\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 137, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/137\hfil Positivity and stability] {Positivity and stability for a system of transport equations with unbounded boundary perturbations} \author[C. D'Apice, B. El habil, A. Rhandi\hfil EJDE-2009/137\hfilneg] {Ciro D'Apice, Brahim El habil, Abdelaziz Rhandi} % in alphabetical order \address{Ciro D'Apice \newline Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Universit\`a degli Studi di Salerno, Via Ponte Don Melillo 84084 Fisciano (Sa), Italy} \email{dapice@diima.unisa.it} \address{Brahim El habil \newline Department of Mathematics, Faculty of Science Semlalia, Cadi Ayyad University, B.P. 2390, 40000, Marrakesh, Morocco} \email{b.elhabil@ucam.ac.ma} \address{Abdelaziz Rhandi \newline Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Universit\`a degli Studi di Salerno, Via Ponte Don Melillo 84084 Fisciano (Sa), Italy and Department of Mathematics, Faculty of Science Semlalia, Cadi Ayyad University, B.P. 2390, 40000, Marrakesh, Morocco} \email{rhandi@diima.unisa.it} \thanks{Submitted September 16, 2009. Published October 25, 2009.} \subjclass[2000]{47D06, 46B42, 34D05, 34G10, 47A10, 47A55, 47B65} \keywords{System of transport equations; $C_0$-semigroup; irreducibility; \hfill\break\indent dominant eigenvalue; asymptotic properties; unbounded boundary perturbation} \begin{abstract} This article concerns wellposedness, positivity and spectral properties of the solution of a system of transport equations with unbounded boundary perturbations. In particular we obtain that the rescaled solution converges to the unique steady-state solution as time approaches infinity on a weighted $L^1$-space. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Inspired from a queueing network model studied by \cite{Gup}, \cite{HG}, \cite{Haji-Radl}, \cite{Radl}, we propose in this paper to study the qualitative and the quantitative properties of the system of partial differential equations \begin{equation} \begin{gathered} {\frac{\partial p_0(x,t)}{\partial t}} +{\frac{\partial p_0(x,t)}{\partial x}}=\eta {\int_0^1\mu(x) p_1(x,t)dx},\quad t\ge 0,\,x\in (0,1), \\ {\frac{\partial p_1(x,t)}{\partial t}}+{\frac{\partial p_1(x,t)}{\partial x}}=-(\alpha + \mu(x))p_1(x,t),\quad t\ge 0,\,x\in (0,1), \\ {\frac{\partial p_n(x,t)}{\partial t}} +{\frac{\partial p_n(x,t)}{\partial x}} =-(\alpha+ \mu(x))p_n(x,t)+\alpha p_{n-1}(x,t),\\ \text{for } t\ge 0,\;x\in (0,1),\; 2 \leq n \leq N+1, \\ {\frac{\partial p_{N+2}(x,t)}{\partial t}}+{\frac{\partial p_{N+2}(x,t)}{\partial x}}=- \mu(x)p_{N+2}(x,t)+\alpha p_{N+1}(x,t),\\ \text{for } t\ge 0,\; x\in (0,1), \end{gathered} \label{PP} \end{equation} with the boundary conditions \begin{equation} \begin{gathered} p_0(0,t)=p_0(1,t), \quad t \geq 0,\\ p_1(0,t)=\alpha p_0(1,t)+q \overline{\mu} p_1(1,t) +\eta\overline{\mu} p_2(1,t), \quad t \geq 0,\\ p_n(0,t)=q \overline{\mu} p_n(1,t) +\eta \overline{\mu} p_{n+1}(1,t), \quad 2 \leq n \leq N+1, \; t \geq 0, \\ p_{N+2}(0,t)=q \overline{\mu} p_{N+2}(1,t), \quad t \geq 0, \end{gathered} \label{CB} \end{equation} and the initial values \begin{equation} \begin{gathered} p_0(x,0)=f_0(x), \quad x \in (0,1),\\ p_1(x,0)=f_1(x), \quad x \in (0,1),\\ p_n(x,0)=f_n(x), \quad 2 \leq n \leq N+1, \; x \in (0,1),\\ p_{N+2}(x,0)= f_{N+2}(x), \quad x \in (0,1), \end{gathered} \label{CI} \end{equation} where $ f_i \in L^1(0, 1)$ for $i \in \{0, 1,\dots , N+2 \}$. Using the language of operator matrices we see that equations \eqref{PP}-\eqref{CB} are equivalent to \begin{gather} \partial_t\begin{pmatrix} p_0\\ p_1\\ \vdots \\p_{N+2} \end{pmatrix}+\partial_x \begin{pmatrix} p_0\\ p_1\\ \vdots \\p_{N+2} \end{pmatrix} = Q\begin{pmatrix} p_0\\ p_1\\ \vdots \\p_{N+2} \end{pmatrix}+R\begin{pmatrix} p_0\\ p_1\\ \vdots \\p_{N+2} \end{pmatrix} \label{eq33} \\ \begin{pmatrix} p_0(0,t)\\ p_1(0,t)\\ \vdots \\p_{N+2}(0,t) \end{pmatrix} = \Phi \begin{pmatrix} p_0(1,t)\\ p_1(1,t)\\ \vdots \\p_{N+2}(1,t) \end{pmatrix},\label{eq34} \end{gather} where $Q$ is the multiplication operator $$ Q=\begin{pmatrix} 0 & 0 & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & D & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & \alpha & D & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & \alpha & D & .& .& .& 0 & 0 & 0 \\ . & .& .& .& .& . & .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & .