t-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^*). $$ \begin{lemma}\label{lemma:sol} $u(x,t)$ given by the formula \begin{equation}\label{eq:**} u(x,t)=S(x,t)c_r(t-x)v_r(t-x)+S_1(x,t)+S(x,t)Q(t-x), \end{equation} where $v_r(t)$ is determined by (\ref{eq:vr}), is a $\mathcal{D}'(\Omega)$-solution to the problem \eqref{eq:51}--\eqref{eq:53} restricted to $\Omega_{1,\varepsilon}$. \end{lemma} \begin{proof} On the account of (\ref{eq:59}) and the construction of the solution on $\Omega_1$ done in Section 6, it is enough to prove that the restriction of $S(x,t)Q(t-x)$ to $Y=\{0\}\times(0,t_1^*+\varepsilon)$ is well defined and that $S(x,t)Q(t-x)$ satisfies \eqref{eq:51} with $g(x,t)\equiv 0$ on $\Omega_{1,\varepsilon}$ in a distributional sense. The proof of the latter uses the argument as in the proof of Lemma \ref{lemma:0}. To prove the former claim, consider the smooth bijective map $$ \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^* 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 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 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 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-\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 $$ \Omega_{0,1}^{\varepsilon}= \{(x,t)\in\Omega: x-\varepsilon 0$ so that the condition (\ref{eq:br}) is fulfilled. By Base case in the proof of Lemma \ref{lemma:uOmega1}, (\ref{eq:36_0}) is true for $i=0$. Therefore $$ \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 $t_i^*-\varepsilon_i>t_{i-1}^*$, $t_i^*+\varepsilon_i t-t_1^*-\varepsilon_1\}$. 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. Biol. {\bf 22} (1985), 145--173. \bibitem{caswell} H. Caswell, Matrix Population Models: Construction, Analysis, and Interpretation, 2nd Edition, Sinauer Associates, Sunderland, Massachusetts, 2001. \bibitem{cushing} J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, 1998. \bibitem{Elt} N. A. \"Eltysheva, On qualitative properties of solutions of some hyperbolic system on the plane, Mat. Sb. {\bf 134} (1988), No. 2, 186--209. \bibitem{Joshi} G. Friedlander, M. Joshi, Introduction to the Theory of Distributions, second ed., Cambridge University Press, 1998. \bibitem{Horm1} L. H\"ormander, Fourier integral operators I, Acta. Math. {\bf 127} (1971), 79--183. \bibitem{Horm} L. H\"ormander, The Analysis of Linear Partial Differential Operators, Vol. I., second ed., Springer-Verlag, 1990. \bibitem{inaba} H. Inaba, Strong ergodicity for perturbed dual semigroups and application to age-dependent population dynamics, J. of Math. Anal. and Appl. {\bf 165} (1992), 102--132. \bibitem{inaba1} H. Inaba, Weak ergodicity population evolution process, Math. Biosci. {\bf 96} (1989), 195--219. \bibitem{kim} Y. J. Kim, Dynamics of populations with changing rates: generalization of the stable population theory, Theoret. Population Biol. {\bf 31} (1987), 306--322. \bibitem{Kmit} I. Kmit, Delta waves for a strongly singular initial-boundary hyperbolic problem with integral boundary condition, J. for Analysis and its Applications {\bf 24} (2005), No. 1, 29--74. \bibitem{KmitHorm} I. Kmit, G. H\"ormann, Semilinear hyperbolic systems with singular non-local boundary conditions: reflection of singularities and delta waves, J. for Analysis and its Applications {\bf 20} (2001), No. 3, 637--659. \bibitem{LavLyu} M. M. Lavrent'ev Jr, N. A. Lyul'ko, Increasing smoothness of solutions to some hyperbolic problems, Siberian Math. J. {\bf 38} (1997), No. 1, 92--105. \bibitem{lopez} A. Lopez, Problems in Stable Population Theory, Office of Population Research, Princeton, NJ, 1961. \bibitem{metz} J. A. J. Metz, O. Diekmann, The Dynamics of Physiologically Structured Populations. Springer Lecture Notes in Biomathematics, 1968. \bibitem{11} M. Oberguggenberger, Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Math. 259, Longman Scientific \& Technical, 1992. \bibitem{Ober86} M. Oberguggenberger, Propagation of singularities for semilinear hyperbolic initial-boundary value problems in one space dimension, J. Diff. Eqns. {\bf 61} (1986), 1--39. \bibitem{13} M. Oberguggenberger, and Y. G. Wang, Reflection of delta-waves for nonlinear wave equations in one space variable, Nonlinear Analysis {\bf 22} (1994), No.8, 983--992. \bibitem{RauchReed81} J. Rauch, and M. Reed, Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: Creation and propagation, Comm. math. Phys. {\bf 81} (1981), 203--207. \bibitem{RauchReed87} J. Rauch, M. Reed, Nonlinear superposition and absorption of delta waves in one space dimension, Funct. Anal. {\bf 73} (1987), 152--178. \bibitem{91} Ja. A. Rojtberg, Boundary value and mixed problems for general hyperbolic systems in a complete scale of Sobolev-type spaces, Sov. Math. Dokl. {\bf 43} (1991), No. 3, 788--793; translation from Dokl. Akad. Nauk SSSR {\bf 318} (1991), No. 4, 820--824. \bibitem{89} Ja. A. Rojtberg, Boundary value and mixed problems for homogeneous hyperbolic equations in a complete scale of spaces of Sobolev-type, Ukr. Math. J. {\bf 41} (1989), No. 5, 592--596; translation from Ukr. Math. Zh. {\bf 41} (1989), No. 5, 685--690. \bibitem{Sanchez} David. A. Sanchez, Linear age-dependent population growth with harvesting, Bull. Math. Biol. {\bf 40} (1978), 377-385. \bibitem{Shilow} G. E. Shilov, Mathematical Analysis. Second Special Course, second ed., Izdatel'stvo Moskovskogo Universiteta (Russian), Moskva, 1984. \bibitem{song} J. Song,C. H. Tuan, and J. Y. Yu, Population Control in China: Theory and Applications, Praeger, New York, 1985. \bibitem{songyu} J. Song and J. Y. Yu, Population System Control, Springer-Verlag, Berlin, 1988. \bibitem{12}K. E. Travers, Semilinear hyperbolic systems in one space dimension with strongly singular initial data, Electr. J. of Differential Equations {\bf 1997}(1997), No. 14, 1-11. \bibitem{webb} G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985. \bibitem{yu} J. Y. Yu, B.Z.Guo, and G.T.Zhu, Asymptotic expansion in $L[0,r_m]$ for the population evolution and controllability of the population system, J. Systems Sci. Math. Sci. {\bf 7}, No. 2 (1987), 97--104. \end{thebibliography} \end{document}