref{thm4}) and
 $u|_{t=0} = a_r(x)+\delta^{(m)}(x-x_1^*), \quad x\in [0,L)$.
\item The product of $(b_r(x)+
\delta^{(n)}(x-x_1))\otimes 1(t)$ and $u(x,t)$
 exists in $\mathcal{D}'(\Pi)$ in the sense of H\"ormander (see Theorem \ref{thm:times}).
\item $\int_{0}^{L}\left[\left(b_r(x)+\delta^{(n)}(x-x_1)\right)
\otimes 1(t)\right]u\,dx$  is a distribution
$v\in\mathcal{D}'(\mathbb{R}_+)$ defined by
$$
\langle v,\psi(t)\rangle=\langle[(b_r(x)+
\delta^{(n)}(x-x_1))\otimes 1(t)]u,1(x)\otimes
\psi(t)\rangle,\quad \psi(t)\in\mathcal{D}(\mathbb{R}_+),
$$
where $b_r(x)=0$, $x\notin[0,L]$.
\item $v$ is a smooth function in $t_1$.
\item $u$ is restrictable to  $\{0\}\times[0,\infty)$
 in the sense of H\"ormander (see Theorem \ref{thm4}) and
 $u|_{x=0} = (c_r(t)+\delta^{(j)}(t-t_1))v$,\quad $t\in[0,\infty)$.
\item $\mathop{\rm sign}\mathop{\rm supp} u\subset\Omega\setminus\{(x,t): x=t\}$.
\end{enumerate}
\end{definition}

 Our next objective is to define the solution concept for
 \eqref{eq:51}--\eqref{eq:53} on $\Pi$. It is not so obvious how we should
 define the restriction of $u\in\mathcal{D}'(\Pi)$ to the boundary of
 $\Pi$ so that the initial and the boundary conditions are meaningful.
In this respect let us make the following observation.


 Note that $\overline{\Pi}\setminus\{(L,0)\}\subset\Omega$. Let
 $\Omega_0\subset\Omega$ be a domain  such that
 $\overline{\Pi}\setminus\{(L,0)\}\subset\Omega_0$ and $u$ be a
$\mathcal{D}'(\Omega)$-solution  to the problem
\eqref{eq:51}--\eqref{eq:53} in the sense of Definition
\ref{defn:Omega}. Then $u$ restricted to $\Omega_0$ is a
$\mathcal{D}'(\Omega_0)$-solution to the problem
\eqref{eq:51}--\eqref{eq:53} in the sense of the same definition.
This suggests the following definition.

\begin{definition}\label{defn:Pi} \rm
Let $u$ be a $\mathcal{D}'(\Omega)$-solution to the problem
\eqref{eq:51}--\eqref{eq:53} in the sense of Definition
\ref{defn:Omega}. Then $u$ restricted to $\Pi$ is called a
$\mathcal{D}'(\Pi)$-solution to the problem \eqref{eq:51}--\eqref{eq:53}.
\end{definition}

Set
$$
\Omega_+=\{(x,t)\in\Omega: x>0,t>0\}.
$$
We are now prepared to state the existence result.

\begin{theorem}\label{thm:exist}
\begin{enumerate}
\item Let {\rm (A1)--(A4)} hold. Then there exists a
$\mathcal{D}'(\Omega)$-solution $u$ to the problem
\eqref{eq:51}--\eqref{eq:53} in the sense of Definition
\ref{defn:Omega} such that
\begin{equation}\label{eq:r}
\parbox{97mm}{the restriction of $u$ to
 any domain $\Omega_+'\supset\Omega_+$
such that any characteristic of
\eqref{eq:51} intersects $\partial\Omega_+'$
at a single point does not depend
on the values of the functions
$p$ and $g$
on
$\Omega\setminus\Omega_+'$.}
\end{equation}

\item Let {\rm (A1)--(A4)} hold. Then there exists a
$\mathcal{D}'(\Pi)$-solution to the problem \eqref{eq:51}--\eqref{eq:53} in
the sense of Definition \ref{defn:Pi}.
\end{enumerate}
\end{theorem}

Given a domain $G$, set
$$
\mathcal{D}_+'(G)=\{u\in\mathcal{D}'(G): u=0\text{ whenever $x<0$ or $t<0$}\}.
$$


\begin{definition}\label{defn:D+} \rm
 $u\in\mathcal{D}_+'(\Omega)$ is called a $\mathcal{D}_+'(\Omega)$-solution to the problem
\eqref{eq:51}--\eqref{eq:53} if the following conditions are met.
\begin{enumerate}
\item
Items 3--5 of Definition \ref{defn:Omega} hold.
\item  Equation \eqref{eq:51} is satisfied in $\mathcal{D}_+'(\Omega)$: for every
 $\varphi\in\mathcal{D}(\Omega)$
\begin{align*}
&\langle(\partial_t+\partial_x-p(x,t))u,\varphi\rangle\\
&=\langle g(x,t),\varphi\rangle
+\big\langle(a_r(x)+\delta^{(m)}(x-x_1^*))\otimes\delta(t)\\
&\quad +\delta(x)\otimes[(c_r(t)+
\delta^{(j)}(t-t_1))v],\varphi\big\rangle,
\end{align*}
where $a_r(x)=0$ if $x<0$ and $v(t)=0$ if $t<0$.

\item %3
 $\mathop{\rm sign}\mathop{\rm supp} u\setminus\partial\Omega_+\subset\Omega_+\setminus
\{(x,t): x=t\}$.
\end{enumerate}
\end{definition}


\begin{proposition}\label{prop:p}
Let $u$ be a $\mathcal{D}'(\Omega)$-solution to the problem
\eqref{eq:51}--\eqref{eq:53} in the sense of Definition
\ref{defn:Omega} that satisfies (\ref{eq:r}). Then there exists a
$\mathcal{D}_+'(\Omega)$-solution $\tilde u$ to the problem
\eqref{eq:51}--\eqref{eq:53} in the sense of Definition
\ref{defn:D+} such that
$$
u=\tilde u \quad \text{in } \mathcal{D}'(\Omega_+).
$$
\end{proposition}

This proposition is a straightforward consequence of Definitions
\ref{defn:Omega} and \ref{defn:D+}. Since $\Pi\subset\Omega_+$, it
makes sense  to state the uniqueness result in $\mathcal{D}_+'(\Omega)$.
Write
\begin{equation}\label{eq:S}
S(x,t)
=\exp\Big\{\int_{\theta(x,t)}^tp(\tau+x-t,\tau)\,d\tau \Big\},
\end{equation}
where $\theta(x,t)=(t-x)H(t-x)$ with $H(z)$ denoting the Heaviside function.
We write $\hat S$ for the function $S$
given by (\ref{eq:S}), where $p$ is replaced
by $-p$.

\begin{theorem}\label{thm:uniq}
\begin{enumerate}
\item  Let {\rm (A1)--(A4)} hold. Then
a $\mathcal{D}_+'(\Omega)$-solution
to the problem
\eqref{eq:51}--\eqref{eq:53}
is unique.

\item Let {\rm (A1)--(A4)} hold. Then a $\mathcal{D}'(\Pi)$-solution
to the problem
\eqref{eq:51}--\eqref{eq:53}
is unique.
\end{enumerate}
\end{theorem}

 From the construction of a $\mathcal{D}'(\Omega)$-solution presented in
the proof of Theorem \ref{thm:exist} we will see that in general
there appear new singularities stronger than the initial
singularities. In other words, the singular order (cf. \cite[\S
13]{Shilow}) of the distributional solution grows in time. We
state this result in the following theorem.

\begin{theorem}\label{thm:order}
\begin{enumerate}
\item Let $u$ be the $\mathcal{D}'(\Pi)$-solution
to the problem
\eqref{eq:51}--\eqref{eq:53}, where $n\ge 1$.
Then for each $i\ge 1$ there exist
$k>i$ and $n'\ge 1$  such that
the singular order of $u$ is equal
to $n'$ in a
neighborhood of  $x=t-t_i^*$ and
the singular order of $u$ is equal to
$n'+n$ in a
neighborhood of $x=t-t_k^*$.

\item If $n=j=m=0$, then the singular order of $u$ on
$\Pi$ is equal to 1.
\end{enumerate}
\end{theorem}

We now start with the proof of Theorem \ref{thm:exist} which will
take Sections 4--9. By our construction of the set $I$, we have
$t_1\in\{t_1^*,t_2^*,\dots\}$. Let, say, $t_1=t_2^*$ (for any
other $t_1=t_i^*$ the proof is virtually the same, see Footnotes 1
and 2).
It is sufficient to solve the problem in the domain
$$
\Omega^T= \{(x,t)\in\Omega: t-T<x, -T<t<T\}
$$
 for an arbitrary fixed $T>0$. Observe that $\Omega^T$ is the intersection of the
strip $\mathbb{R}\times(-T,T)$ and the domain of determinacy of \eqref{eq:51}
with respect to the set
$([0,L)\times\{0\})\cup(\{0\}
\times[0,T))$.
Fix $T>0$ and start with a subdomain
$$
\Omega_0=\{(x,t)\in\Omega^T: t<x<t+L\}
$$
(see Figure 1). To abuse notation, we do not indicate the
dependence of $\Omega_0$ on $T$.