& D & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & . & \alpha & D & 0 \\ 0 & 0 & 0 & 0 & . & . & . & 0 & \alpha & -\mu(.) \end{pmatrix}, $$ and $R$ the integral operator $$ R=\begin{pmatrix} 0 & \eta \Psi & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & .& .& .& 0 & 0 & 0 \\ . & .& .& .& .& . & .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & . & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & . & 0 & 0 & 0 \end{pmatrix} $$ with $ \Psi(\varphi)= {\int_0^1 \varphi(x)\mu(x)dx}$ and $D\varphi= -( \alpha+ \mu(.))\varphi$ for $\varphi \in L^1(0, 1)$. The $(N+3)\times (N+3)$-matrix $\Phi $ is $$ \Phi= \begin{pmatrix} 1 & 0 & 0 & 0 & .& .& .& 0 & 0 & 0 \\ \alpha & q& \eta& 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & q& \eta& .& .& .& 0 & 0 & 0 \\ 0 & 0 & 0 & q& .& .& .& 0 & 0 & 0 \\ . & .& .& .& .& . & .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & .& . & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & . & 0 & q& \eta \\ 0 & 0 & 0 & 0 & . & . & . & 0 & 0 & q \end{pmatrix}. $$ Here and in the sequel we suppose that $ \mu \in L^{\infty}((0,1),\mathbb{R}_+ )$, $\eta \in (0, 1)$, $q:=1- \eta $, $\lambda_0> 0 $ and take without loss of generality $ {\int_0^1 \mu(x)dx}=\overline{\mu}=1$. Hence, equations (\ref{eq33})-(\ref{eq34}) are similar to a model describing the growth of a cell population proposed by Rotenberg \cite{Rotenberg} (see also \cite{Boulanouar00}, \cite{Boulanouar01}). On the Banach space $ X:= [L^1(0, 1)]^{N+3},\; N \ge 1$, endowed with the usual norm $$ \|\varphi\| := {\sum_{i=0}^{N+2}\|\varphi_i\|_{L^1(0,1)}},\quad \varphi \in X, $$ one can see that $Q,\,R\in \mathcal{L}(X)$. Then the problem (\ref{CI})-(\ref{eq34}) can be written as the Cauchy problem \begin{equation} \begin{gathered} P'(t)=A_m P(t)+BP(t):= L_m P(t), \quad t \geq 0, \\ \Gamma_0P(t)= \Phi \Gamma_1 P(t):=\overline{\Phi} P(t), \\ P(0)= (f_0,\ldots, f_{N+2})^T \in X, \end{gathered} \label{PC} \end{equation} where $ B=R+Q $, the operator $A_m $ and the trace application $ \Gamma_0 $ and $ \Gamma_1 $ are respectively defined by $$ A_m = -{\frac{\partial}{\partial x}Id_X}, \quad \Gamma_0 = \gamma_ 0 Id_X, \quad \Gamma_1 = \gamma_1 Id_X , $$ where $ \gamma_i: L^1(0,1): \to \mathbb{C}$, $\gamma_i(\varphi)=\varphi(i)$ for $i \in \{ 0, 1 \} $ and $ \varphi \in L^1(0,1) $. In Section 2 below we construct the semigroup solution $S_\Phi (\cdot)$ of the Cauchy problem \eqref{PC} and give the explicit expression of the unperturbed semigroup $T_\Phi (\cdot)$ corresponding to $A_m$ (i.e. B=0). In Section 3 we prove the irreducibility of the semigroups $S_\Phi (\cdot)$ and $T_\Phi (\cdot)$, and show that the growth bound of $T_\Phi (\cdot)$ is $\omega_0(T_\Phi)=0$. In the last section we investigate the spectrum of the generator $L_\Phi$ of the semigroup $S_\Phi (\cdot)$ and we prove in particular that the spectral bound $s(L_\Phi)$ of $L_\Phi$ is a dominant eigenvalue and a first order pole of the resolvent of $L_\Phi$. As a consequence we obtain that the rescaled semigroup $(e^{-s(L_\Phi)t}S_\Phi(t))_{t\ge 0}$ converges to the unique steady-state solution as $t$ goes to infinity on a weighted $L^1$-space. \section{Construction of the semigroup solution of \eqref{PC}} In this section we prove that the operator \begin{gather*} L_{\Phi} \varphi = (A_\Phi +B)\varphi=(A_m +B)\varphi ,\\ D(L_{\Phi})= D(A_\Phi):= \{ \varphi \in [W^1(0, 1)]^{N+3}, \;\; \Gamma_0\varphi = \Phi \Gamma_1 \varphi = \overline{\Phi} \varphi \} \end{gather*} generates a $C_0$-semigroup $S_\Phi(\cdot)$ on $X$. Thus the Cauchy problem \eqref{PC} is wellposed. Here $ W^1(0, 1)= {\{\varphi \in L^1(0,1): \frac{\partial \varphi }{\partial x } } \in L^1(0, 1)\} $ is the first Sobolev space equipped with the norm $$ \| \varphi \|_{W^1(0,1)}:= \| \varphi \|_{L^1(0, 1)}+\| \frac{ \partial \varphi }{ \partial x }\|_{L^1(0, 1)}. $$ First, it is known that the operator $ A_0 $, defined by $$ A_0 \varphi = A_m \varphi, \quad D(A_0)= \{ \varphi \in [W^1(0,1)]^{N+3}, \; \Gamma_0\varphi = 0 \}, $$ generates the positive $C_0$-semigroup $ (T_0(t))_{t \geq 0}$, given by $$ T_0(t)\varphi(x)=\chi_{(t,1)}(x) \varphi(x-t) $$ with $\chi_{(t,1)}(x) := \begin{cases} 1 , & \text{if } x \geq t,\\ 0 , & \text{otherwise}. \end{cases}$ We show now that the operator $ A_{\Phi} $ generates a $ C_0$-semigroup $ (T_{\Phi}(t))_{t \geq 0}$ on $X$. To this purpose we give the expression of the resolvent of $A_\Phi$. \begin{lemma}\label{resolvante} For $ \lambda > \log(1 + \alpha) $, the resolvent $R(\lambda ,A_\Phi)$ of $ A_{\Phi} $ is given by \begin{equation}\label{eq35} R(\lambda ,A_\Phi)g=(\lambda - A_{\Phi})^{-1}g =e^{- \lambda .