\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \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 $\Pi=\{(x,t)\in\mathbb{R}^2:00\}$. 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. $$\label{eq:0} \begin{gathered} a(x)=a_r(x)+\sum_{i=1}^md_{1i}\delta^{(m_i)}(x-x_i)\quad \text{for some }d_{1i}\in\mathbb{R},\; m_i\in\mathbb{N}_0, x_i\in(0,L), \\ b(x)=b_r(x)+\sum_{k=1}^sd_{2i}\delta^{(n_k)}(x-x_k)\quad \text{for some }d_{2i}\in\mathbb{R},\; n_k\in\mathbb{N}_0, x_k\in(0,L), \\ c(t)=c_r(t)+\sum_{j=1}^qd_{3i} \delta^{(l_j)}(t-t_j)\quad \text{for some } d_{3i}\in\mathbb{R},\; l_j\in\mathbb{N}_0, t_j\in(0,\infty). \end{gathered}$$ 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 $$\label{eq:M} \begin{gathered} (\partial_t + \partial_x)u = -p(x)u \\ u|_{t=0} = u_0(x)\\ u|_{x=0} = \int_{0}^{L}b(x)u\,dx. \end{gathered}$$ 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 $x_1^*0$ 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^*0,t>0\}. $$We are now prepared to state the existence result. \begin{theorem}\label{thm:exist} \begin{enumerate} \item Let {\rm (A1)--(A4)} hold. Then there exists a \mathcal{D}'(\Omega)-solution u to the problem \eqref{eq:51}--\eqref{eq:53} in the sense of Definition \ref{defn:Omega} such that $$\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_+'.}$$ \item Let {\rm (A1)--(A4)} hold. Then there exists a \mathcal{D}'(\Pi)-solution to the problem \eqref{eq:51}--\eqref{eq:53} in the sense of Definition \ref{defn:Pi}. \end{enumerate} \end{theorem} Given a domain G, set$$ \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_+). $$\end{proposition} This proposition is a straightforward consequence of Definitions \ref{defn:Omega} and \ref{defn:D+}. Since \Pi\subset\Omega_+, it makes sense to state the uniqueness result in \mathcal{D}_+'(\Omega). Write $$\label{eq:S} S(x,t) =\exp\Big\{\int_{\theta(x,t)}^tp(\tau+x-t,\tau)\,d\tau \Big\},$$ where \theta(x,t)=(t-x)H(t-x) with H(z) denoting the Heaviside function. We write \hat S for the function S given by (\ref{eq:S}), where p is replaced by -p. \begin{theorem}\label{thm:uniq} \begin{enumerate} \item Let {\rm (A1)--(A4)} hold. Then a \mathcal{D}_+'(\Omega)-solution to the problem \eqref{eq:51}--\eqref{eq:53} is unique. \item Let {\rm (A1)--(A4)} hold. Then a \mathcal{D}'(\Pi)-solution to the problem \eqref{eq:51}--\eqref{eq:53} is unique. \end{enumerate} \end{theorem} From the construction of a \mathcal{D}'(\Omega)-solution presented in the proof of Theorem \ref{thm:exist} we will see that in general there appear new singularities stronger than the initial singularities. In other words, the singular order (cf. \cite[\S 13]{Shilow}) of the distributional solution grows in time. We state this result in the following theorem. \begin{theorem}\label{thm:order} \begin{enumerate} \item Let u be the \mathcal{D}'(\Pi)-solution to the problem \eqref{eq:51}--\eqref{eq:53}, where n\ge 1. Then for each i\ge 1 there exist k>i and n'\ge 1 such that the singular order of u is equal to n' in a neighborhood of x=t-t_i^* and the singular order of u is equal to n'+n in a neighborhood of x=t-t_k^*. \item If n=j=m=0, then the singular order of u on \Pi is equal to 1. \end{enumerate} \end{theorem} We now start with the proof of Theorem \ref{thm:exist} which will take Sections 4--9. By our construction of the set I, we have t_1\in\{t_1^*,t_2^*,\dots\}. Let, say, t_1=t_2^* (for any other t_1=t_i^* the proof is virtually the same, see Footnotes 1 and 2). It is sufficient to solve the problem in the domain$$ \Omega^T= \{(x,t)\in\Omega: t-T0. 