\section{The solution on $\Omega_0$}

Observe that $\Omega_0$ is the intersection of the strip
$\mathbb{R}\times(-T,T)$ with the domain of  determinacy of the
problem \eqref{eq:51}--(\ref{eq:52}).
In the case that the initial data are functions, a unique solution to the
problem \eqref{eq:51}--(\ref{eq:52})
on $\Omega_0$
can be written in the form
\begin{equation}\label{eq:54}
u(x,t)=S_1(x,t)+S(x,t)a_r(x-t)+S(x,t)\delta^{(m)}(x-t-x_1^*)
\end{equation}
with the functions $S(x,t)$ given by
(\ref{eq:S}) and
\begin{equation}\label{eq:S1}
\begin{aligned}
S_1(x,t)
&=\exp\Big\{\int_{\theta(x,t)}^tp(\tau+x-t,\tau)\,d\tau\Big\} \\
&\quad \times \int_{\theta(x,t)}^t
\exp\Big\{-\int_{\theta(x,t)}^{\tau}p(\tau_1+x-t,\tau_1)\,d\tau_1
\Big\}g(\tau+x-t,\tau)\,d\tau.
\end{aligned}
\end{equation}
Let $A_i(x,t)=\delta^{(i)}(x)\otimes 1(t)$ and
$B_i(x,t)=1(x)\otimes \delta^{(i)}(t)$ be  the distributions in $\mathbb{R}^2$
that are derivatives of the Dirac measure $\delta^{(i)}(x)$ and
$\delta^{(i)}(t)$ supported along the $t$-axis and the $x$-axis, respectively.
They are defined by the equalities
\begin{gather*}
\langle A_i(x,t),\varphi(x,t)\rangle=(-1)^i\int\varphi_x^{(i)}(0,t)\,dt,
\\
\langle B_i(x,t),\varphi(x,t)\rangle=(-1)^i
\int\varphi_t^{(i)}(x,0)\,dx
\end{gather*}
for all $\varphi\in\mathcal{D}(\mathbb{R}^2)$.
When $i=0$, then we have the Dirac measure supported along the
respective axes.

Let $f$ be the smooth map
$$
f:(x,t)\to (x,x-t-x_1^*).
$$
Then its inverse
$$
f^{-1}:(x,t)\to (x,x-t-x_1^*)
$$
is unique and maps the $x$-axis to the line $t=x-x_1^*$ and
the $t$-axis onto itself.
Moreover,
$$
  f'(x,t)=  \begin{pmatrix}
    1&0\\
    1&-1   \end{pmatrix}.
$$
 For the Jacobian of $f$ we hence have
$J(f)=|f'|=-1\ne 0$
and  $f^*B_m=\delta^{(m)}(x-t-x_1^*)$, the pullback of
$B_m$ by $f$ (see Definition \ref{def3}), is well defined. Therefore the distribution
$\delta^{(m)}(x-t-x_1^*)$ acts on test functions
$\varphi\in\mathcal{D}(\mathbb{R}^2)$ in the following way:
\begin{align*}
\langle\delta^{(m)}(x-t-x_1^*),\varphi(x,t)\rangle
&=\langle f^*B_m,\varphi(x,t)\rangle \\
&=-\langle B_m,\varphi(x,t)\circ f^{-1}(x,t)\rangle \\
&=(-1)^{m+1}\int\partial_t^m\varphi(x,x-t-x_1^*)\big|_{t=0}\,dx\\
&=-\int \partial_t^m\varphi(x,t)|_{t=x-x_1^*}\,dx.
\end{align*}
Hence, similarly to  $B_m$, $f^*B_m$ is the $m$-th derivative of the
Dirac measure supported along the line $t=x-x_1^*$.

 \begin{definition}\label{defn:Omega0} \rm
 A distribution $u$ is called a $\mathcal{D}'(\Omega_0)$-solution to
 the problem \eqref{eq:51},
(\ref{eq:52}) if Items 1 and 2 of Definition \ref{defn:Omega} with
$\Omega$ replaced by $\Omega_0$ hold.
\end{definition}


\begin{lemma}\label{lemma:0} The function
$u(x,t)$ given by the formula (\ref{eq:54}) is a  $\mathcal{D}'(\Omega_0)$-solution
to the problem \eqref{eq:51}--(\ref{eq:52}).
\end{lemma}

\begin{proof}
A straightforward verification shows that the sum of the first two
summands in (\ref{eq:54}) is a smooth  (and, therefore,
distributional) solution to the problem
\eqref{eq:51}--(\ref{eq:52}) with the singular part of the initial
condition (\ref{eq:52})  identically equal to 0. Our  goal is now
to prove that the third summand in (\ref{eq:54}) is a
distributional solution to the homogeneous equation \eqref{eq:51}
with singular initial condition $\delta^{(m)}(x-x_1^*)$. Indeed,
for all $\varphi\in\mathcal{D}(\Omega_0)$, we have
\begin{align*}
&\langle(\partial_t+\partial_x)(S\delta^{(m)}(x-t-x_1^*)),\varphi\rangle \\
&=-\langle S\delta^{(m)}(x-t-x_1^*),\partial_t\varphi+\partial_x\varphi\rangle\\
&=-\langle\delta^{(m)}(x-t-x_1^*),S\partial_t\varphi+S\partial_x\varphi\rangle\\
&=-\langle\delta^{(m)}(x-t-x_1^*),\partial_t(S\varphi)+\partial_x(S\varphi)
  -\partial_tS\varphi-\partial_xS\varphi\rangle.
\end{align*}
Since $w=\delta^{(m)}(x-t-x_1^*)$ is a distribution in $x-t$, this
is a weak solution to the equation $(\partial_t+\partial_x)w=0$. Note that
$S\varphi\in\mathcal{D}(\Omega_0)$. Therefore,
$$
\langle\delta^{(m)}(x-t-x_1^*),\partial_t(S\varphi)+\partial_x(S\varphi)\rangle=0.
$$
By (\ref{eq:S}), we have $\partial_tS+\partial_xS=pS$. The desired assertion
is therewith proved.

It remains to prove that $S(x,t)\delta^{(m)}(x-t-x_1^*)$ can be
restricted to the initial interval $X=[0,L)\times\{0\} $. For this
purpose we use  Theorems \ref{thm4} and \ref{thm6}.
Observe that $f$ restricted to
$\Omega_0$ is a diffeomorphism. We check the condition
\begin{equation}\label{eq:57}
\mathop{\rm WF}(Sf^*B_m)\cap N(X)=\emptyset,
\end{equation}
where the normal bundle $N(X)$ to $X$
is defined by the formula
$$
N(X)=\{(x,t,\xi,\eta): (x,t)\in X, \langle
T_{(x,t)}(X),(\xi,\eta)\rangle=0\}
$$
and $T_{(x,t)}(X)$ is the space of all tangent vectors to $X$ at
$(x,t)$. It is clear that in our case
$$
N(X)=\{(x,0,0,\eta),\eta\ne 0\}.
$$
Let us now look at $\mathop{\rm WF}(Sf^*B_m)$. By Proposition \ref{prop2}, we have
$$
\mathop{\rm WF}(Sf^*B_m)\subset \mathop{\rm WF}(f^*B_m).
$$
By definition,
\begin{equation}\label{eq:fxt}
\mathop{\rm WF}(f^*B_m)=\left\{(x,t,df_x^t\cdot(\xi,\eta)): (f(x,t),\xi,\eta)
\in \mathop{\rm WF}(B_m)\right\}.
\end{equation}
We also have
$$
\mathop{\rm WF}(B_m)\subset \mathop{\rm WF}(B_0)=\{(x,0,0,\eta),\eta\ne 0\}.
$$
It follows that  $f(x,t)=(x,0)$ in (\ref{eq:fxt}) and therefore
$(x,t)=(x,x-x_1^*)$. Furthermore,
$$
df_x^t=  \begin{pmatrix}
    1&1\\
    0&-1
  \end{pmatrix},
\quad
df_x^t\cdot(0,\eta)=  \begin{pmatrix}
   \eta\\ -\eta
  \end{pmatrix}.
$$
As a consequence,
$$
\mathop{\rm WF}(Sf^*B_m)\subset
  \{(x,x-x_1^*,\eta,-\eta), \eta\ne 0\}.
$$
This implies that $S(x,t)\delta^{(m)}(x-t-x_1^*)$ is restrictable to
$X$. Consider the distribution $\delta^{(m)}(x-t-x_1^*)$ to be
smooth in $t$ with distributional values in $x$. Then initial
condition (\ref{eq:57}) follows from (\ref{eq:54}), completing the
proof.
\end{proof}


We have proved that $u$ defined by (\ref{eq:54}) satisfies Items 1
and 2 of Definition \ref{defn:Omega} with $\Omega$ replaced by
$\Omega_0$. Items 4--7 on $\Omega_0$ do not need any proof. Item 3
will be given by Lemma \ref{lemma:item4} in the next section.