}(Id - e^{- \lambda }\Phi)^{-1} \Phi \Gamma_1(\lambda - A_0)^{-1}g + (\lambda - A_0)^{-1}g, \end{equation} for $g\in X$. \end{lemma} \begin{proof} Let $ \lambda > \log |\Phi|=\log(1 + \alpha)$, $\psi\in \mathbb{C}^{N+3}$ and $ g \in X$. The general solution of the equation \begin{equation} \begin{gathered} \lambda \varphi + {\frac{\partial}{\partial x } \varphi} = g,\\ \Gamma_0\varphi = \psi. \end{gathered}\label{equ.resolvante} \end{equation} is \begin{equation} \varphi(x)=e^{- \lambda x}\psi + (\lambda - A_0)^{-1}g(x). \label{solution} \end{equation} We have to show that the solution of \eqref{equ.resolvante} satisfies the boundary condition $ \psi = \Phi \Gamma_1 \varphi $. So, by \eqref{solution} we obtain $$ \psi = e^{- \lambda }\Phi\psi + \Phi \Gamma_1 (\lambda - A_0)^{-1}g. $$ Hence, $ [Id- e^{- \lambda }\Phi]\psi = \Phi \Gamma_1 (\lambda - A_0)^{-1}g $. Since $ e^{-\lambda}|\Phi| < 1$, it follows that the equation \eqref{equ.resolvante} with the boundary condition $\Gamma_0\varphi =\Phi \Gamma_1\varphi$ has a unique solution given by $$ \varphi(x)=e^{- \lambda x}(Id - e^{- \lambda }\Phi)^{-1} \Phi \Gamma_1(\lambda - A_0)^{-1}g + (\lambda - A_0)^{-1}g(x). $$ Moreover, $ \varphi $ is in $ (W^1(0, 1))^{N+3}$ which implies that $\varphi \in D(A_\Phi)$ and this proves (\ref{eq35}). \end{proof} Now, we show that operator $ A_{\Phi}$ generates a $C_0$-semigroup on $X$. \begin{theorem} \label{thm2.2} On $X$ the operator $A_{\Phi} $ generates a $C_0$-semigroup $ (T_{\Phi}(t))_{t \geq 0}$ satisfying \begin{equation}\label{Tfi} \|T_{\Phi}(t)\|_{\mathcal{L}(X)}\leq (1 + \alpha) {e^{t \log(1 + \alpha)}}. \end{equation} \end{theorem} \begin{proof} On $ X $ we define a new norm $$ \|| \varphi \||:= {\int_0^1(1 + \alpha)^x|\varphi(x)|dx },\quad \varphi \in X. $$ Since \begin{equation}\label{norm-equiv} \|\varphi \| \leq \|| \varphi \|| \leq (1 + \alpha)\|\varphi\|, \quad \varphi \in X, \end{equation} these two norms are equivalent. Take $\lambda >\log(1+\alpha),\,g\in X$ and set $\varphi =R(\lambda ,A_\Phi)g$. Multiplying \eqref{equ.resolvante} by $(1 + \alpha)^x sign(\varphi)(x)$ and integrating by parts, we find \begin{align*} \lambda \||\varphi\|| &=\lambda {\int_0^1 (1 + \alpha)^x |\varphi(x)|dx} \\ &\leq - {\int_0^1(1 + \alpha)^x{\frac{\partial}{\partial x }|\varphi(x)|}dx}+{\int_0^1 (1 + \alpha)^x |g(x)|dx}\\ &\leq \||g\||+ \log (1 + \alpha) \||\varphi\||+ |\Gamma_0 \varphi|-(1 + \alpha) |\Gamma_1\varphi|\\ &= \||g\||+ \log (1 + \alpha) \||\varphi\|| +|\Gamma_0 \varphi|-|\Phi | |\Gamma_1\varphi|\\ &\leq \||g\||+ \log (1 + \alpha) \||\varphi\||. \end{align*} Consequently, $$ \||R(\lambda ,A_\Phi)g\|| \leq { \frac{1}{\lambda - \log(1 + \alpha)}}\||g\||. $$ Since $D(A_\Phi)$ is dense in $X$, the Hille-Yosida theorem implies that $A_\Phi$ generates a $C_0$-semigroup $T_\Phi(\cdot)$ satisfying $$ \||T_\Phi(t)\||\le e^{t\log(1+\alpha)},\quad t\ge 0. $$ Now the estimate (\ref{Tfi}) follows from (\ref{norm-equiv}) and this completes the proof. \end{proof} Since $B\in {\mathcal L}(X)$, by the bounded perturbation theorem (cf. \cite[Theorem III.1.3]{Engel-Nagel}) we obtain the following generation result for the operator $L_\Phi$. \begin{theorem} \label{thm2.3} The operator $ L_{\Phi} $ generates a $ C_0$-semigroup $ (S_{\Phi}(t))_{t \geq 0}$ on $X$ satisfying $$ \|S_{\Phi}(t)\|_{\mathcal{L}(X)} \leq (1 + \alpha)e^{t(\log(1 + \alpha)+(1 + \alpha)\|B\|)}. $$ \end{theorem} In the remainder part of this section, we give an explicit formula for the semigroup $T_\Phi(\cdot)$. For this purpose we define, on the space $[W^1(0,1)]^{N+3}$, the linear operator $\mathcal{T}_{\Phi}(t)$ by \begin{equation} \mathcal{T}_{\Phi}(t)\varphi(x):=\chi_{[0,t]}(x)\Phi\Gamma_1T_0(t-x)\varphi, \quad x \in (0, 1), \, 0 \leq t \leq 1 \label{41} \end{equation} for $\varphi \in [W^1(0,1)]^{N+3}$, where $ \chi_{[0,t]}$ is the characteristic function of the interval $[0,t]$ defined by $$ \chi_{[0,t]}(x)= \begin{cases} 0, & \text{if } t < x,\\ 1, & \text{otherwise}. \end{cases} $$ For $ \varphi \in [W^1(0,1)]^{N+3}$ we have \begin{equation} \label{eq+} \begin{aligned} \|\mathcal{T}_{\Phi}(t)\varphi\| & = {\int_0^1 | \chi_{[0, t]}(x)\Phi\Gamma_1T_0(t-x)\varphi|\,dx}\\ &\leq (1+\alpha) {\int_0^t |\Gamma_1T_0(t-x)\varphi|\,dx}\\ &\leq (1+\alpha) {\int_0^t |\chi(1,t-x)\varphi(1-t+x)|\,dx}\\ &\leq (1+\alpha) {\int_0^1 |\varphi(1-x)|\,dx}\\ & = (1+\alpha)\|\varphi \|. \end{aligned} \end{equation} Since $[W^1(0,1)]^{N+3}$ is dense in $X $, the operator $\mathcal{T}_{\Phi}(t)$, $t\in [0,1]$, can be extended to a