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 $$\Omega_0=\{(x,t)\in\Omega^T: t0. Indeed, the second summand is smooth due to a_r^{(i)}(0)=0 for 0\le i\le n (see (A1)). The third summand is smooth due to b_r^{(i)}(L)=0 for 0\le i\le m (see (A2)). \subsection*{Further plan of the solution construction} We split \Omega^T\setminus\overline{\Omega_0} into subdomains$$ \Omega_i=\big\{(x,t)\in\Omega^T\setminus\overline{\Omega_0}: t-t_i^*0, set $$\Omega_1^s=\{(x,t)\in\Omega_1: 00. We apply the contraction principle to (\ref{eq:57_2}). Comparing the difference of two continuous functions u and \tilde u satisfying (\ref{eq:57_2}), we have$$ |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 \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} 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_1t-t_1^*-\varepsilon\right\} for a fixed\varepsilon>0$such that$t_1^*-\varepsilon>0$,$t_1^*+\varepsilont_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 $$\label{eq:**} u(x,t)=S(x,t)c_r(t-x)v_r(t-x)+S_1(x,t)+S(x,t)Q(t-x),$$ 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$\Omega_2$: $$\label{eq:59_1} u(x,t)=S(x,t)u(0,t-x)+S_1(x,t).$$ 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$t_1^*0$in the sense of Definition \ref{defn:Omega} with$\Omega$replaced by$\Omega^T$and$\Pi$replaced by$\Pi^T=\{(x,t)\in\Pi: ti$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: t0\}=\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 $$\label{eq:phi0} \varphi_0(\xi)= -\frac{\hat S_1(\xi+T(\psi),T(\psi))} {\hat S(\xi+T(\psi),T(\psi))}.$$ 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 $$\label{eq:t2} \langle v(t)-\tilde v(t),\psi(t)\rangle=0 \quad\text{for all } \psi(t)\in\mathcal{D}(0,t_1^*),$$ 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 $00\} \ne\emptyset$. 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 $$\label{eq:phi1} \varphi_1(\xi)= -\frac{\hat S_1(T(\psi)-\xi,T(\psi))} {\hat S(T(\psi)-\xi,T(\psi))}$$ 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 $$\label{eq:36_0} \langle v-\tilde v,\psi(t)\rangle=0 \quad\text{for all } \psi(t)\in\mathcal{D}(\varepsilon i,\varepsilon i+2\varepsilon),$$ for each $0\le i\le t_1^*/\varepsilon-2$, where $\varepsilon>0$ is chosen so that $t_1^*/\varepsilon$ is an integer and $$\label{eq:br} b_r(x)=0\quad \text{for }x\in[0,2\varepsilon].$$ 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+}: $$\label{eq:u01} \begin{gathered} u=u_0+u_1\quad \text{in } \mathcal{D}'(\Omega_+),\\ \tilde u=\tilde u_0+\tilde u_1\quad \text{in } \mathcal{D}'(\Omega_+), \end{gathered}$$ 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 $$\label{eq:t3} \langle v-\tilde v,\psi(t)\rangle=\langle u_1-\tilde u_1,b_r(x)\psi(t)\rangle \quad\text{for all } \psi(t)\in\mathcal{D}(0,4\varepsilon).$$ According to Item 1 of Definition \ref{defn:D+}, \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} 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 $$\label{eq:I} \langle(u_0-\tilde u_0)(x,t)b(x),1(x)\otimes\psi(t)\rangle= \langle J_0(t)-\tilde J_0(t),\psi(t)\rangle,$$ 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 $0t-\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-\varepsilon0$ 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 $t_i^*-\varepsilon_i>t_{i-1}^*$, $t_i^*+\varepsilon_it-t_1^*-\varepsilon_1\}$. 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. \begin{thebibliography}{10} \bibitem{bu-ia} S. Busenberg, M. Iannelli, Separable models in age-dependent population dynamics, J. Math. 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