\section{Multiplication of distributions under the integral
in (\protect\ref{eq:3})}

In the further sections we will extend
the solution to $ \Omega^T\setminus\overline{\Omega_0}. $ We use
the fact that any $\mathcal{D}'(\Omega)$-solution $u$ to our problem is
representable as
\begin{equation}\label{eq:u}
u(x,t)=u_0(x,t)+u_1(x,t),
\end{equation}
where $u_0=u$ in $\mathcal{D}'(\Omega_0)$,
$u_0$ is identically equal to 0 on
$\overline{\Omega^T}\setminus\Omega_0$,
$u_1=u$ in $\mathcal{D}'(\Omega^T\setminus\overline{\Omega_0})$,  and
$u_1$ is identically equal to 0 on
$\overline{\Omega_0}$.
 Indeed,  if $u$ is a solution, then it is a smooth function
in a neighborhood of the line $x=t$ (see Item 7 of Definition
\ref{defn:Omega}). Given an arbitrary $\varphi\in\mathcal{D}(\Omega^T)$,
consider a  representation $
\varphi(x,t)=\varphi_1(x,t)+\varphi_2(x,t)+\varphi_3(x,t) $ such
that $\varphi_i(x,t)\in\mathcal{D}(\Omega^T)$,
$\mathop{\rm supp}\varphi_1\subset\Omega_0$, $\mathop{\rm supp}\varphi_2\cap\mathop{\rm sign}\mathop{\rm supp}
u=\emptyset$, and
$\mathop{\rm supp}\varphi_3\subset\Omega^T\setminus\overline{\Omega_0}$. We
have
\begin{align*}
\langle u_0+u_1,\varphi\rangle
&=\langle u_0,\varphi_1+\varphi_2\rangle+\langle u_1,\varphi_2+\varphi_3\rangle
\\
&=\langle u,\varphi_1\rangle+\langle u_0,\varphi_2\rangle+
\langle u_1,\varphi_2\rangle+\langle u,\varphi_3\rangle \\
&=\langle u,\varphi_1+\varphi_2+\varphi_3\rangle=\langle u,\varphi\rangle.
\end{align*}
Using (\ref{eq:u}), we rewrite $v(t)$ (see Item 4 of Definition
\ref{defn:Omega}) in the form
$$
v(t)=\int_{0}^Lb(x)u_0(x,t)\,dx+\int_{0}^Lb(x)u_1(x,t)\,dx.
$$
In this section we compute
the integral
\begin{equation}\label{eq:I_0}
J_0(t)=\int_{0}^Lb(x)u_0(x,t)\,dx,\quad 0<t<T,
\end{equation}
that will be used in the construction. We have to tackle
the multiplication of distributions
involved in the integrand. For technical reasons
we extend $a_r(x)$ and $b_r(x)$ over all $\mathbb{R}$ defining them to be $0$
outside $[0,L]$.
By (\ref{eq:54}),
we rewrite (\ref{eq:I_0}) as follows
\begin{align*}
J_0(t)
&=\int_t^Lb_r(x)\left(S_1(x,t)+S(x,t)a_r(x-t)\right)\,dx\\
&\quad + \int_{0}^L
\delta^{(n)}(x-x_1)\left(S_1(x,t)+S(x,t)a_r(x-t)\right)\,dx\\
&\quad +\int_{0}^L b_r(x)S(x,t)\delta^{(m)}(x-t-x_1^*)\,dx\\
&\quad +\int_{0}^L\delta^{(n)}(x-x_1)S(x,t)\delta^{(m)}(x-t-x_1^*)\,dx.
\end{align*}
To evaluate the second and the third integrals, we take a test
function $\psi(t)\in\mathcal{D}(0,T)$ and compute the actions (see
Definition \ref{defn:Omega}, Item 4),
\begin{align*}
&\langle\delta^{(n)}(x-x_1)\left(S_1(x,t)+S(x,t)a_r(x-t)\right),1(x)
\otimes\psi(t)\rangle \\
&=(-1)^n\langle \partial_x^n\left(S_1(x,t)+S(x,t)a_r(x-t)\right)
\big|_{x=x_1},\psi(t)\rangle,
\end{align*}
and
\begin{align*}
&\langle S(x,t)b_r(x) \delta^{(m)}(x-t-x_1^*), 1(x)\otimes\psi(t)\rangle \\
&=(-1)^m\langle \partial_x^m\left(S(x+t+x_1^*,t)b_r(x+t+x_1^*)\right)
\big|_{x=0},\psi(t)\rangle.
\end{align*}
To evaluate the last integral in the expression for $J_0(t)$ we need
 the following fact.

\begin{lemma}\label{lemma:vw}
The product of two distributions $v=\delta^{(n)}(x-x_1)\otimes
1(t)$ and $w=\delta^{(m)}(x-t-x_1^*)$ exists in the sense of
H\"ormander (see Theorem \ref{thm:times}).
\end{lemma}

\begin{proof}
 Recall that
\begin{gather*}
\mathop{\rm WF}(v)=\{(x_1,t,\xi_1,0),\xi_1\ne 0\}, \\
\mathop{\rm WF}(w)\subset\{(x,x-x_1^*,\xi_2,-\xi_2),\xi_2\ne 0\}.
\end{gather*}
Thus all the conditions of Theorem \ref{thm:times} are true and
the lemma follows.
\end{proof}

We have proved the following lemma.

\begin{lemma}\label{lemma:item4}
The distribution $u$ defined by (\ref{eq:54}) satisfies Item 3 of
Definition \ref{defn:Omega} with $\Pi$ replaced by
$\Pi\cap\Omega_0$.
\end{lemma}


Turning back to computing the last integral in $J_0(t)$,
consider the map
$$
H:(x,t)\to(x-x_1,x-t-x_1^*)
$$
and the inverse map
$$
H^{-1}:(x,t)\to(x+x_1,x-t+x_1-x_1^*).
$$
Define $H^*A_n=\delta^{(n)}(x-x_1)\otimes 1(t)$ and
$H^*B_m=\delta^{(m)}(x-t-x_1^*)$. Let us compute the actions of
$H^*A_n$ and
$H^*B_m$ on a test function
$\varphi\in\mathcal{D}(\mathbb{R}^2)$ explicitly,
\begin{align*}
\langle H^*A_n,\varphi(x,t)\rangle
&=\langle A_n,\varphi(x+x_1,x-t+x_1-x_1^*)\rangle \\
&=\langle\delta^{(n)}(x),\int \varphi(x+x_1,x-t+x_1-x_1^*)\,dt\rangle\\
&=\langle\delta^{(n)}(x),\int \varphi(x+x_1,\tau)\,d\tau\rangle\\
&=(-1)^n\int \varphi_x^{(n)}(x_1,\tau)\,d\tau
\end{align*}
and similarly with $H^*B_m$.

We are now in a position to compute the product of two distributions
$\delta^{(n)}(x-x_1)$ and
$\delta^{(m)}(x-t-x_1^*)$: For any $\varphi\in\mathcal{D}(\mathbb{R}^2)$ we have
\begin{align*}
&\langle S(x,t)\delta^{(n)}(x-x_1)\delta^{(m)}(x-t-x_1^*),\varphi(x,t)\rangle\\
&=\langle H^*A_nH^*B_m,S(x,t)\varphi(x,t)\rangle\\
&=\langle H^*(A_nB_m),S(x,t)\varphi(x,t)\rangle\\
&=\langle A_nB_m,(S\varphi)(x+x_1,x-t+x_1-x_1^*)\rangle\\
&=\langle\delta^{(n)}(x)\otimes\delta^{(m)}(t),(S\varphi)(x+x_1,x-t+x_1-x_1^*)\rangle\\
&=(-1)^{n+m}\partial_x^n\partial_t^m(S\varphi)(x+x_1,x-t+x_1-x_1^*)\big|_{x=0,t=0}\\
&=\sum_{j=0}^n \sum_{i=0}^{n+m}F_{ji}(x,t)\partial_x^j\partial_t^i\varphi(x+x_1,t+x_1-x_1^*)\big|_{x=0,t=0}\\
&=\sum_{j=0}^n \sum_{i=0}^{n+m}F_{ji}(0,0)\partial_x^j\partial_t^i\varphi(x_1,t_1^*)\\
&=\sum_{j=0}^n \sum_{i=0}^{n+m}(-1)^{j+i}F_{ji}(0,0)\langle\delta^{(j)}(x-x_1)\otimes\delta^{(i)}(t-t_1^*),
\varphi(x,t)\rangle.
\end{align*}
Here $F_{ji}(x,t)$ are known smooth functions of  $S$ and of all
its derivatives up to the order $n+m$.
Hence, for all $\psi(t)\in\mathcal{D}(0,T)$ we get
\begin{align*}
&\big\langle\int_0^L\delta^{(n)}(x-x_1)S(x,t)\delta^{(m)}(x-t-x_1^*)\,dx,\psi(t)
\big\rangle\\
&=\sum_{j=0}^n \sum_{i=0}^{n+m}(-1)^{j+i}F_{ji}(0,0)
\big\langle\int_0^L\delta^{(j)}(x-x_1)\otimes\delta^{(i)}(t-t_1^*)\,dx,\psi(t)
\big\rangle\\
&=\sum_{j=0}^n \sum_{i=0}^{n+m}(-1)^{j+i}F_{ji}(0,0)
\langle\delta^{(j)}(x-x_1)\otimes\delta^{(i)}(t-t_1^*),1(x)\otimes\psi(t)\rangle\\
&=\sum_{i=0}^{n+m}(-1)^{i}F_{0i}(0,0)\langle\delta^{(i)}(t-t_1^*),\psi(t)\rangle.
\end{align*}
As a consequence,
\begin{equation}\label{eq:57_1}
\begin{aligned}
J_0(t)&=\int_{t}^Lb_r(x)(S(x,t)a_r(x-t)+S_1(x,t))\,dx \\
&\quad +(-1)^n\partial_x^m\left(S(x,t)a_r(x-t)
+S_1(x,t)\right)\big|_{x=x_1}\\
&\quad +(-1)^m
\partial_x^m\left(S(x+t+x_1^*,t)b_r(x+t+x_1^*)\right)
\big|_{x=0} \\
&\quad +\sum_{i=0}^{n+m}(-1)^{i}F_{0i}(0,0)\delta^{(i)}(t-t_1^*).
\end{aligned}
\end{equation}
Observe that the first three summands in (\ref{eq:57_1}) are
smooth for $t>0$. Indeed, the second summand is smooth due to
$a_r^{(i)}(0)=0$ for $0\le i\le n$ (see (A1)). The third
summand is smooth due to $b_r^{(i)}(L)=0$ for $0\le i\le m$ (see
(A2)).