bounded linear operator on $X$ which will be also denoted by $\mathcal{T}_{\Phi}(t) $. \begin{lemma} \label{lem2.4} The family $(\mathcal{T}_{\Phi}(t))_{0\le t\le 1}$ satisfies: \begin{itemize} \item[(i)] $\mathcal{T}_{\Phi}(0)=0 $, and $\|\mathcal{T}_{\Phi}(t)\|_{\mathcal{L}(X)} \leq (1+\alpha)$ for all $t \in [0, 1] $, \item[(ii)] for all $ t, s \in [0,1]$ such that $ s+t \in [0,1]$, $\mathcal{T}_{\Phi}(t)\mathcal{T}_{\Phi}(s)=0$. \end{itemize} \end{lemma} \begin{proof} (i) It is easy to see that $ \mathcal{T}_{\Phi}(0)=0 $. The estimate has been proved above (see (\ref{eq+})). (ii) Let $ \varphi \in [W^1(0, 1)]^{N+3},\,t,s\in [0,1]$ such that $s+t\in [0,1]$, and set $\psi=\mathcal{T}_\Phi(s)\varphi$. Then \begin{align*} \psi(x) &= \chi_{[0, s]}(x) \Phi (T_0(s-x)\varphi)(1)\\ &= \chi_{[0, s]}(x)\Phi \varphi(1-s+x)\\ & =: \chi_{[0, s]}(x)\Phi y(x) \end{align*} with $y(x) :=\varphi(1-s+x) \in \mathbb{C}^{N+3}$. Hence, \begin{align*} \mathcal{T}_{\Phi}(t)\psi(x) &= (\mathcal{T}_{\Phi}(t)\chi_{[0,s]} \Phi y(\cdot))(x)\\ &= \chi_{[0,t]}(x)\Phi \Gamma_1T_0(t-x)\chi_{[0,s]}\Phi y(\cdot)\\ &= \chi_{[0,t]}(x)\Phi \chi_{[0,s]}(1-t+x)\Phi y(1-t+x) = 0, \end{align*} since $\chi_{[0,s]}(1-t+x)=0$ for all $x \in (0, 1)$. The denseness of $ [W^1(0, 1)]^{N+3}$ in $X $ completes the proof. \end{proof} To show the main result of this section, we define some auxiliary operators. For any $t\ge 0$ there exists $n\in \mathbb{N}$ and $r\in [0,\frac{1}{2})$ such that $t=\frac{n}{2}+r$. We define the operators $\overline{B}_{\Phi}(t),\,t\ge 0$, by $$ \overline{B}_{\Phi}(t):= (B_{\Phi}(1/2))^n B_{\Phi}(r), $$ where $B_\Phi(t)=T_0(t)+\mathcal{T}_{\Phi}(t)$ for $t\in [0,1]$. \begin{lemma}\label{lem 25} The family $ (\overline{B}_{\Phi}(t))_{t\geq 0}$ is a $C_0$-semigroup on $ X $. \end{lemma} \begin{proof} The uniqueness of the decomposition $t=\frac{n}{2}+r$ with $n\in \mathbb{N}$ and $r\in [0,\frac{1}{2})$ implies that the operators $\overline{B}_{\Phi}(t),\,t\ge 0$, are well defined. Moreover, from the boundedness of $B_\Phi(t)$ follows that $\overline{B}_{\Phi}(t),\,t\ge 0$, are bounded linear operators on $X$, and the following holds $$ \overline{B}_\Phi(0)=B_\Phi(0)=T_0(0)+\mathcal{T}_\Phi(0)=Id. $$ We propose now to show the semigroup property. First, we start with the case $t,s\in [0,1]$ with $s+t\in [0,1]$ and prove that \begin{equation}\label{r1} B_{\Phi}(t)B_{\Phi}(s)\varphi=B_\Phi(t+s)\varphi \end{equation} for $\varphi \in X$. In fact, for $\varphi \in [W^1(0,1)]^{N+3}$ (and hence by density for $\varphi \in X$), we have \begin{align*} &B_{\Phi}(t)B_{\Phi}(s)\varphi(x)\\ &=(T_0(t)+\mathcal{T}_{\Phi}(t))(T_0(s)+ \mathcal{T}_{\Phi}(s))\varphi(x)\\ &= T_0(t+s)\varphi(x)+ \mathcal{T}_{\Phi}(t)T_0(s)\varphi(x)+ T_0(t)\mathcal{T}_{\Phi}(s)\varphi(x) \\ &= T_0(t+s)\varphi(x)+\chi_{[0, t]}(x) \Phi \Gamma_1T_0(t+s-x)\varphi +\chi_{[ t,1]}(x)\mathcal{T}_{\Phi}(s)\varphi(x-t) \\ &= T_0(t+s)\varphi(x)+[\chi_{[0, t]}(x)\chi_{[t+s,1]}(x)+ \chi_{[0, t]}(x)\chi_{[0, t+s]}(x)]\Phi\Gamma_1T_0(t+s-x)\varphi \\ &\quad + \chi_{[t, 1]}(x)\chi_{[0, t+s]}(x)\Phi \Gamma_1 T_0(t+s-x)\varphi \\ &= B_\Phi(t+s)\varphi(x). \end{align*} Next, by an easy computation one sees that \begin{align*} \Big(\mathcal{T}_\Phi(r)T_0(\frac{1}{2})\varphi +T_0(r)\mathcal{T}_\Phi(\frac{1}{2})\varphi \Big)(x) &= \Big(T_0(\frac{1}{2})\mathcal{T}_\Phi(r)\varphi +\mathcal{T}_\Phi(\frac{1}{2})T_0(r)\varphi \Big)(x)\\ &= \chi_{[0,r+\frac{1}{2}]}(x)\Phi\Gamma_1T_0(r+\frac{1}{2}-x)\varphi \end{align*} for all $\varphi \in X$. This shows that \begin{equation}\label{r2} B_{\Phi}(r)B_{\Phi}(1/2)=B_{\Phi}(1/2)B_{\Phi}(r)\quad \text{for all }r\in [0,\frac{1}{2}]. \end{equation} Now, the semigroup property $$ \overline{B}_\Phi(t+s)=\overline{B}_\Phi(t)\overline{B}_\Phi(s), \quad t,s\ge 0 $$ follows from (\ref{r1}) and (\ref{r2}). For the strong continuity, let us consider $t\in (0,\frac{1}{2})$ and $\varphi \in X$. Then $\overline{B}_\Phi(t)\varphi -\varphi =(T_0(t)\varphi -\varphi)+\mathcal{T}_\Phi(t)\varphi \to 0$ as $t\to 0^+$, since $T_0(\cdot)$ is strongly continuous and $\|\mathcal{T}_\Phi(t)\varphi\|\le (1+\alpha)\int_{1-t}^1|\varphi(x)|\,dx$. \end{proof} \begin{theorem} \label{thm2.6} The semigroups $ T_{\Phi}(\cdot)$ and $\overline{B}_{\Phi}(\cdot)$ coincide. \end{theorem} \begin{proof} We denote by $ C $ the generator of the $C_0$-semigroup $ \overline{B}_{\Phi}(\cdot)$. Let $\varphi \in D(A_{\Phi})$, $t\in (0,1)$ and set $\psi=\varphi-\Gamma_0\varphi$. Then \begin{align*} &\frac{1}{t}(\overline{B}_\Phi(t)\varphi -\varphi)+\varphi ' \\ &= \frac{1}{t}(T_0(t)\psi -\psi)+\psi '+\frac{1}{t}(\chi_{(t,1)} (\cdot)-1)\Gamma_0\varphi +\frac{1}{t}\mathcal{T}_\Phi(t)\varphi \\ &= \frac{1}{t}(T_0(t)\psi -\psi)+\psi '-\frac{1}{t}\chi_{(0,t)}(\cdot) \Gamma_0\varphi +\frac{1}{t} \chi_{(0,t)}(\cdot)\Phi \varphi(1-t+\cdot). \end{align*} Since $\psi \in D(A_0)$ and $\Gamma_0\varphi =\Phi \Gamma_1\varphi$, it follows that $$ \lim_{t\to 0^+}\frac{1}{t}(\overline{B}_\Phi(t)\varphi -\varphi) +\varphi '=0. $$ Hence, $D(A_\Phi)\subset D(C)$ and $C|_{D(A_\Phi)}=A_\Phi$. Since $C$ and $A_\Phi$ are both generators, we deduce that $A_\Phi=C$ and therefore $T_\Phi(\cdot)=\overline{B}_\Phi(\cdot)$. \end{proof} \section{Irreducibility and some spectral properties} In this section we study the irreducibility of the semigroups $T_\Phi(\cdot)$ and $S_\Phi(\cdot)$, and we characterize the growth bound $\omega_0(T_\Phi)$. We begin by proving the irreducibility. To this purpose we need the following lemma. \begin{lemma}\label{irred} Assume that $A$ generates an irreducible $C_0$-semigroup $T(\cdot)$ on a Banach lattice $X$ and $B\in \mathcal{L}(X)$ is such that $e^{tB}\ge 0,\,t\ge 0$. Then the perturbed semigroup $S(\cdot)$ is irreducible. \end{lemma} \begin{proof} Since the semigroup $(e^{tB})_{t\ge 0}$ is positive, it follows that $B+\|B\|Id\ge 0$ (cf. \cite[Theorem 1.11.C-II]{Nagel}). Hence the semigroup generated by $A+B+\|B\|Id$ satisfies $$ e^{t\|B\|}S(t)\ge T(t),\quad t\ge 0. $$ Thus the irreducibility of $T(\cdot)$ implies that the semigroup $(e^{t\|B\|}S(t))_{t\ge 0}$ is irreducible. Hence, $S(\cdot)$ is irreducible too. \end{proof} As a consequence we obtain the following result. \begin{proposition}\label{lem 4.1} The semigroups $(T_{\Phi}(t))_{t \geq 0}$ and $ (S_{\Phi}(t))_{t \geq 0} $ are irreducible. \end{proposition} \begin{proof} Let $ \lambda \geq \ln (1 + \alpha)$ and $ \varphi > 0 $. By Lemma \ref{resolvante} we have \begin{align*} (\lambda - A_{\Phi})^{-1}\varphi &= e^{- \lambda .}(Id - e^{- \lambda }\Phi)^{-1} \Phi \Gamma_1(\lambda - A_0)^{-1}\varphi + (\lambda - A_0)^{-1}\varphi\\ & \geq e^{- \lambda .}(Id - e^{- \lambda }\Phi )^{-1} \Phi \Gamma_1(\lambda - A_0)^{-1}\varphi \\ & \geq e^{- \lambda .}{\sum_{n=0}^{\infty}}(e^{- \lambda } \Phi )^{n} \Phi \Gamma_1(\lambda - A_0)^{-1}\varphi \\ & \geq e^{- \lambda .}\Phi \Gamma_1(\lambda-A_0)^{-1}\varphi \\ &= e^{-\lambda \cdot}\Phi \Big(\int_0^1e^{\lambda(s-1)}\varphi(s)\,ds\Big)>0, \end{align*} since $(\lambda -A_0)^{-1}\varphi(x) =\int_0^xe^{\lambda(s-x)}\varphi(s)\,ds$ and $\Phi>0$. Hence $(\lambda - A_{\Phi})^{-1}$ is irreducible and therefore $ T_{\Phi}(\cdot) $ is irreducible. Now, we decompose $B$ as $ B = B_0+B_1$ with \begin{gather*} B_0= \begin{pmatrix} 0 & \eta \Psi & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & \alpha & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & \alpha & 0 & .& .& .& 0 & 0 & 0 \\ . & .& .& .& .& . & .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & .& .& .& \alpha & 0 & 0 \\ 0 & 0 & 0 & 0 & .& .& .& 0 & \alpha & 0 \end{pmatrix}, \\ B_1= \begin{pmatrix} 0 & 0 & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & D & 0 & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & D & 0 & .& .& .& 0 & 0 & 0 \\ 0 & 0 & 0 & D & .& .& .& 0 & 0 & 0 \\ . & .& .& .& .& . & .& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & .& D & 0 & 0 \\ 0 & 0 & 0 & 0 & . & . & . & 0 & D & 0 \\ 0 & 0 & 0 & 0 & . & . & . & 0 & 0 & -\mu(.) \end{pmatrix}. \end{gather*} Since $B_1$ is a real multiplication operator on $X$, it follows that $(e^{tB_1})_{t\ge 0}$ is a positive semigroup on $X$. Thus, by the positivity of $B_0$, we get the positivity of $(e^{tB})_{t\ge 0}$ on $X$. Hence, the irreducibility of $S_\Phi(\cdot)$ follows now from Lemma \ref{irred}. \end{proof} \begin{proposition}\label{prop 4.3} The growth bound of the semigroups $T_\Phi(\cdot)$ satisfies $$ \omega_0( T_{\Phi}) =0. $$ \end{proposition} \begin{proof} Since $\sigma(A_0)=\emptyset$, it follows from the proof of Lemma \ref{resolvante} that $$ \lambda \in \sigma(A_\Phi)\Longleftrightarrow 1\in \sigma(e^{-\lambda}\Phi). $$ An easy computation shows that $$ \det(Id-e^{-\lambda}\Phi)=(1-e^{-\lambda})(1-qe^{-\lambda})^{N+2}. $$ Hence, $1\in \sigma(e^{-\lambda}\Phi)\Leftrightarrow e^{\lambda}=1$ or $e^{\lambda} =q$. This implies that $\{\Re \lambda :\lambda \in \sigma(A_\Phi)\}=\{0,\log q\}$ and thus $$ s(A_\Phi)=\omega_0(T_\Phi)=0, $$ since $q\in (0,1)$. \end{proof} \section{The spectral bound of the generator of $S_\Phi(\cdot)$} In this section