\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 131, pp. 1--21.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/131\hfil Null controllability] {Null controllability of population dynamics with interior degeneracy} \author[I. Boutaayamou, Y. Echarroudi \hfil EJDE-2017/131\hfilneg] {Idriss Boutaayamou, Younes Echarroudi} \address{Idriss Boutaayamou \newline Facult\'e polydisciplinaire de Ouarzazate, Universit\'e Ibn Zohr, B.P. 638, Ouarzazate 45000, Morocco} \email{dsboutaayamou@gmail.com} \address{Younes Echarroudi \newline Private University of Marrakesh, Km 13 Route d'Amizmiz, Marrakesh, Morocco} \email{yecharroudi@gmail.com, y.echarroudi@upm.ac.ma} \dedicatory{Communicated by Jerome A. Goldstein} \thanks{Submitted January 9, 2017. Published May 12, 2017.} \subjclass[2010]{35K65, 92D25, 93B05, 93B07} \keywords{Population dynamics model; interior degeneracy; Carleman estimate; \hfill\break\indent observability inequality; null controllability} \begin{abstract} In this article, we study the null controllability of population model with an interior degenerate diffusion. To this end, we proved first a new Carleman estimate for the full adjoint system and then we deduce a suitable observability inequality which will be needed to establish the existence of a control acting on a subset of the space which lead the population to extinction in a finite time. \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}\label{first-section} Consider the system \begin{equation}\label{405} \begin{gathered} \frac{\partial y}{\partial t} + \frac{\partial y}{\partial a}-(k(x)y_{x})_{x} +\mu(t,a, x)y = \vartheta\chi_{\omega} \quad \text{in } Q,\\ y(t,a, 1)=y(t,a, 0)=0 \quad\text{on }(0,T)\times(0,A),\\ y(0, a, x)=y_0(a, x) \quad\text{in }Q_{A},\\ y(t, 0, x)=\int_0^{A} \beta (t,a, x)y (t,a, x)da \quad\text{in } Q_{T}, \end{gathered} \end{equation} where $Q=(0,T)\times(0,A)\times(0,1)$, $Q_{A}=(0,A)\times(0,1)$, $Q_{T}=(0,T)\times(0,1)$ and we will denote $q=(0,T)\times(0,A)\times \omega$, where $\omega=(x_1, x_2) \subset\subset (0,1)$ is the region where the control $\vartheta$ is acting. This control corresponds to an external supply or to removal of individuals on the subdomain $\omega$. Since \eqref{405} models the dispersion of gene of a given population, $x$ represents the gene type and $y(t,a,x)$ is the distribution of individuals of age $a$ at time $t$ and of gene type $x$. The parameters $\beta(t,a,x)$ and $\mu(t,a,x)$ are respectively the natural fertility and mortality rates, $A$ is the maximal age of life and $k$ is the gene dispersion coefficient. $y_0\in L^2(Q_{A})$ is the initial distribution of population. Finally, $\int_0^{A} \beta (t,a, x)y (t,a, x)da $ is the distribution of the newborns of the population that are of gene type $x$ at time $t$. As usual, we will suppose that no individual reaches the maximal age $A$. We note that in the most works concerned with the diffusion population dynamics models, $x$ is viewed as the space variable. The population models in their different aspects attracted many authors that investigated them from many sides (see for example \cite{genni7, genni6, genni4, genni5, Pia, Poz}). Among