\subsection*{Further plan of the solution construction}
We  split $\Omega^T\setminus\overline{\Omega_0}$ into subdomains
$$
\Omega_i=\big\{(x,t)\in\Omega^T\setminus\overline{\Omega_0}:
t-t_i^*<x<t-t_{i-1}^*\big\}
$$
(see Figure 1) and construct the solution separately in each
$\Omega_i$ and in a neighborhood of each border between $\Omega_i$
and $\Omega_{i+1}$. Here $t_0^*=0$, $1\le i\le k(T)$, where $k(T)$
is defined by  inequalities $t_{k(T)}^*<T$ and
$t_{k(T)+1}^*\ge T$. The finiteness of $k(T)$  is obvious.

\section{Existence of the smooth solution on $\Omega_1$}

\begin{lemma}\label{lemma:11}
There exists a  smooth solution to the problem
\eqref{eq:51}--\eqref{eq:53} on $\Omega_1$.
\end{lemma}

\begin{proof}
Under the assumption that $x_1^*<x_1$, we have $t_1^*<L$. Hence
$(x_1,t_1^*)\in\Omega_0$.
 Therefore any  solution which
is given by (\ref{eq:54}) on $\Omega_0$, is smooth on $\Omega_1$,
and has the property given by Item 7 of Definition
\ref{defn:Omega}, satisfies the Volterra integral equation of the
second kind
\begin{equation}\label{eq:57_2}
u(x,t)=S_3(x,t)+S_2(x,t)\int_0^{t-x}b_r(\xi)u(\xi,t-x)\,d\xi,
\end{equation}
where
$S_2(x,t)=S(x,t)c_r(t-x)$ and
$$
S_3(x,t)=S_2(x,t)J_0(t-x)+S_1(x,t)
$$
are known by  (\ref{eq:57_1}).
The smoothness of $J_0(t-x)$  at every point $(x,t)\in\Omega_1$ follows from
the facts that
$t-x<t_1^*$ and that $J_0(t)$ restricted to the
interval $(0,t_1^*)$ is smooth. Therefore $S_2$ and $S_3$ are smooth.

The lemma will follow from two claims. Given $s>0$, set
$$
\Omega_1^s=\{(x,t)\in\Omega_1: 0<t<s\}.
$$

\subsection*{Claim 1:}
 Given $m\in\mathbb{N}_0$, there exists a unique solution
$u\in C^m(\overline{\Omega_1^{s_m}})$ to the problem
\eqref{eq:51}--\eqref{eq:53}
for some $s_m>0$. We apply the contraction principle to (\ref{eq:57_2}).
Comparing the difference of two continuous functions $u$ and
$\tilde u$ satisfying (\ref{eq:57_2}), we have
$$
|u-\tilde u|\le s_0q
\max_{(x,t)\in\overline{\Omega_1^{s_0}}}|u-\tilde u|,
$$
where
$$
q=\max_{(x,t)\in\overline{\Omega_1}}|S|
\max_{t\in[0,t_1^*]}|c_r|
\max_{x\in[0,L]}|b_r|.
$$
Choosing $s_0<1/q$, we obtain the contraction property for the operator
defined by the right-hand side of (\ref{eq:57_2}).
The claim for $m=0$ follows.

Our next concern is the existence and uniqueness of a
$C^1(\overline{\Omega_1^{s_1}})$-solution for some $s_1$. Let us consider
the problem
\begin{equation} \label{eq:ux}
\begin{aligned}
\partial_xu(x,t)&=\partial_xS_3(x,t)+\partial_xS_2(x,t)\int_0^{t-x}
b_r(\xi)u(\xi,\theta(x,t))\,d\xi \\
&\quad -b_r(t-x)u(t-x,t-x)-S_2(x,t)\int_0^{t-x}
b_r(\xi)(\partial_tu)(\xi,t-x)\,d\xi.
\end{aligned}
\end{equation}
 From \eqref{eq:51} we have $\partial_tu=p(x,t)u+g(x,t)-\partial_xu$.
We  choose an arbitrary $s_1\le s_0$.
Since $u$ is a known $C(\Omega_1^{s_1})$-function,
(\ref{eq:ux})  on $\overline{\Omega_1^{s_1}}$ is a Volterra integral
equation of the second kind with respect
to $\partial_xu$. Assuming in addition to the condition $s_1\le s_0$ that
$s_1<q$,
we obtain the contraction property for (\ref{eq:ux}). On the account of
\eqref{eq:51}, the claim for $m=1$
follows.

Proceeding further by induction and using in parallel
\eqref{eq:51}, (\ref{eq:57_2}), and their suitable differentiations, we
complete the proof of the claim.

\subsection*{Claim 2:} In the domain
$\Omega_1^{t_1^*}$ there exists a unique smooth solution
 to the problem \eqref{eq:51}--\eqref{eq:53}.
Given $m\in\mathbb{N}_0$, we prove that there exists a unique
$u\in C^m(\Omega_1^{t_1^*})$
in at most $\lceil t_1^*/s_m\rceil$ steps by iterating the local existence
and uniqueness result in domains
$$
\Omega_1^{ks_m}\setminus\overline{\Omega_1^{(k-1)s_m}},\quad
1\le k\le\lceil T/s_m\rceil.
$$
In particular, for $m=0$ in the $k$-th step of the proof  we have
\begin{equation}\label{eq:uuu}
\begin{aligned}
u(x,t)&=S_3(x,t)+ S_2(x,t)\int_0^{t-x-(k-1)s_m}b_r(\xi)
u(\xi,t-x)\,d\xi \\
&\quad +S_2(x,t)\int_{t-x-(k-1)s_m}^{t-x} b_r(\xi)
u(\xi,t-x)\,d\xi
\end{aligned}
\end{equation}
on $\{(x,t)\in\Omega_1^{ks_m}: x\le t-(k-1)s_m\}$,
 and
\begin{equation}\label{eq:k}
u(x,t)=S(x,t)u(0,t-x)+S_1(x,t)\,\,\text{on}\,\,
\{(x,t)\in\Omega_1: x\ge t-(k-1)s_m\}.
\end{equation}
As in the latter formula $t-x\le(k-1)s_m$,
the function $u$ defined by
(\ref{eq:k}) is smooth and
 known from the previous steps. This implies that
  the last summand in (\ref{eq:uuu})  is known and smooth. Hence
(\ref{eq:uuu}) is a Volterra integral equation of the second kind.
Applying now the argument used to prove Claim 1, we obtain the
existence and uniqueness of a continuous solution $u$ to
(\ref{eq:uuu}) on
$\Omega_1^{ks_m}\setminus\overline{\Omega_1^{(k-1)s_m}}$. Since
$k$ is an arbitrary integer in the range $1\le k\le\lceil
T/s_m\rceil$, we have $u\in C(\Omega_1^{t_1^*})$.
Further we similarly proceed with all derivatives of $u$. Claim 2
is therewith proved.

The solution on the whole $\Omega_1$ is now uniquely determined by the
formula
$$
u(x,t)=S(x,t)u(0,t-x)+S_1(x,t),
$$
where   $u(0,t-x)$ is a known smooth
function.
The latter is true due to
$0<t-x<t_1^*$ and Claim 2.
The proof of the lemma is complete.
\end{proof}

 From formulas (\ref{eq:54}) and (\ref{eq:57_2}), Lemma \ref{lemma:11},
and (A1) it follows  that $u$
is smooth in a  neighborhood of the characteristic line $x=t$.
This ensures that $u$ we  construct satisfies Item 7 of Definition
\ref{defn:Omega}.