we are interested in studying some spectral properties of the generator $L_\Phi$ of the semigroup $S_\Phi(\cdot)$ on $X$. In particular we show that $00$ is a dominant eigenvalue and a first order pole of the resolvent of $L_\Phi$. \\ Here, as in \cite{Radl}, we use an abstract framework developed by Greiner \cite{Gre87}. On the product space $\mathcal{X}:= X \times \mathbb{C}^{N+3}$, we define the operators \begin{gather*} \mathcal{A}_0 := \begin{pmatrix} L_m & 0 \\ -\Gamma_0 & 0 \\ \end{pmatrix}\quad \text{with } D(\mathcal{A}_0):= D(L_m) \times \{0\}, \\ \mathcal{B}:= \begin{pmatrix} 0 & 0 \\ \overline{\Phi} & 0 \\ \end{pmatrix}\quad \text{with } D(\mathcal{B}):= D(L_m) \times \mathbb{C}^{N+3}, \\ \mathcal{A}:= \mathcal{A}_0 + \mathcal{B} =\begin{pmatrix} L_m & 0 \\ \overline{\Phi}-\Gamma_0 & 0 \\ \end{pmatrix} \quad \text{with } D(\mathcal{A}):= D(L_m) \times \{0\}. \end{gather*} Set $ \mathcal{X}_0:= X \times \{0\}= \overline{D(\mathcal{A}_0)}$. Since $\Gamma_0\in \mathcal{L}(D(A_m),\mathbb{C}^{N+3})$ is surjective one can define for $\gamma \in \rho(L_0)$ the operator $\mathcal{D}_\gamma:=\big(\Gamma_0|_{\ker (\gamma -L_m)}\big)^{-1}\in \mathcal{L}(\mathbb{C}^{N+3},\ker (\gamma -L_m))$ called the {\it Dirichlet} operator. Moreover, $$ R(\gamma,\mathcal{A}_0)= \begin{pmatrix} R(\gamma,L_0) & D_{\gamma} \\ 0 & 0 \end{pmatrix}. $$ The part $ \mathcal{A}|_{\mathcal{X}_0} $ of $ \mathcal{A} $ in $ \mathcal{X}_0 $ is given by $$ D(\mathcal{A}|_{\mathcal{X}_0)}= D(L_{\Phi}) \times \{0 \} \quad\text{and}\quad \mathcal{A}|_{\mathcal{X}_0}= \begin{pmatrix} L_{\Phi} & 0 \\ 0 & 0 \\ \end{pmatrix}. $$ Thus, $ \mathcal{A}|_{\mathcal{X}_0} $ can be identified with the operator $ ( L_{\Phi}, D(L_{\Phi}))$. Furthermore, for $\gamma \in \rho(L_0)$, the following characteristic equation holds (cf. \cite[Page 11]{Radl}) \begin{equation}\label{carac-equa0} \gamma \in \sigma_p(L_\Phi)\Leftrightarrow 1\in \sigma_p(\overline{\Phi}\mathcal{D}_\gamma)= \sigma(\overline{\Phi}\mathcal{D}_\gamma) \end{equation} and if in addition there exists $\beta \in \mathbb{C} $ such that $1\in \rho(\overline{\Phi}\mathcal{D}_\beta)$, then \begin{equation}\label{carac-equa1} \gamma \in \sigma(L_\Phi)\Leftrightarrow 1\in \sigma(\overline{\Phi}\mathcal{D}_\gamma). \end{equation} Let us consider the operators $D_0,\,D_1$ and $D_2$ defined on $W_0^{1,1}(0,1):=\{\varphi \in W^{1,1}(0,1):\varphi(0)=0\}$ by $D_0\varphi =-\varphi'$, $D_1\varphi =-\varphi'-(\alpha+\mu(\cdot))\varphi$ and $D_2\varphi =-\varphi'-\mu(\cdot)\varphi$, $\varphi \in W_0^{1,1}(0,1)$. Then, for any $\gamma \in \mathbb{C}$, we have \begin{gather*} (R(\gamma, D_0)\varphi)(x) = e^{-\gamma x}{\int_0^x}e^{\gamma s}\varphi(s)ds, \\ (R(\gamma, D_1)\varphi)(x) = e^{-(\gamma + \alpha)x - \int_0^x\mu(\sigma)d \sigma}{\int_0^x}e^{(\gamma + \alpha)s + \int_0^s\mu(\sigma)d \sigma}\varphi(s)ds , \\ (R(\gamma, D_2)\varphi)(x) = e^{-\gamma x - \int_0^x\mu(\sigma)d \sigma}{\int_0^x}e^{\gamma s + \int_0^s\mu(\sigma)d \sigma}\varphi(s)ds \end{gather*} for $\varphi \in L^1(0,1)$ and $x\in [0,1]$. Set \begin{gather*} r_{1,1}= R(\gamma, D_0), \\ r_{1,2} = \eta R(\gamma, D_0)\Psi R(\gamma, D_1),\\ r_{j,k} = \alpha^{j-k}R(\gamma, D_1)^{j-k+1},\quad 2 \leq k \leq j \leq N+ 2,\\ r_{N+3,k} = \alpha^{N+3-k}R(\gamma, D_2)R(\gamma, D_1)^{N+3-k},\quad 2 \leq k \leq N+ 3. \end{gather*} Then the resolvent of $L_0$ can be computed explicitly as the following lemma shows. \begin{lemma}\label{lem4.1} For the operator $(L_0,D(L_0))$ we have $\rho(L_0)=\mathbb{C}$ and $$ R(\gamma, L_0)= \begin{pmatrix} r_{1,1}&r_{1,2} & 0 & 0 & \dots & 0& 0 \\ 0 & r_{2,2} & 0 & 0 & \dots & 0& 0 \\ 0 & r_{3,2} & r_{3,3} & 0 & \dots & 0 & 0 \\ 0 & r_{4,2} & r_{4,3} & r_{4,4} & \dots & 0 & 0 \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 0 & r_{N+2,2} & r_{N+2,3} & r_{N+2,4} & \dots & r_{N+2,N+2} & 0 \\ 0 & r_{N+3,2}& r_{N+3,2} & r_{N+3,4} & \dots & r_{N+3,N+2} & r_{N+3,N+3} \end{pmatrix}. $$ \end{lemma} One can also characterize $\ker (\gamma -L_m)$ for any $\gamma \in \mathbb{C}$ and therefore one obtains an explicit formula for the Dirichlet operator $\mathcal{D}_\gamma$. To this purpose, for $\gamma \in \mathbb{C}$, set \begin{gather*} \epsilon^{\gamma}_k(x):= \frac{\alpha^k}{k!