those questions, we find the null controllability problem or in general the controllability problems for age and space structured population dynamics models which were studied in a intensive literature basing, in general, on the references interested on the controllability of heat equation (see for instance \cite{Can2, Can3, Can4, Can5, Can6, genni3} for a different controllability problems of heat equation). In this context, we can cite the pioneering items of Barbu and al.\ \cite{Marcheva}, Ainseba and Anita \cite{Ain4, Ain3, Ain2, Ain1}. In \cite{Marcheva}, the authors proved the null controllability for a population dynamics model without diffusion both in the cases of migration and birth control for $T\geq A$ showing directly an appropriate observability inequality for the associated adjoint system and they concluded that in the case of the migration control, only a classes of age was controlled in contrary with the birth control which allows to steer all population to extinction. In \cite{Ain4, Ain3, Ain2, Ain1}, the diffusion was taken into account in a age-space structured model and the null controllability of \eqref{405} for classes of age was established in the case where $k=1$ and for any dimension $n$ by means of a weighted estimates called Carleman estimates and exploiting the results gotten for heat equation in \cite{Fursikov}. Ainseba et al.\ \cite{ech} studied a more general case allowing the dispersion coefficient to depend on the variable $x$ and satisfies $k(0)=0$ (i.e, the coefficient of dispersion $k$ degenerates at 0). The authors tried to obtain \eqref{null-contr} in such a situation with $\beta \in L^{\infty}$ basing on the work done in \cite{Bouss} for the degenerate heat equation to establish a new Carleman estimate for the full adjoint system \eqref{adj-sys} and afterwards his observability inequality. However, the null controllability property of this paper was showed under the condition $T\geq A$ (as in \cite{Marcheva}) and this constitutes a restrictiveness on the ``optimality'' of the control time $T$ since it means, for example, that for a pest population whose the maximal age $A$ may equal to a many days (may be many months or years) we need much time to bring the population to the zero equilibrium. In the same trend and to overcome the condition $T\geq A$, Maniar et al.\ \cite{man} suggested the fixed point technique in which the birth rate $\beta$ must be in $C^2(Q)$ specially in the proof of \cite[Proposition 4.2]{man}. Such a technique consists briefly to demonstrate in a first time the null controllability for an intermediate system with a fertility function $b\in L^2(Q_{T})$ instead of $ \int_0^{A} \beta (t,a, x)y (t,a, x)da $ and to achieve the task via a Leray-Schauder Theorem. Thereby, the main goal of this article is to sutdy the null controllability property with a minimum of regularity of $\beta$ (see \eqref{hyp-beta-mu}) and a positive small control time $T$ taking into account that $k$ depends on the gene type and degenerates at a point $x_0\in\omega$, i.e $k(x_0)=0$, e.g $k(x)=|x-x_0|^{\alpha}$. To be more accurate, for a fixed $T\in (0, \delta)$ with $\delta\in(0,A)$ small enough, we investigate the existence of a suitable control $\vartheta\in L^2(q)$ which depends on $y_0$ and $\delta$ and such that the associated solution $y$ of \eqref{405} satisfies \begin{equation}\label{null-contr} y(T,a, x)=0, \quad \text{a.e. in } (\delta, A)\times(0,1). \end{equation} If $k(x_0)=0$ in a point $x_0\in\omega$, we say that \eqref{405} is a population dynamics model with interior degeneracy. Genetically speaking, the meaning of $k(x_0)=0$ is that the gene of type $x_0\in(0,1)$ can not be transmitted from the studied population to its offspring. This objective will be attained via the classical procedure following the strategy in \cite{genni}. On other words, we will establish an appropriate observability inequality for the full adjoint system of \eqref{405} which is an outcome of a suitable Carleman estimate. We highlight that such a result can be shown if we replace the homogeneous Dirichlet boundary conditions by the ones of Neumann, i.e., $y_{x}(t,a,0)=y_{x}(t,a,1)=0, (t,a)\in(0,T)\times(0,A)$ using the same way done in \cite{bout}. Another interesting null controllability problem of \eqref{405} can be elaborate using the work of Fragnelli et al.\ \cite{genni2} arising in the case when the potential term admits an interior singularity belonging to gene type domain. The remainder of this paper is organized as follows: in Section \ref{second-section}, we will provide the well-posedness of \eqref{405} and give the proof of the Carleman estimate of its adjoint system. The Section \ref{third-section} will be devoted to the observability inequality and hence we obtain the null controllability result \eqref{null-contr}. The last section will take the form of an appendix where we will bring out a Caccioppoli's inequality which plays an important role to show the desired Carleman estimate. \section{Well-posedness and Carleman estimate results}\label{second-section} \subsection{Well-posedness} For this section and for the sequel, we assume that the dispersion coefficient $k$ satisfies \begin{equation}\label{hyp-k} \begin{gathered} \exists x_0 \in (0,1),\; k \in C([0,1])\cap C^{1}([0,1]\backslash \{x_0\}),\; k>0 \text{ in }[0,1]\backslash \{x_0\} \text{ and } k(x_0)=0,\\ \exists \gamma \in [0,1): (x-x_0)k'(x)\leq\gamma k(x), \; x \in [0,1]\backslash \{x_0\}. \end{gathered} \end{equation} It is well-known in the literature of degenerate problems that there exist two kinds of degeneracy namely the weakly degenerate and the strong degenerate problems, in our study we will restrict ourselves to the first one and this fact explains the choice of $\gamma\in [0,1)$ which in fact are associated to the Dirichlet boundary conditions (see \cite[Hypothesis 1.1]{genni}). On the other hand, the last hypothesis on $k$ means in the case of $k(x)=|x-x_0|^\alpha$ that $0\leq \alpha<1$. The investigation of \eqref{null-contr} needs also the following assumptions on the natural rates $\beta$ and $\mu$: \begin{equation}\label{hyp-beta-mu} \begin{gathered} \mu, \beta \in L^{\infty}(Q),\quad \beta(t,a,x), \mu(t,a,x)\geq0, \quad \text{a.e. in }Q,\\ \beta(\cdot,0,\cdot)\equiv0 \quad \text{a.e. in } (0,T)\times(0,1). \end{gathered} \end{equation} The last assumption in \eqref{hyp-beta-mu} is natural since the newborns are not fertile. Also, it is worth mentioning to point out that, as in \cite{ech} we do not need to require that $\mu$ satisfies an hypotheses like $\int_0^{A}\mu(t-s,A-s,x)ds=+\infty, \quad (t,x) \in [0,T]\times[0,1]$ since it does not play any role on the well-posedness result and the computations concerning the proofs of our controllability result as well. However, we will suppose that no individual can reach the maximal age $A$ as mentioned in the introduction. In the same context, we emphasize that in \cite{man}, the $L^{\infty}-$regularity of $\beta$ is sufficient to prove the well posedness of the studied model which is exactly our case. To this end, we introduce the following weighted Sobolev spaces: %\label{Sob-spaces} \begin{gather*} \begin{aligned} H^{1}_{k}(0, 1):=\big\{&u \in L^2(0,1): u \text{ is abs. cont. on } [0,1]:\\ &\quad \sqrt{k}u_{x} \in L^2(0,1), u(1)=u(0)=0\big\}, \end{aligned} \\ H^2_{k} (0, 1):=\big\{ u \in H^1_{k}(0, 1) : k(x)u_x \in H^1(0, 1)\big\}, \end{gather*} endowed respectively with the norms \begin{gather*} %\label{norme} \|u\|^2_{H^{1}_{k}(0, 1)} :=\|u\|^2_{L^2(0,1)}+ \|\sqrt{k}u_{x}\|^2_{L^2(0,1)}, \quad u \in H^{1}_{k}(0, 1),\\ \|u\|^2_{H^2_{k}} := \|u\|^2 _{H^1_{k}(0, 1)} + \|(k(x)u_x)_x\|^2_{ L^2(0,1)}, \quad u \in H^2_{k}(0, 1). \end{gather*} We recall from \cite[Theorem 2.2]{genni1} that the operator $(P,D(P))$ defined by $Pu := (k(x)u_x)_x$, $u \in D(P) = H^2_{k}(0, 1)$, is closed self-adjoint and negative with dense domain in $L^2(0, 1)$. Consequently, from \cite[Theorem 5]{web} the operator $\mathcal{A}:=-\frac{\partial }{\partial a}+P$ generates a $C_0$-semigroup on the space $L^2((0, A)\times(0, 1))$. Then, the following well-posedness result holds. \begin{theorem}\label{410} Under assumptions \eqref{hyp-k} and \eqref{hyp-beta-mu}, for all $\vartheta\ \in L^2(Q)$ and $y_0$ in $L^2(Q_{A})$, system \eqref{405} admits a unique solution $y$. This solution belongs to \begin{align*} E:=&C([0, T], L^2((0, A)\times(0, 1)))\cap C([0, A], L^2((0, T)\times(0, 1)))\\ &\cap L^2((0, T)\times(0, A), H^{1}_{k}(0, 1)). \end{align*} Moreover, the solution of \eqref{405} satisfies \begin{align*} %\label{well-posedness} &\sup_{t\in [0,T]}\|y(t)\|^2_{L^2(Q_{A})} + \sup_{a\in [0,A]}\|y(a)\|^2_{L^2(Q_{T})} +\int_0^1\int_0^{A}\int_0^{T} (\sqrt{k(x)}y_x)^2\,dt\,da\,dx\\ &\leq C\Big(\int_{q}\vartheta^2+\int_{Q_{A}}y_0^2\,dx\,dx\Big). \end{align*} \end{theorem} \subsection{Carleman estimates} As we said in the introduction, we will show the main key of this paper namely the Carleman type inequality. In general, it is well-known that to prove a controllability result of a studied model through this a priori estimate, we must show this last for the associated adjoint system. In our case, this adjoint system takes the form \begin{equation} \label{adj-sys} \begin{gathered} \frac{\partial w}{\partial t} + \frac{\partial w}{\partial a} +(k(x)w_{x})_{x}-\mu(t,a, x)w =-\beta(t,a,x)w(t,0,x),\\ w(t,a, 1)=w(t,a, 0)=0, \\ w(T,a, x)=w_{T}(a, x),\\ w(t,A, x)=0, \end{gathered} \end{equation} where $T>0$ and assume that $w_{T} \in L^2(Q_{A})$. Of course, assumptions \eqref{hyp-k} and \eqref{hyp-beta-mu} on $k$, $\mu$ and $\beta$ are perpetuated. To attaint our goal, we will introduce the weight functions \begin{equation}\label{weight-functions} \begin{gathered} \varphi(t,a, x):=\Theta (t, a)\psi(x),\\ \Theta (t, a):= \frac{1}{(t(T-t))^{4}a^{4}},\\ \psi(x):= c_1\Big(\int_{x_0}^{x}\frac{r-x_0}{k(r)} dr-c_{2}\Big). \end{gathered} \end{equation} For the moment, we will assume that $c_{2}>\max\{\frac{(1-x_0)^2}{k(1)(2-\gamma)}, \frac{x_0^2}{k(0)(2-\gamma)} \}$ and $c_1>0$. A more precise restriction on $c_1$ will be given later. On the other hand, using the relation satisfied by $c_{2}$ and with the aid of \cite[Lemma 2.1]{genni} one can prove that $\psi(x)<0$ for all $x\in [0,1]$. Observe also that $\Theta(a,t)\to +\infty$ as $t\to T^{-}, 0^{+}$ and $a\to 0^{+}$. To demonstrate our Carleman estimate, we require that $k$ fulfills, besides \eqref{hyp-k} the following hypothesis \begin{equation}\label{H-P} \begin{gathered} \exists \theta\in (0, \gamma] \text{ such that } x\mapsto\frac{k(x)}{|x-x_0|^{\theta}} \text{ is non-increasing on the left of }\\ x=x_0 \text{ and nondecreasing on the right of } x=x_0, \end{gathered} \end{equation} where $\gamma$ is defined by \eqref{hyp-k}. The first Carleman estimate result is the following result. \begin{proposition}\label{Intermediate-Carleman estimate} Consider the following two systems with $h \in L^2(Q)$, \begin{equation}\label{inter-adjoint sys-1} \begin{gathered} \frac{\partial w}{\partial t} + \frac{\partial w}{\partial a} +(k(x)w_{x})_{x} = h,\\ w(a, t, 1)=w(a, t, 0)=0, \\ w(T, a, x)=w_{T}(a, x),\\ w(t, A, x)=0, \end{gathered} \end{equation} and \begin{equation}\label{inter-adjoint sys-2} \begin{gathered} \frac{\partial w}{\partial t} + \frac{\partial w}{\partial a} +(k(x)w_{x})_{x}-\mu(t, a, x)w = h,\\ w(t, a, 1)=w(t, a, 0)=0, \\ w(T, a, x)= w_{T}(a, x),\\ w(t, A, x)= 0. \end{gathered} \end{equation} Then, there exist two positive constants $C$ and $s_0$, such that every solution of \eqref{inter-adjoint sys-1} and \eqref{inter-adjoint sys-2} satisfy, for all $s\geq s_0$, the inequality \begin{equation}\label{Inter-Carl} \begin{aligned} & s^{3}\int_{Q} \Theta^{3}\frac{(x-x_0)^2}{k(x)}w^2e^{2s\varphi} \,dt\,da\,dx + s\int_{Q} \Theta k(x)w_{x}^2e^{2s\varphi} \,dt\,da\,dx \\ &\leq C\Big(\int_{Q}\mid h\mid^2e^{2s\varphi} \,dt\,da\,dx +s \int_0^{A}\int_0^{T}[k\Theta e^{2s\varphi} (x-x_0)w_{x}^2]_{x=0}^{x=1} \,dt\,da\Big). \end{aligned} \end{equation} \end{proposition} \begin{proof} Firstly, we prove \eqref{Inter-Carl} for system \eqref{inter-adjoint sys-1} and replacing $h$ by $h+\mu w$ we will get the same inequality for \eqref{inter-adjoint sys-2}. So, let $w$ be the solution of \eqref{inter-adjoint sys-1} and put $$ \nu(t, a, x):=e^{s\varphi(t, a, x)}w(t, a, x). $$ Then, $\nu$ satisfies the system \begin{equation}\label{Carl-1} \begin{gathered} L^{+}_{s}\nu + L^{-}_{s}\nu = e^{s\varphi(t, a, x)}h, \\ \nu(t, a, 1)=\nu(t, a, 0) = 0, \\ \nu(T, a, x)=\nu(0, a, x) = 0,\\ \nu(t, A, x)=\nu(t, 0, x) = 0, \end{gathered} \end{equation} where \begin{gather*} L^{+}_{s}\nu:=(k(x)\nu_{x})_{x}-s(\varphi_{a}+\varphi_{t})\nu+s^2 \varphi^2_{x}k(x)\nu,\\ L^{-}_{s}\nu:=\nu_{t}+\nu_{a}-2sk(x)\varphi_{x}\nu_{x}-s(k(x)\varphi_{x})_{x}\nu. \end{gather*} Passing to the norm in \eqref{Carl-1}, one has $$ \|L^{+}_{s}\nu\|^2_{L^2(Q)} + \|L^{-}_{s}\nu\|^2_{L^2(Q)} + 2 \langle L^{+}_{s}\nu,L^{-}_{s}\nu\rangle = \|e^{s\varphi(a, t, x)}h\|^2_{L^2(Q)}, $$ where $\langle\cdot,\cdot\rangle$ denotes here the inner product in $L^2(Q)$. Then, the proof of step one is based on the calculus of the inner product $\langle L^{+}_{s}\nu,L^{-}_{s}\nu\rangle $ whose a first expression is given in the following lemma. \begin{lemma}\label{Inter-Carl-1} The following identity holds $$ \langle L^{+}_{s}\nu,L^{-}_{s}\nu\rangle =S_1+S_{2}, $$ with \begin{equation} \label{20} \begin{aligned} S_1&=s\int_{Q}(k(x)\nu_{x})^2\varphi_{xx}\,dt\,da\,dx -s^{3}\int_{Q}(k(x)\varphi_{x})_{x}k(x)\varphi^2_{x}\nu^2\,dt\,da\,dx \\ &\quad +s^2\int_{Q}(\varphi_{a}+\varphi_{t})(k(x)\varphi_{x})_{x}\nu^2\,dt\,da\,dx \\ &\quad +s\int_{Q}k(x)\nu_{x}((k(x)\varphi_{x})_{xx}\nu+(k(x)\varphi_{x})_{x}\nu_{x}) \ ,dt\,da\,dx \\ &\quad +s^{3}\int_{Q}(k^2\varphi^{3}_{x})_{x}\nu^2\,dt\,da\,dx -s^2\int_{Q}(k(x)(\varphi_{a}+\varphi_{t})\varphi_{x})_{x}\nu^2\,dt\,da\,dx \\ &\quad +\frac{s}{2}\int_{Q}(\varphi_{at}+\varphi_{tt})\nu^2\,dt\,da\,dx -\frac{s^2}{2}\int_{Q}(\varphi^2_{x})_{t}k(x)\nu^2\,dt\,da\,dx\\ &\quad + \frac{s}{2}\int_{Q}(\varphi_{at}+\varphi_{aa})\nu^2\,dt\,da\,dx -\frac{s^2}{2}\int_{Q}(\varphi^2_{x})_{a}k(x)\nu^2\,dt\,da\,dx, \end{aligned} \end{equation} and \begin{equation} \label{21} \begin{aligned} &S_{2} \\ &=\int_0^{A}\int_0^{T}[k(x)\nu_{x}\nu_{a}]_0^{1}\,dt\,da +\int_0^{A}\int_0^{T}[k(x)\nu_{x}\nu_{t}]_0^{1}\,dt\,da \\ &\quad + s^2\int_0^{A}\int_0^{T}[k(x)\varphi_{x}(\varphi_{a} +\varphi_{t})\nu^2]_0^{1}\,dt\,da -s^{3}\int_0^{A}\int_0^{T}[k^2(x)\varphi^{3}_{x}\nu^2]_0^{1}\,dt\,da \\ &\quad -s\int_0^{A}\int_0^{T}[k(x)\nu\nu_{x}(k(x)\varphi_{x})_{x}]_0^{1}\,dt\,da -s\int_0^{A}\int_0^{T}[(k(x)\nu_{x})^2\varphi_{x}]_0^{1}\,dt\,da. \end{aligned} \end{equation} \end{lemma} For the proof of Lemma \ref{Inter-Carl-1}, see \cite[Lemma 3.2]{man}. The previous expressions of $S_1$ and $S_{2}$ can be simplified using the functions $\varphi$ and $\psi$ given in \eqref{weight-functions} and also the homogeneous Dirichlet boundary conditions satisfied by $\nu$. Hence, one has \begin{equation}\label{S1} \begin{aligned} S_1&=\frac{s}{2}\int_{Q}(\Theta_{aa}+\Theta_{tt})\psi\nu^2dx\,dt\,da +s\int_{Q}\Theta_{ta}\psi\nu^2\,dt\,da\,dx \\ &\quad +sc_1\int_{Q}\Theta (2k(x)-(x-x_0)k'(x))\nu_{x}^2\,dt\,da\,dx \\ &\quad -2s^2\int_{Q}\Theta c_1^2\frac{(x-x_0)^2}{k(x)} (\Theta_{a}+\Theta_{t})\nu^2\,dt\,da\,dx\\ &\quad +s^{3}\int_{Q}\Theta^{3}c_1^{3}\Big(\frac{x-x_0}{k(x)})^2(2k(x) -(x-x_0)k'(x)\Big)\nu^2\,dt\,da\,dx, \end{aligned} \end{equation} and $$ S_{2}=-sc_1\int_0^{A}\int_0^{T}[k\Theta e^{2s\varphi} (x-x_0)\nu_{x}^2]_{x=0}^{x=1} \,dt\,da. $$ Accordingly, \begin{equation}\label{scalar-product} \begin{aligned} &\langle L^{+}_{s}\nu,L^{-}_{s}\nu\rangle \\ &= \frac{s}{2}\int_{Q}(\Theta_{aa}+\Theta_{tt})\psi\nu^2dx\,dt\,da +s\int_{Q}\Theta_{ta}\psi\nu^2\,dt\,da\,dx \\ &\quad +sc_1\int_{Q}\Theta (2k(x)-(x-x_0)k'(x))\nu_{x}^2\,dt\,da\,dx \\ &\quad -2s^2\int_{Q}\Theta c_1^2\frac{(x-x_0)^2}{k(x)}(\Theta_{a} +\Theta_{t})\nu^2\,dt\,da\,dx\\ &\quad + s^{3}\int_{Q}\Theta^{3}c_1^{3}\big(\frac{x-x_0}{k(x)}\big)^2 (2k(x)-(x-x_0)k'(x))\nu^2\,dt\,da\,dx \\ &\quad -sc_1\int_0^{A}\int_0^{T}[k\Theta e^{2s\varphi} (x-x_0)w_{x}^2]_{x=0}^{x=1} \,dt\,da. \end{aligned} \end{equation} Thanks to the third assumption in \eqref{hyp-k}, we have \begin{equation}\label{S1-1} \begin{aligned} S_1&\geq\frac{s}{2}\int_{Q}(\Theta_{aa}+\Theta_{tt})\psi\nu^2\,dt\,da\,dx +s\int_{Q}\Theta_{ta}\psi\nu^2\,dt\,da\,dx \\ &\quad +sc_1\int_{Q}\Theta k(x)\nu_{x}^2\,dt\,da\,dx \\ &\quad -2s^2\int_{Q}\Theta c_1^2\frac{(x-x_0)^2}{k(x)}(\Theta_{a} +\Theta_{t})\nu^2\,dt\,da\,dx \\ &\quad + s^{3}\int_{Q}\Theta^{3}c_1^{3}\frac{(x-x_0)^2}{k(x)}\nu^2\,dt\,da\,dx. \end{aligned} \end{equation} Observe that $|\Theta(\Theta_{a}+\Theta_{t})|\leq c\Theta^{3}$; for $s$ large we infer that \begin{equation}\label{S1-2} \begin{aligned} &\big|-2s^2\int_{Q}\Theta c_1^2\frac{(x-x_0)^2}{k(x)}(\Theta_{a} +\Theta_{t})\nu^2\,dt\,da\,dx \big|\\ & \leq 2s^2c_1^2c\int_{Q}\frac{(x-x_0)^2}{k(x)}\Theta^{3}\nu^2\,dt\,da\,dx\\ &\leq \frac{c_1^{3}}{4}s^{3}\int_{Q}\frac{(x-x_0)^2}{k(x)}\Theta^{3}\nu^2 \,dt\,da\,dx. \end{aligned} \end{equation} On the other hand, the mapping $r\mapsto\frac{|r-x_0|^{\gamma}}{k(r)}$ is nondecreasing at the right of $x_0$. Then \begin{equation}\label{S1-3} \begin{aligned} |\psi(x)|&=|c_1l(x)-c_1c_{2}| \leq c_1\left|\int_{x_0}^{x}\frac{r-x_0}{k(r)}dr\right|+c_1c_{2} \\ &\leq c_1c_{2}+c_1\frac{(1-x_0)^2}{k(1)(2-\gamma)} \leq \frac{c_1}{(2-\gamma)k(1)}+c_1c_{2}. \end{aligned} \end{equation} A simple computations shows that $\Theta_{aa}+\Theta_{tt}+|\Theta_{ta}|\leq C_1 \Theta^{3/2}$. This yields \begin{equation}\label{S1-4} \begin{aligned} &\big|\frac{s}{2}\int_{Q}(\Theta_{aa}+\Theta_{tt})\psi\nu^2\,dt\,da\,dx +s\int_{Q}\Theta_{ta}\psi\nu^2\,dt\,da\,dx\big| \\ &\leq s\Big(\frac{c_1}{(2-\gamma)k(1)}+c_1c_{2}\Big) \int_{Q}\Big(\frac{\Theta_{aa}+\Theta_{tt}}{2}+|\Theta_{ta}|\Big)\nu^2\,dt\,da\,dx\\ &\leq M s\Big(\frac{c_1}{(2-\gamma)k(1)}+c_1c_{2}\Big) \int_{Q}\Theta^{3/2}\nu^2\,dt\,da\,dx. \end{aligned} \end{equation} It remains now to bound the term $|\int_{Q}\Theta^{3/2}\nu^2\,dt\,da\,dx|$. Using the generalized Young inequality we obtain \begin{equation}\label{S1-5} \begin{aligned} \big|\int_0^{1}\Theta^{3/2}\nu^2dx\big| &= \Big|\int_0^{1}\Big(\Theta\frac{k^{1/3}}{|x-x_0|^{\frac{2}{3}}}\nu^2 \Big)^{3/4} \Big(\Theta^{3}\frac{|x-x_0|^2}{k}\nu^2\Big)^{1/4}dx\Big|\\ &\leq\frac{3\epsilon}{4}\int_0^{1}\Theta \frac{k^{1/3}}{|x-x_0|^{\frac{2}{3}}}\nu^2dx +\frac{1}{4\epsilon}\int_0^{1}\Theta^{3}\frac{|x-x_0|^2}{k}\nu^2dx \end{aligned} \end{equation} Put \begin{equation}\label{p} p(x)=(k(x)|x-x_0|^{4})^{1/3}. \end{equation} By hypothesis \eqref{H-P}, one can check that $x\mapsto\frac{p(x)}{|x-x_0|^{q}}$, with $q:=\frac{4+\theta}{3}\in (1,2)$ is nonincreasing on the left of $x=x_0$ and nondecreasing on the right of $x=x_0$. Furthermore, we have $\frac{k^{1/3}}{|x-x_0|^{\frac{2}{3}}}=\frac{p(x)}{(x-x_0)^2}$ and there exists $C_{2}>0$ such that $p(x)0$ is the constant of