Under the assumption that $\Omega_2$ is nonempty,
in the next section we give the formula of the solution on
$$
\Omega_{1,\varepsilon}=\Omega_1\cup
\left\{(x,t)\in\overline{\Omega_2}: x>t-t_1^*-\varepsilon\right\}
$$
for a fixed $\varepsilon>0$ such that $t_1^*-\varepsilon>0$,
$t_1^*+\varepsilon<t_2^*$, and
\begin{equation}\label{eq:br2}
 b_r(x)=0,\quad x\in[0,2\varepsilon].
\end{equation}
Such $\varepsilon$ exists by (A2).

\section{The  solution on $\Omega_{1,\varepsilon}$}

Write now
\begin{equation}\label{eq:v}
v(t)=\int_{0}^{L}\left(b_r(x)+\delta^{(n)}(x-x_1)\right)u\,dx=v_r(t)+v_s(t),
\end{equation}
where $v_r(t)$ and $v_s(t)$ are, respectively, the
regular (smooth) and singular parts of $v(t)$. On the account of
(\ref{eq:u}), (\ref{eq:57_1}), (\ref{eq:br2}), and the fact that $x_1^*<x_1$,
we have on $[0,t_1^*+\varepsilon]$:
\begin{equation} \label{eq:vr}
\begin{aligned}
v_r(t)&=\int_{2\varepsilon}^{t}b_r(x)u(x,t)\,dx+
\int_{t}^Lb_r(x)\left(S(x,t)a_r(x-t)+S_1(x,t)\right)\,dx \\
&\quad +(-1)^n\partial_x^n(S(x,t)a_r(x-t)
+S_1(x,t))\big|_{x=x_1} \\
&\quad +(-1)^m\partial_x^m\left(S(x+t+x_1^*,t)b_r(x+t+x_1^*)\right)\big|_{x=0}
\end{aligned}
\end{equation}
and
\begin{equation}\label{eq:vs}
v_s(t)=\sum_{i=0}^{n+m}(-1)^{i}F_{0i}(0,0)\delta^{(i)}(t-t_1^*).
\end{equation}
Note that the first summand in (\ref{eq:vr}) is a known smooth
function. This follows from the inclusion
$[t-t_1^*+\varepsilon,t]\times\{t\}\subset\Omega_1
\cup\{(x,t): x=t\}$, Lemma \ref{lemma:11}, and (A1).


On the account of (\ref{eq:v})--(\ref{eq:vs}) and the fact that
$x_1-x_1^*=t_1^*$, we derive the following formula
for $u(0,t)$
on $(0,t_1^*+\varepsilon)$:
\begin{equation}\label{eq:59}
\begin{aligned}
u(0,t)&=c_r(t)\sum_{i=0}^{n+m}(-1)^{i}F_{0i}(0,0)\delta^{(i)}(t-t_1^*)
+c_r(t)v_r(t)\\
&=\sum_{i=0}^{n+m}E_{i}\delta^{(i)}(t-t_1^*)+c_r(t)v_r(t),
\end{aligned}
\end{equation}
where $E_i$ are constants depending on $F_{0k}(0,0)$ and
$c_r^{(k)}(t_1^*)$ for $0\le k\le i$.
Note that if  $t_1^*=t_1$, then  $x_1-x_1^*>t_1^*$ by (A4). 
This implies
$v(t)=v_r(t)$ on $[0,t_1^*+\varepsilon]$. Thus, Item 6 of
Definition \ref{defn:Omega} for $u$ we  construct is fulfilled.
Furthermore, we have an expression for $u(0,t)$ on
$(0,t_1^*+\varepsilon)$ similar to (\ref{eq:59}), namely,
$u(0,t)=(\delta^{(j)}(t-t_1^*)+c_r(t))
v_r(t)= v_r^{(j)}(t_1^*)\delta^{(j)}(t-t_1^*)+c_r(t)v_r(t)$.

Set
$$
Q(t)=\sum_{i=0}^{n+m}E_{i}\delta^{(i)}(t-t_1^*).
$$


\begin{lemma}\label{lemma:sol}
$u(x,t)$ given by the formula
\begin{equation}\label{eq:**}
u(x,t)=S(x,t)c_r(t-x)v_r(t-x)+S_1(x,t)+S(x,t)Q(t-x),
\end{equation}
where $v_r(t)$ is determined by (\ref{eq:vr}),
is  a $\mathcal{D}'(\Omega)$-solution to the problem \eqref{eq:51}--\eqref{eq:53}
restricted to $\Omega_{1,\varepsilon}$.
\end{lemma}

\begin{proof}
On the account of (\ref{eq:59}) and
the construction of the solution on $\Omega_1$  done in
 Section 6, it is enough to prove that the restriction of $S(x,t)Q(t-x)$
to $Y=\{0\}\times(0,t_1^*+\varepsilon)$
is well defined and that $S(x,t)Q(t-x)$ satisfies \eqref{eq:51}
with $g(x,t)\equiv 0$ on $\Omega_{1,\varepsilon}$
in a distributional sense. The proof of the latter uses the argument as
in the proof of Lemma \ref{lemma:0}. To prove the former claim,
consider the smooth bijective map
$$
\Phi:(x,t)\to (x,t-x-t_1^*)
$$
and its inverse
$$
\Phi^{-1}:(x,t)\to (x,x+t+t_1^*).
$$
 Applying Theorem \ref{thm6}, we have
$$
\mathop{\rm WF}(\Phi^*B_i)\subset \{(0,t+t_1^*,-\eta,
\eta),\eta\ne 0\}.
$$
Furthermore, $N(Y)=\{(0,t,\xi,0)\}$ and therefore
$$
\mathop{\rm WF}(\Phi^*B_i)\cap N(Y)=\emptyset \quad
\text{for all }  0\le i\le n+m.
$$
By Theorem \ref{thm4}, the restriction of $S(x,t)Q(\theta(x,t))$ to $Y$ is
well defined. The lemma is therewith proved.
\end{proof}

\section{Construction of the smooth solution on $\Omega_2$}

To shorten notation, without loss of generality we assume that
$t_2^*\le T$.

\begin{lemma}\label{lemma:12}
There exists a  smooth solution to the problem
\eqref{eq:51}--\eqref{eq:53} on $\Omega_2$.
\end{lemma}

\begin{proof}
 We start from the general formula of a smooth solution on $\Omega_2$:
\begin{equation}\label{eq:59_1}
u(x,t)=S(x,t)u(0,t-x)+S_1(x,t).
\end{equation}
Since $S$ and $S_1$ are smooth, our task is  to prove that
there exists a smooth function identically equal to
$u(0,t-x)$  on $\Omega_2$. Since
$t_1^*<t-x<t_2^*$ if $(x,t)\in\Omega_2$ and $c(t)=c_r(t)$ if
$t\in(t_1^*,t_2^*)$,
it suffices to show the existence of a smooth function
$v_r(t)$ identically equal to $v(t)$
on $(t_1^*,t_2^*)$.
 From the formula (\ref{eq:vs}) for $v_s(t)$ on $(0,t_1^*+\varepsilon)$
it follows that $v(t)=v_r(t)$
if $t\in(t_1^*,t_1^*+\varepsilon)$, where $\varepsilon$ is as in
Section 7 and $v_r(t)$ is known and determined by (\ref{eq:vr}).
To prove the lemma, it is sufficient to show
that there exists a smooth extension of $v_r(t)$ from $(0,t_1^*+\varepsilon)$
to $[t_1^*+\varepsilon,t_2^*)$ such that
$v_r(t)=v(t)$ if $t\in[t_1^*+\varepsilon,t_2^*)$.
By (\ref{eq:**}), such an extension must satisfy the following
integral equation
on $[t_1^*+\varepsilon,t_2^*)$:
\begin{equation}\label{eq:rv}
v_r(t) =\int_0^{t-t_1^*-\varepsilon}b_r(x)S(x,t)c_r(t-x)v_r(t-x)\,dx
+R(t),
\end{equation}
where
\begin{equation}\label{eq:R}
\begin{aligned}
R(t)&=\int_{t-t_1^*-\varepsilon}^{P(t)}b_r(x)S(x,t)c_r(t-x)v_r(t-x)\,dx
+\int_{0}^{P(t)}b_r(x)S_1(x,t)\,dx\\
&\quad +J_0(t)+\int_0^Lb_r(x)S(x,t)Q(t-x)\,dx
\end{aligned}
\end{equation}
and
$$
P(t)=\begin{cases}
t &\text{if }L\le t, \\
 L &\text{if } L \ge t.
\end{cases}
$$
Here $b_r(x)$ is defined to be $0$ outside $[0,L]$, and
$v_r$ in the formula (\ref{eq:R}) is known and defined by (\ref{eq:vr}).
One can easily see that the
first three summands in (\ref{eq:R}) are smooth functions on
$[t_1^*+\varepsilon,t_2^*)$. We now show that the last summand is a
$C^{\infty}[t_1^*+\varepsilon,t_2^*)$-function as well.
Indeed, take $\psi(t)\in\mathcal{D}(t_1^*+\varepsilon/2,t_1^*)$ and compute
\begin{align*}
&\big\langle\int_0^L b_r(x)S(x,t)\delta^{(j)}(t-x-t_1^*)\,dx,\psi(t)\big\rangle \\
&= \langle\delta^{(j)}(t-x-t_1^*),b_r(x)S(x,t)\psi(t)\rangle\\
&=-\langle\delta^{(j)}(x)\otimes 1(t),
b_r(t-x-t_1^*)S(t-x-t_1^*,t)\psi(t)\rangle\\
&=(-1)^{j+1}\langle\partial_x^j\left(b_r(t-x-t_1^*)S(t-x-t_1^*,t)\right)
\big|_{x=0},\psi(t)\rangle.
\end{align*}
The desired assertion follows.
As follows from (\ref{eq:br2}), the functions $v_r(t)$
defined by (\ref{eq:vr}) and
(\ref{eq:rv}) coincide at  $t=t_1^*+\varepsilon$.
The same is true with respect to all the  derivatives of $v_r$.