}x^ke^{-(\gamma + \alpha)x - \int_0^x\mu(s)d s},\quad 0\le k \le N ,\\ d^{\gamma}_{1,1} := {\frac{ \eta}{\gamma}}(1-e^{-\gamma x}) {\int_0^1\mu(x)\epsilon^{\gamma}_0(x)dx }, \\ d^{\gamma}_{N+3, k} := \exp(-\gamma \cdot -\int_0^\cdot \mu(s)ds)- {\sum_{n=0}^{N+1-k}}\epsilon^{\gamma}_n ,\quad 1\leq k \leq N+1, \\ d^{\gamma}_{N+3, N+2} := \exp(-\gamma \cdot -\int_0^\cdot \mu(s)ds). \end{gather*} \begin{lemma}\label{lem4.2} For $\gamma \in \mathbb{C}$, the Dirichlet operator $\mathcal{D}_\gamma$ is given by $$ \mathcal{D}_{\gamma}= \begin{pmatrix} e^{-\gamma x}& d^{\gamma }_{1,1}& 0 & 0 & \dots & 0 & 0 \\0& \epsilon^{\gamma }_0 & 0 & 0 & \dots & 0 & 0 \\ 0& \epsilon^{\gamma }_1 & \epsilon^{\gamma }_0 & 0 & \dots & 0 & 0 \\ 0& \epsilon^{\gamma }_2 & \epsilon^{\gamma }_1 & \epsilon^{\gamma }_0 & \dots & 0& 0 \\ .&. & . & . & . & . & . \\ . &.& . & . & . & . & . \\ .&. & . & . & . & . & . \\ 0& \epsilon^{\gamma }_N & \epsilon^{\gamma }_{N-1} & \epsilon^{\gamma }_{N-2} & \dots & \epsilon^{\gamma }_0 & 0 \\ 0& d^{\gamma }_{N+3,1} & d^{\gamma }_{N+3,2}& d^{\gamma }_{N+3,3} & \dots & d^{\gamma }_{N+3,N+1} & d^{\gamma }_{N+3,N+2} \end{pmatrix}. $$ \end{lemma} By setting \begin{gather*} a^{\gamma}_{k,j}= 0 \; \mbox{if } \; 0\leq k \leq N \; \mbox{and} \; j \geq k+2, \\a^{\gamma}_{0,0}= e^{-\gamma }, \\a^{\gamma}_{1,0}= \alpha e^{-\gamma }, \\ a^{\gamma}_{0,1}= d^{\gamma}_{1,1}(1), \\ a^{\gamma}_{1,1} = \alpha d^{\gamma}_{1,1}(1)+ q\epsilon^{\gamma}_0(1) + \eta \epsilon^{\gamma}_1(1), \\ a^{\gamma}_{1,2} = \eta \epsilon^{\gamma}_0(1), \\ a^{\gamma}_{2,2} = q\epsilon^{\gamma}_0(1) + \eta\epsilon^{\gamma}_1(1), \\ a^{\gamma}_{k,1} = q\epsilon^{\gamma}_{k-1}(1) + \eta \epsilon^{\gamma}_k(1) ,\quad \quad \quad \quad 2 \leq k \leq N \,\text{\ if }N\ge 2, \\ a^{\gamma}_{N+2,k}= qd^{\gamma}_{N+3,k}(1), \quad \quad \quad \quad \quad \quad \quad \,\,1 \leq k \leq N+2, \\ b^{\gamma}_{N+1,k} = q\epsilon^{\gamma}_{N-k+1}(1) + \eta d^{\gamma}_{N+3,k}(1),\quad 1 \leq k \leq N+1, \\ b^{\gamma}_{N+1,N+2} = \eta d^{\gamma}_{N+3,N+2}(1), \end{gather*} one deduces the expression of $\overline{\Phi}\mathcal{D}_\gamma$. \begin{lemma}\label{lem4.3} For $\gamma \in \mathbb{C}$, the matrix $\overline{\Phi}\mathcal{D}_\gamma$ is equal to $$ \begin{pmatrix} a^{\gamma}_{0,0}& a^{\gamma}_{0,1}& 0& 0&\dots&.&0&0&0\\ a^{\gamma}_{1,0}& a^{\gamma}_{1,1}& a^{\gamma}_{1,2} & 0 &\dots&.&0&0&0\\ 0& a^{\gamma}_{2,1} & a^{\gamma}_{2,2} & a^{\gamma}_{1,2} &\dots &.&0&0&0 \\ 0& a^{\gamma}_{3,1} & a^{\gamma}_{2,1} &a^{\gamma}_{2,2} &\dots &.&0&0&0 \\ 0&a^{\gamma}_{4,1} & a^{\gamma}_{3,1} &a^{\gamma}_{2,1} &\dots &.&0&0&0\\ .&.&.&.&\dots &.&.&.&. \\ .&.&.&.&\dots &.&.&.&. \\ 0&a^{\gamma}_{N-1,1}&.&. &\dots &a^{\gamma}_{2,2}&a^{\gamma}_{1,2}&0&0\\ 0&a^{\gamma}_{N,1}&a^{\gamma}_{N-1,1}&.&\dots &a^{\gamma}_{2,1} &a^{\gamma}_{2,2}&a^{\gamma}_{1,2}& 0 \\ 0&b^{\gamma}_{N+1,1}&b^{\gamma}_{N+1,2}&b^{\gamma}_{N+1,3} &\dots &.&b^{\gamma}_{N+1,N}&b^{\gamma}_{N+1,N+1}&b^{\gamma}_{N+1,N+2}\\ 0 &a^{\gamma}_{N+2,1}& a^{\gamma}_{N+2,2}&a^{\gamma}_{N+2,3} &\dots &.&a^{\gamma}_{N+2,N}&a^{\gamma}_{N+2,N+1}&a^{\gamma}_{N+2,N+2} \end{pmatrix}. $$ \end{lemma} \begin{remark}\label{rem4.4} \rm By setting $\overline{\Phi}\mathcal{D}_\gamma =(\alpha^{(\gamma)}_{ij})_{1\le i,j\le N+3},\,\gamma >0$, we have $\lim_{\gamma \to +\infty}\alpha^{(\gamma)}_{ij}=0$. Hence, there is $\beta >0$ such that $r(\overline{\Phi}\mathcal{D}_\beta)<1$. This implies that $1\in \rho(\overline{\Phi}\mathcal{D}_\beta)$. So, by (\ref{carac-equa0}), (\ref{carac-equa1}) and Lemma \ref{lem4.1}, we get, for any $\gamma \in \mathbb{C}$, \begin{equation}\label{carac-equa2} \gamma \in \sigma(L_\Phi)\Leftrightarrow 1\in \sigma(\overline{\Phi}\mathcal{D}_\gamma)= \sigma_p(\overline{\Phi}\mathcal{D}_\gamma)\Leftrightarrow \gamma \in \sigma_p(L_\Phi). \end{equation} In particular we obtain $$\sigma(L_\Phi)=\sigma_p(L_\Phi)$$ and if $1\in \rho(\overline{\Phi}\mathcal{D}_\gamma)$, then \begin{equation}\label{resolvent-L} R(\gamma ,L_\Phi)=R(\gamma ,L_0)+\mathcal{D}_\gamma(Id_{\mathbb{C}^{N+3}}-\overline{\Phi}\mathcal{D}_\gamma)^{-1}\overline{\Phi}R(\gamma ,L_0) \end{equation} (cf. \cite[Proposition 1.8]{Radl}). \end{remark} The following result shows that $s(L_\Phi)>0$. \begin{proposition}\label{prop4.4} There exists $\gamma_0 >0$ such that $1 = r(\overline{\Phi}\mathcal{D}_{\gamma_0})$ and therefore $$ s(L_\Phi)=\gamma_0 >0.