Our task is therefore reduced to
show that there exists a  $C^{\infty}[t_1^*+\varepsilon,t_2^*)$-function
$v_r(t)$  satisfying (\ref{eq:rv}).
This follows from the fact that (\ref{eq:rv}) is a
 Volterra integral equation of the second kind with respect to $v_r(t)$
(for  details see the proof of Lemma \ref{lemma:11}). The  proof
is complete.
\end{proof}


\section{Completion of the construction}

Continuing our construction in this fashion,
we extend  $u$ over  a
neighborhood of each subsequent border between
$\Omega_{i-1}$ and $\Omega_i$  and over
$\Omega_{i}$ for all $3\le i\le k(T)$.
 Eventually
we construct $u$ on $\Omega^T$ for any $T>0$  in the sense of
Definition \ref{defn:Omega} with $\Omega$ replaced by $\Omega^T$
and $\Pi$ replaced by $\Pi^T=\{(x,t)\in\Pi: t<T\}$. As easily
seen from our construction, the condition (\ref{eq:r}) is
fulfilled with $\Omega_+$ and $\Omega_+'$ replaced by
$\Omega^T\cap\Omega_+$ and $\Omega^T\cap\Omega_+'$, respectively.
Since $T$ is arbitrary, the proof of Item 1 of Theorem
\ref{thm:exist} is complete. On the account of Definition
\ref{defn:Pi} and the definition of the restriction
$u\in\mathcal{D}'(\Omega)$ to a subset of $\Omega$ (see \cite[Section
5]{Horm}), Item 2 of Theorem \ref{thm:exist} is a straightforward
consequence of Item 1.
 Theorem \ref{thm:exist} is therewith proved.

By (\ref{eq:**})
it follows from the  construction, that if the singular part of
$b(x)$ is the derivative of the Dirac measure of order $n$, then
for each $i\ge 1$ there exist $k>i$ and $n'\ge 1$  such that $u$
is the derivative of the Dirac measure of order $n'$ along the
characteristic line $t-t_i^*$ and $u$ is the derivative of the
Dirac measure of order $n'+n$ along the characteristic line
$t-t_k^*$. In contrast, this is not so if singular parts of the
initial and the boundary data are Dirac measures. In the latter
case the solution preserves the same order of regularity in time.
Furthermore, the assumption  $b_r^{(i)}(L)=0$ for all $i\in\mathbb{N}_0$
can be weakened to $b_r(L)=0$. Since $u$ restricted to
$\Pi\setminus I$ is smooth, Theorem \ref{thm:order} follows from
Item 2 of Theorem \ref{thm:uniq}.

\section{Uniqueness of the  solution (Proof of
Theorem \ref{thm:uniq})}

In this section we reuse notation
$\Omega_i$, $i\ge 0$, by setting
\begin{gather*}
\Omega_0=\{(x,t)\in\Omega: t<x<t+L\}, \\
\Omega_i=\{(x,t)\in\Omega: t-t_i^*<x<t-t_{i-1}^*\}, \quad i\ge 1.
\end{gather*}
Recall that $t_0^*=0$.

Without loss of generality, we make the same assumption as in the
proof of Theorem \ref{thm:exist}, namely, that $t_1=t_2^*$. The
proof of Theorem \protect\ref{thm:uniq} is based on five lemmas.

\begin{lemma}\label{lemma:uOmega0}
A $\mathcal{D}_+'(\Omega)$-solution $u$ to the problem \eqref{eq:51}--\eqref{eq:53}
is unique on $\Omega_0$.
\end{lemma}

\begin{proof}
Note that any $\mathcal{D}_+'(\Omega)$-solution $u$ to the problem
\eqref{eq:51}--\eqref{eq:53} on $\Omega_0$
is a $\mathcal{D}_+'(\Omega_0)$-solution to the problem \eqref{eq:51}--(\ref{eq:52}).
Let $u$ and $\tilde u$ be two $\mathcal{D}_+'(\Omega_0)$-solutions
to the problem \eqref{eq:51}--(\ref{eq:52}). Then
\begin{equation}\label{eq:L1}
\langle L(u-\tilde u),\varphi\rangle
=\langle u-\tilde u,L^*\varphi\rangle=0
\quad \text{for all } \varphi\in\mathcal{D}(\Omega_0),
\end{equation}
where
\begin{equation}\label{eq:LL}
  L=\partial_t+\partial_x-p,\quad L^*=-(\partial_t+\partial_x+p).
\end{equation}
Our goal is to show that
\begin{equation}\label{eq:L5}
 \langle u-\tilde u,\psi\rangle=0\quad \text{for all }
\psi\in\mathcal{D}(\Omega_0).
\end{equation}
Using the definition of
$\mathcal{D}_+'(\Omega_0)$ and (\ref{eq:L1}), it is sufficient to  prove that
for every $\psi\in\mathcal{D}(\Omega_0)$
there exists $\varphi\in\mathcal{D}(\Omega_0)$ such that
\begin{equation}\label{eq:L4}
L^*\varphi=\psi\quad \text{on}\quad
\{(x,t)\in\Omega_0: t\ge 0\}.
\end{equation}
Fix $\psi\in\mathcal{D}(\Omega_0)$. If $\mathop{\rm supp}\psi\cap \{(x,t): t>0\}=\emptyset$, then
(\ref{eq:L5}) follows immediately from the definition of $\mathcal{D}_+'(\Omega_0)$.
We therefore assume that $\mathop{\rm supp}\psi\cap \{(x,t): t>0\}\ne\emptyset$.
Consider the problem
\begin{gather*}
\varphi_t+\varphi_x=-p\varphi-\psi,\quad (x,t)\in
\{(x,t)\in\Omega_0: t>0\}, \\
\varphi|_{t=0}=\varphi_0(x),\quad x\in(0,L),
\end{gather*}
where $\varphi_0(x)\in\mathcal{D}(0,L)$ will be specified below.
This problem has a unique smooth solution given by the formula
$$
\varphi(x,t)=\hat S(x,t)\varphi_0(x-t)+\hat S_1(x,t),
$$
where $\hat S_1$ is given by
(\ref{eq:S1}) with $p$ and $g$ replaced by $-p$ and $-\psi$,
respectively.

Fix $T(\psi)>0$ so that
$\mathop{\rm supp}\psi\cap\{(x,t): t\ge T(\psi)\}=\emptyset$
for all $x$ with
$(x,T(\psi))\in\Omega_0$.
Set
$$
\varphi_0(x-T(\psi))=
-\frac{\hat S_1(x,T(\psi))}{\hat S(x,T(\psi))}
$$
for $x$ such that $(x,T(\psi))\in\Omega_0$.
Changing coordinates $x\to\xi=x-T(\psi)$, we obtain
\begin{equation}\label{eq:phi0}
\varphi_0(\xi)= -\frac{\hat S_1(\xi+T(\psi),T(\psi))}
{\hat S(\xi+T(\psi),T(\psi))}.
\end{equation}
We  construct the desired function $\varphi(x,t)$ by the formula
$$
\varphi(x,t)=
\begin{cases}0 &\text{if $(x,t)\in\Omega_0$ and $ t\ge T(\psi)$}, \\
 \hat S(x,t)\varphi_0(x-t)+\hat S_1(x,t)
&\text{if $(x,t)\in\Omega_0$ and $0\le t\le T(\psi)$},\\
 \tilde\varphi(x,t)
&\text{if $(x,t)\in\Omega_0$ and $t\le 0$},
\end{cases}
$$
where $\tilde\varphi(x,t)$ is chosen so that $\varphi\in\mathcal{D}(\Omega_0)$.
The proof is complete.
\end{proof}

\begin{lemma}\label{lemma:uOmega1}
A $\mathcal{D}_+'(\Omega)$-solution to the problem \eqref{eq:51}--\eqref{eq:53}
is unique on $\Omega_1$.
 \end{lemma}

\begin{proof}
Assume that there exist two $\mathcal{D}_+'(\Omega)$-solutions
$u$ and $\tilde u$. We will show that
\begin{equation}\label{eq:t2}
\langle v(t)-\tilde v(t),\psi(t)\rangle=0
\quad\text{for all }
\psi(t)\in\mathcal{D}(0,t_1^*),
\end{equation}
where $v(t)$ is defined by Item 5 of Definition \ref{defn:Omega}
and $\tilde v(t)$ is defined similarly with $u$ replaced by
$\tilde u$.  Postponing the proof, assume that (\ref{eq:t2}) is
true. Taking into account Item 2 of Definition \ref{defn:D+}
and the fact that $c(t)=c_r(t)$ if $0<t<t_1^*$, we have
$$
 \langle L(u-\tilde u),\varphi\rangle=
\langle u-\tilde u,L^*\varphi\rangle=0\quad \text{for all }
 \varphi\in\mathcal{D}(\Omega_1).
$$
Let us prove that
\begin{equation}\label{eq:psi}
 \langle u-\tilde u,\psi\rangle=0\quad \text{for all }
\psi\in\mathcal{D}(\Omega_1).
\end{equation}
Following the argument used in the proof of Lemma
\ref{lemma:uOmega0},
 it is sufficient to show that,
given $\psi\in\mathcal{D}(\Omega_1)$, there exists $\varphi\in\mathcal{D}(\Omega_1)$
such that
$$
L^*\varphi=\psi\,\,\text{on}\,\,
\{(x,t)\in\Omega_1: x\ge 0\}.
$$
We concentrate on the case that $\mathop{\rm supp}\psi\cap\{(x,t): x>0\}
\ne\emptyset$.
Otherwise (\ref{eq:psi}) is immediate
because $u-\tilde u\in\mathcal{D}_+'(\Omega_1)$.
Consider the problem
\begin{gather*}
\varphi_t+\varphi_x=-p\varphi-\psi,\quad (x,t)\in
\{(x,t)\in\Omega_1: x>0\}, \\
\varphi|_{x=0}=\varphi_1(t),\quad t\in(0,t_1^*),
\end{gather*}
where $\varphi_1(t)\in\mathcal{D}(0,t_1^*)$ is a fixed function. Let
$T(\psi)>0$ be the same as in the proof of Lemma
\ref{lemma:uOmega0}. We specify $\varphi_1(\xi)$ by
\begin{equation}\label{eq:phi1}
\varphi_1(\xi)=
-\frac{\hat S_1(T(\psi)-\xi,T(\psi))}
{\hat S(T(\psi)-\xi,T(\psi))}
\end{equation}
and construct the desired $\varphi$ similarly to the construction
of $\varphi$ in the proof of Lemma \ref{lemma:uOmega0}. To finish
the proof of the lemma, it remains to show that
\begin{equation}\label{eq:36_0}
\langle v-\tilde v,\psi(t)\rangle=0
\quad\text{for all }
\psi(t)\in\mathcal{D}(\varepsilon i,\varepsilon i+2\varepsilon),
\end{equation}
for each $0\le i\le t_1^*/\varepsilon-2$, where $\varepsilon>0$
is chosen so that
$t_1^*/\varepsilon$ is an integer and
\begin{equation}\label{eq:br}
b_r(x)=0\quad \text{for }x\in[0,2\varepsilon].
\end{equation}
Such $\varepsilon$ exists by  (A2). We prove (\ref{eq:36_0}) by
induction on $i$.

\noindent\emph{Base case:} (\ref{eq:36_0}) is true for $i=0$.
 We will use
the following representations for $u$ and $\tilde u$ on $\Omega_+$
which  are possible owing to  Item 3 of Definition \ref{defn:D+}:
\begin{equation}\label{eq:u01}
\begin{gathered}
u=u_0+u_1\quad \text{in } \mathcal{D}'(\Omega_+),\\
\tilde u=\tilde u_0+\tilde u_1\quad \text{in } \mathcal{D}'(\Omega_+),
\end{gathered}
\end{equation}
where $u_0=u$ and $\tilde u_0=\tilde u$ in $\mathcal{D}'(\Omega_0\cap\Omega_+)$,
$u_0=\tilde u_0\equiv 0$ on $\left(\Omega\setminus\Omega_0\right)\cap\Omega_+$,
$u_1=u$ and $\tilde u_1=\tilde u$ in
$\mathcal{D}'\left(\left(\Omega\setminus\overline{\Omega_0}\right)\cap\Omega_+\right)$,
$u_1=\tilde u_1\equiv 0$  on $\overline{\Omega_0}\cap\Omega_+$.

We first prove that
\begin{equation}\label{eq:t3}
\langle v-\tilde v,\psi(t)\rangle=\langle u_1-\tilde u_1,b_r(x)\psi(t)\rangle
\quad\text{for all }
\psi(t)\in\mathcal{D}(0,4\varepsilon).
\end{equation}
According to  Item 1 of Definition \ref{defn:D+},
\begin{equation}\label{eq:uu}
\begin{aligned}
\langle v-\tilde v,\psi(t)\rangle
&=\langle \left(u-\tilde u\right)b(x),1(x)\otimes\psi(t)\rangle
\\
&=\langle (u_0-\tilde u_0)b(x),1(x)\otimes\psi(t)\rangle
+\langle (u_1-\tilde u_1)b(x),1(x)\otimes\psi(t)\rangle,
\end{aligned}
\end{equation}
where $b_r(x)=0$, $x\not\in[0,L]$. By Lemma \ref{lemma:uOmega0},
$u_0=\tilde u_0$ in $\mathcal{D}'(\Omega_0\cap\Omega_+)$. Applying in
addition Item 1 of Theorem \ref{thm:exist} and Proposition
\ref{prop:p}, we have
\begin{equation}\label{eq:I}
\langle(u_0-\tilde u_0)(x,t)b(x),1(x)\otimes\psi(t)\rangle=
\langle J_0(t)-\tilde J_0(t),\psi(t)\rangle,
\end{equation}
where $J_0(t)$ is defined by (\ref{eq:I_0}) and $\tilde J_0(t)$ is
defined by (\ref{eq:I_0})  with $u_0$ replaced by $\tilde u_0$.
 From (\ref{eq:57_1}) we have $J_0(t)=\tilde J_0(t)$ for
$0<t<4\varepsilon$. Hence the right-hand side of (\ref{eq:I}) is equal to 0.
On the  account of the inclusions
$\mathop{\rm supp}(u_1-\tilde u_1)\subset\Omega\setminus\Omega_0$ and
$\mathop{\rm supp}\psi(t)\subset[0,2\varepsilon]$, (\ref{eq:uu}) does not depend
on $b(x)$ outside  $[0,2\varepsilon]$.
Since  $x_1^*<x_1$, $b(x)=b_r(x)$ on $[0,2\varepsilon]$. Therefore
(\ref{eq:uu}) implies (\ref{eq:t3}).   The base case now
follows from (\ref{eq:br}).

Assume that (\ref{eq:36_0}) is true for
$i=k-1$, where $k\ge 1$,  and prove that it
is true for $i=k$.


\noindent\emph{Induction step:} (\ref{eq:36_0}) is true for
$i=k$, $k\ge 1$.
The proof is similar to the proof of the base case.
Based on the induction assumption and applying the argument used in the
proof of (\ref{eq:psi}), we obtain
\begin{equation}\label{eq:eq}
u=\tilde u\quad\text{in }\mathcal{D}_+'(G_{k-1}),
\end{equation}
where
$$
G_k=\Omega_1\cap\{(x,t): x>t-\varepsilon k-2\varepsilon\}.
$$
Applying in addition Item 1 of Theorem \ref{thm:exist},
Proposition \ref{prop:p}, and Lemma \ref{lemma:11}, we conclude
that $u$ is smooth on $G_{k-1}\cap\Omega_+$. Owing to
(\ref{eq:eq}) and the latter fact,
 the following  representations for
$u$ and $\tilde u$ on $\Omega_+$ are possible:
\begin{gather*}
u=u_0+u_{k-1}+u_k\quad\text{in }\mathcal{D}'(\Omega_+), \\
\tilde u=u_0+u_{k-1}+\tilde u_k\quad\text{in }\mathcal{D}'(\Omega_+),
\end{gather*}
where $u_0$ is the same as in (\ref{eq:u01}),
$u_{k-1}=u$  in $\mathcal{D}'(G_{k-1}\cap\Omega_+)$,
$u_{k-1}\equiv 0$  on
$\Omega_+\setminus G_{k-1}$, $u_k=u$ and
$\tilde u_k=\tilde u$ in
$\mathcal{D}'\left(\Omega_+\setminus(\overline{G_{k-1}}\cup\overline{\Omega_0})\right)$,
$u_k=\tilde u_k\equiv 0$  on
$\Omega_+\cap(\overline{G_{k-1}}\cup\overline{\Omega_0})$.
Similarly to (\ref{eq:t3}), we derive the equality
$$
\langle v-\tilde v,\psi(t)\rangle=\langle u_k-\tilde u_k,b_r(x)\psi(t)\rangle
\quad\text{for all }
\psi(t)\in\mathcal{D}(\varepsilon k,\varepsilon k+2\varepsilon).
$$
The induction step follows from the
 support properties of $u_k-\tilde u_k$, $\psi(t)$,
and $b_r$ given by (\ref{eq:br}).
The proof is complete.
\end{proof}