$$ \end{proposition} \begin{proof} Since $\overline{\Phi}\mathcal{D}_0=(\alpha^{(0)}_{ij})_{1\le i,j\le N+3}$ is an irreducible matrix, it follows from \cite[Proposition 6.3., Chap.I]{Schaefer} that $r(\overline{\Phi}\mathcal{D}_0)>\max_{1\le i\le N+3}\alpha^{(0)}_{ii}$. In particular, \begin{equation}\label{radius} r(\overline{\Phi}\mathcal{D}_0)>a_{0,0}^0=1. \end{equation} On the other hand, by the explicit expression of $\overline{\Phi}\mathcal{D}_\beta$ one can see that the function $0<\beta \mapsto r(\overline{\Phi}\mathcal{D}_\beta)$ is decreasing and $\lim_{\beta \to +\infty}r(\overline{\Phi}\mathcal{D}_\beta)=0$. Thus, by continuity and (\ref{radius}), there exists a unique $\gamma_0 >0$ such that $r(\overline{\Phi}\mathcal{D}_{\gamma_0})=1 \in \sigma(\overline{\Phi}\mathcal{D}_{\gamma_0})$. Hence, from (\ref{carac-equa2}) we get $\gamma_0 \in \sigma(L_\Phi)$. Now, take $\lambda >\gamma_0$ and set $\overline{\Phi}\mathcal{D}_\lambda=(\alpha^{(\lambda)}_{ij})_{1\le i,j\le N+3}$. Since $0\le \alpha^{(\lambda)}_{ij}\le \alpha^{(\gamma_0)}_{ij}$ and $\alpha^{(\lambda)}_{11}<\alpha^{(\gamma_0)}_{11}$, it follows from \cite[Page 22]{Schaefer} that $$ r(\overline{\Phi}\mathcal{D}_\lambda) s(L_\Phi)$, that $$ r_{\rm ess}(R(\gamma ,L_\Phi))=r_{\rm ess}(R(\gamma ,L_0)),\quad \Re \gamma >s(L_\Phi). $$ Since $\sigma(L_0)=\emptyset $, we deduce from the spectral theorem for the resolvent (cf. \cite{Engel-Nagel}) that $r_{\rm ess}(R(\gamma ,L_0))=0$ and hence $$ r_{\rm ess}(R(\gamma ,L_\Phi))=0,\quad \Re \gamma >s(L_\Phi). $$ This implies that $\frac{1}{\lambda -s(L_\Phi)}$ is a pole of finite algebraic multiplicity for any $\lambda >s(L_\Phi)$. By \cite[Proposition 2.5.A-III]{Nagel} we deduce that $s(L_\Phi)$ is a pole of finite algebraic multiplicity and the first assertion is proved by applying \cite[Proposition 3.5.C-III]{Nagel}, since $S_\Phi(\cdot)$ is irreducible (see Proposition \ref{lem 4.1}). For the second assertion we note first that, by Proposition \ref{prop4.4}, $s(L_\Phi)=\gamma_0>0$. Let us consider $a\in \mathbb{R}$ such that $$ |a|>\sqrt{\frac{4\gamma_0^2}{(1-e^{-\gamma_0})^2}-\gamma_0^2}=:\xi_0. $$ Then, it is easy to see that $$ |d_{1,1}^{\gamma_0+ia}(1)| < d_{1,1}^{\gamma_0}(1). $$ Hence, $$ |\alpha_{ij}^{(\gamma_0+ia)}|\le \alpha_{ij}^{(\gamma_0)} \quad \text{and}\quad |\alpha_{12}^{(\gamma_0+ia)}|< \alpha_{12}^{(\gamma_0)} $$ for all $i,j=1,\dots , N+3$, where $(\alpha_{ij}^{(\gamma)})_{1\le i,j\le N+3}=\overline{\Phi}\mathcal{D}_\gamma$, $\gamma \in \mathbb{C}$. So, by \cite[Page 22]{Schaefer} and Proposition \ref{prop4.4} we obtain $$ r(\overline{\Phi}\mathcal{D}_{\gamma_0+ia}) \xi_0$. This means that $\sigma_b(L_\Phi)$ is bounded. On the other hand, using \cite[Proposition 2.9.C-III]{Nagel} and \cite[Proposition 2.10.C-III]{Nagel}, we obtain that $\sigma_b(L_\Phi)$ is cyclic, i.e., if $a+ib\in \sigma_b(L_\Phi),\,a,b\in \mathbb{R}$, then $a+ikb\in \sigma_b(L_\Phi)$ for all $k\in \mathbb{Z}$. Now, the boundedness of $\sigma_b(L_\Phi)$ gives the second assertion. \end{proof} Now, we deduce the asymptotic behavior of the semigroup $(S_\Phi(t))_{t\ge 0}$. \begin{theorem}\label{exponential-growth} There exists $0\ll w\in [L^\infty(0,1)]^{N+3}$ such that the rescaled semigroup $(e^{-s(L_\Phi)t}S_\Phi(t))_{t\ge 0}$ converges to the unique steady-state solution as $t$ goes to infinity in the weighted space $L^1_w:=[L^1(0,1;wdx)]^{N+3}$; i.e., there is $0\ll \psi \in L^1_w$ and $0\ll \widehat{w}\in (L^1_w)^\ast$ such that $$ \lim_{t\to \infty}e^{-s(L_\Phi)t}S_\Phi(t)\varphi = \langle \widehat{w},\varphi \rangle_{L^1_w}\psi $$ for all $\varphi \in L^1_w$, where the limit is in $L^1_w$ equipped with the weighted norm $$ \|\varphi \|_w:=\sum_{i=0}^{N+2}\int_0^1\varphi_i(x)w_i(x)\,dx. $$ \end{theorem} \begin{proof} Since, by Theorem \ref{boundary-spectrum}, $s(L_\Phi)$ is a first order pole of the resolvent, it follows from \cite[Proposition 3.5.C-III]{Nagel} that there is a strictly positive eigenvector $w$ of $L_\Phi^\ast$ corresponding to $s(L_\Phi)$. Hence, $e^{-s(L_\Phi)t}S_\Phi(t)^\ast w=w$ and therefore $$ \|e^{-s(L_\Phi)t}S_\Phi(t)\|_w\le 1 \quad \text{\ for all }t\ge 0. $$ On the other hand, we know from Theorem \ref{boundary-spectrum}, Remark \ref{rem4.4} and Proposition \ref{lem4.1} that $s(L_\Phi)\in \sigma_p(L_\Phi)$ and $S_\Phi(\cdot)$ is irreducible. So, we deduce that the set $\{e^{-s(L_\Phi)t}S_\Phi(t): t\ge 0\}$ is relatively weakly compact in $L^1_w$ (cf. \cite[Lemma 3.10]{Haji-Radl}). Now, the assertion follows as in \cite[Theorem 3.11]{Haji-Radl}. \end{proof} \begin{thebibliography}{00} \bibitem{Boulanouar00} M. Boulanouar; \emph{Une \'etude Math\'ematique d'un modele de Rotenberg}, J. Math. Anal. Appl. {\bf 79} (2000), 1029-1055. \bibitem{Boulanouar01} M. Boulanouar; \emph{A mathematical study in the theory of dynamic population}, J. Math. Anal. Appl. {\bf 255} (2001), 230-259. \bibitem{Engel-Nagel} K. J. Engel, R. 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