Set
$$
\Omega_{0,1}^{\varepsilon}=
\{(x,t)\in\Omega: x-\varepsilon<t<x+\varepsilon)\}.
$$

\begin{lemma}\label{lemma:uOmega0Omega1}
A $\mathcal{D}_+'(\Omega)$-solution to \eqref{eq:51}--\eqref{eq:53}
is unique on $\Omega_{0,1}^{\varepsilon}$ provided $\varepsilon$
is small enough.
 \end{lemma}

\begin{proof}
Let $u$ and $\tilde u$ be two $\mathcal{D}_+'(\Omega)$-solutions to the
problem \eqref{eq:51}--\eqref{eq:53}. Fix $\varepsilon>0$ so that
the condition (\ref{eq:br}) is fulfilled. By Base case in the
proof of Lemma \ref{lemma:uOmega1}, (\ref{eq:36_0}) is true for
$i=0$. Therefore
$$
\langle L(u-\tilde u),\varphi\rangle=\langle u-\tilde u,L^*\varphi\rangle=0\quad
\text{for all }
\varphi\in\mathcal{D}(\Omega_{0,1}^{\varepsilon}).
$$
Our task is to prove (\ref{eq:psi}) with $\Omega_1$ replaced by
$\Omega_{0,1}^{\varepsilon}$. In fact, we prove that,
given $\psi\in\mathcal{D}(\Omega_{0,1}^{\varepsilon})$ with
$\mathop{\rm supp}\psi\cap\{(x,t): x>0\}\ne\emptyset$,
there exists $\varphi\in\mathcal{D}(\Omega_{0,1}^{\varepsilon})$ satisfying the initial
boundary  problem
\begin{gather*}
\varphi_t+\varphi_x=-p\varphi-\psi,\quad
(x,t)\in\Omega_{0,1}^{\varepsilon}\cap\Omega_+,
\\
\varphi|_{t=0}=\varphi_0(x),\quad x\in[0,\varepsilon),
\\
\varphi|_{x=0}=\varphi_1(t),\quad t\in[0,\varepsilon).
\end{gather*}
Here $\varphi_0(x)\in C^{\infty}[0,\varepsilon)$  is a fixed
function identically equal to 0 in a neighborhood of
$\varepsilon$, $\varphi_1(t)\in C^{\infty}[0,\varepsilon)$  is a
fixed function identically equal to 0 in a neighborhood of
$\varepsilon$, and $\varphi_0^{(i)}(0)=\varphi_1^{(i)}(0)$ for all
$i\in\mathbb{N}_0$. We construct $\varphi(x,t)$, combining the
constructions of $\varphi(x,t)$ in the proofs of Lemmas
\ref{lemma:uOmega0} and \ref{lemma:uOmega1}. Thus we fix
$T(\psi)>0$ to be the same as in the proof of Lemma
\ref{lemma:uOmega0} and specify $\varphi_0(x)$ and $\varphi_1(t)$
by (\ref{eq:phi0}) and (\ref{eq:phi1}), respectively. Let
$$
\varphi(x,t)=\begin{cases}
0 &\text{if $(x,t)\in\Omega_{0,1}^{\varepsilon}$ and $t\ge T(\psi)$},\\
\hat S(x,t)\varphi_0(x-t)+\hat S_1(x,t)
&\text{if $(x,t)\in\overline{\Omega_0}\cap\Omega_{0,1}^{\varepsilon}$
 and $0\le t\le T(\psi)$},\\
\hat S(x,t)\varphi_1(t-x)+\hat S_1(x,t)
&\text{if $(x,t)\in\overline{\Omega_1}\cap\Omega_{0,1}^{\varepsilon}$
 and $0\le t\le T(\psi)$},\\
\tilde\varphi(x,t)
&\text{if $(x,t)\in\Omega_{0,1}^{\varepsilon}$ and
$(x\le 0\,\text{or}\,t\le 0)$},
\end{cases}
$$
where $\tilde\varphi(x,t)$ is chosen so that
$\varphi\in\mathcal{D}(\Omega_{0,1}^{\varepsilon})$.
The proof is complete.
\end{proof}

For every $i\ge 1$ fix $\varepsilon_i$  such that
 $t_i^*-\varepsilon_i>t_{i-1}^*$, $t_i^*+\varepsilon_i<t_{i+1}^*$, and
\begin{equation}\label{eq:bri}
b_r(x)=0\quad\text{for }x\in[0,4\varepsilon_i].
\end{equation}
Set
$$
Q_i=\{(x,t): t-t_i^*-\varepsilon_i<x<t-t_i^*+\varepsilon_i\}.
$$

\begin{lemma}\label{lemma:uOmega12}
A $\mathcal{D}_+'(\Omega)$-solution
to  problem \eqref{eq:51}--\eqref{eq:53}
is unique on $Q_1$.
\end{lemma}

\begin{proof}
Assume that there exist two $\mathcal{D}_+'(\Omega)$-solutions
$u$ and $\tilde u$
and show that
\begin{equation}\label{eq:L25}
\langle v-\tilde v,\psi(t)\rangle=0\quad\text{for all }
\psi(t)\in\mathcal{D}(t_1^*-\varepsilon_1,t_1^*+\varepsilon_1).
\end{equation}
By Lemmas \ref{lemma:11} and \ref{lemma:uOmega1}, Item 1 of
Theorem \ref{thm:exist}, and Proposition \ref{prop:p}, any
solution to      \eqref{eq:51}--\eqref{eq:53} restricted to
$\Omega_1$ is smooth. Based on this fact and on Lemmas
\ref{lemma:uOmega0}--\ref{lemma:uOmega0Omega1}, similarly to
(\ref{eq:t3}), we derive the equality
$$
\langle v-\tilde v,\psi(t)\rangle=
\langle u_1-\tilde u_1,b_r(x)\psi(t)\rangle
\quad \text{for all }
\psi(t)\in\mathcal{D}(t_1^*-\varepsilon_1,t_1^*+\varepsilon_1),
$$
where $u_1=u$ and $\tilde u_1=
\tilde u$ in $\mathcal{D}'(G)$,  $u_1$ and $\tilde u_1$ are identically
equal to zero on $\Omega_+\setminus G$. Here
$$
G=\{(x,t)\in\Omega_+: x<t-t_1^*+\varepsilon_1\}.
$$
The equality (\ref{eq:L25}) now
follows
from the support properties of $u_1-\tilde u_1$, $\psi$, and $b_r$
given by (\ref{eq:bri}) for $i=1$.

Note that $c(t)=c_r(t)$ for $t$ in the range
$t_1^*-\varepsilon_1<t<t_1^*+\varepsilon_1$. Applying
(\ref{eq:L25}) and Item 2 of Definition \ref{defn:D+}, we have
\begin{equation}\label{eq:Lu}
L(u-\tilde u)=0\quad\text{in }
\mathcal{D}'(Q_1).
\end{equation}
Note that if $t_1^*=t_1$, then $c(t)=\delta^{(j)}(t-t_1)+c_r(t)$.
By Item 5 of Definition \ref{defn:Omega}, $v-\tilde v$ is smooth
in a neighborhood of $t_1^*$. Combining the latter with
\eqref{eq:L25}, we get \eqref{eq:Lu}.

For the rest of the proof we proceed as in  the proof of
 Lemma \ref{lemma:uOmega1}.
\end{proof}



\begin{lemma}\label{lemma:uOmega2}
A $\mathcal{D}_+'(\Omega)$-solution
to  problem \eqref{eq:51}--\eqref{eq:53}
is unique on $\Omega_2$.
\end{lemma}

\begin{proof}
We follow the proof of Lemma \ref{lemma:uOmega1} with $\Omega_1$
replaced by $\Omega_2$ and with minor changes caused by the fact
that due to
 Lemmas \ref{lemma:12} and \ref{lemma:uOmega12},
$u$ and $\tilde u$ are smooth on
$\Omega_2\cap\Omega_+\cap\{(x,t):
x>t-t_1^*-\varepsilon_1\}$. Hence
(\ref{eq:t2}) is true with $\mathcal{D}(0,t_1^*)$ replaced by
$\mathcal{D}(t_1^*+\varepsilon_1/2,t_2^*)$.
\end{proof}

Continuing in this fashion, we eventually prove the uniqueness
over subsequent $\Omega_i$ and $Q_i$ for any desired  $i\in\mathbb{N}$.
Combining it with Lemmas \ref{lemma:uOmega0} and
\ref{lemma:uOmega0Omega1} and Theorem \ref{thm:shilow}, we obtain
Item 1 of Theorem \ref{thm:uniq}.


Item 2 of Theorem \ref{thm:uniq} is a straightforward consequence
of Item 1 of Theorem \ref{thm:uniq}, Item 2 of Theorem
\ref{thm:exist}, and  Proposition \ref{prop:p}.

\subsection*{Acknowledgments}
I  thank to the members of the DIANA group  for their kind
hospitality during my stay at the Vienna university.
I am grateful to the referee for an important correction, and the
valuable comments and suggestions.

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\end{document}