Hardy-Poincar\'e. Combining \eqref{S1-5} and \eqref{S1-6}, we obtain \begin{equation}\label{S1-7} \big|\int_0^{1}\Theta^{3/2}\nu^2dx\big| \leq \frac{3\epsilon}{4}C_{3}\int_{Q}\Theta k \nu_{x}^2\,dt\,da\,dx +\frac{1}{4\epsilon}\int_{Q}\Theta^{3}\frac{|x-x_0|^2}{k}\nu^2\,dt\,da\,dx. \end{equation} Hence, \eqref{S1-4} and \eqref{S1-7} lead to \begin{equation}\label{S1-8} \begin{aligned} &\big|\frac{s}{2}\int_{Q}(\Theta_{aa}+\Theta_{tt})\psi\nu^2\,dt\,da\,dx +s\int_{Q}\Theta_{ta}\psi\nu^2\,dt\,da\,dx\big| \\ &\leq sc_1C_{4}\epsilon\int_{Q}\Theta k \nu_{x}^2\,dt\,da\,dx +\frac{sc_1C_{5}}{4\epsilon}\int_{Q}\Theta^{3}\frac{|x-x_0|^2}{k} \nu^2\,dt\,da\,dx. \end{aligned} \end{equation} Taking $\epsilon$ small enough and $s$ large, we conclude that \begin{equation}\label{S1-9} \begin{aligned} &\big|\frac{s}{2}\int_{Q}(\Theta_{aa}+\Theta_{tt})\psi\nu^2\,dt\,da\,dx +s\int_{Q}\Theta_{ta}\psi\nu^2\,dt\,da\,dx\big| \\ &\leq\frac{sc_1}{4}\int_{Q}\Theta k \nu_{x}^2\,dt\,da\,dx +\frac{s^{3}c_1^{3}}{4}\int_{Q}\Theta^{3}\frac{|x-x_0|^2}{k}\nu^2\,dt\,da\,dx. \end{aligned} \end{equation} Taking into account relations \eqref{S1-1} and \eqref{S1-2} we arrive at \begin{equation}\label{S1-10} S_1\geq K_1s^{3}\int_{Q}\Theta^{3}\frac{|x-x_0|^2}{k}\nu^2\,dt\,da\,dx +K_{2}s\int_{Q}\Theta k \nu_{x}^2\,dt\,da\,dx \end{equation} Hence, \begin{equation}\label{S1-11} \begin{aligned} 2\langle L^{+}_{s}\nu,L^{-}_{s}\nu\rangle &\geq m\Big(s^{3}\int_{Q}\Theta^{3}\frac{|x-x_0|^2}{k}\nu^2\,dt\,da\,dx +s\int_{Q}\Theta k \nu_{x}^2\,dt\,da\,dx\Big)\\ &\quad -2sc_1\int_0^{A}\int_0^{T}[k\Theta e^{2s\varphi} (x-x_0)\nu_{x}^2]_{x=0}^{x=1} \,dt\,da \end{aligned} \end{equation} This yields to the following Carleman estimate satisfied by the solution $\nu$ of \eqref{Carl-1}, \begin{equation}\label{S1-12} \begin{aligned} &s^{3}\int_{Q}\Theta^{3}\frac{|x-x_0|^2}{k}\nu^2\,dt\,da\,dx +s\int_{Q}\Theta k \nu_{x}^2\,dt\,da\,dx \\ &\leq C_{6}\Big(\int_{Q}h^2e^{2s\varphi}\,dt\,da\,dx+s\int_0^{A} \int_0^{T}[k\Theta e^{2s\varphi} (x-x_0)\nu_{x}^2]_{x=0}^{x=1} \,dt\,da\Big) \end{aligned} \end{equation} By the definition of $\nu$ we infer that \begin{equation}\label{S1-13} \nu_{x}=s\varphi_{x}e^{s\varphi}w+e^{s\varphi}w_{x}, \quad e^{2s\varphi}w_{x}^2\leq2(\nu_{x}^2+s^2\varphi_{x}^2\nu^2). \end{equation} Finally, the Carleman estimate \eqref{Inter-Carl} of \eqref{inter-adjoint sys-1} is obtained. Now, If we apply the same inequality of Hardy-Poincar\'e in a similar way as before to the function $\nu:=e^{s\varphi}w$, taking into account the hypothesis on $\mu$ assumed in \eqref{hyp-beta-mu}, using the Carleman type inequality \eqref{Inter-Carl} for the function $h+\mu w$ and taking $s$ quite we achieve the Proposition \ref{Intermediate-Carleman estimate}. \end{proof} With the aid of the estimate \eqref{Inter-Carl} and Caccioppoli's inequality \eqref{Caccio}, we can now show a $\omega$-local Carleman estimate for the system \eqref{inter-adjoint sys-2}. This result will be useful to show our main Carleman estimate replacing the second term $h$ by $-\beta (t,a,x)w(t,0,x)$. To this end, we introduce the weight functions \begin{equation}\label{poids} \begin{gathered} \Phi(t,a, x):=\Theta (t, a)\Psi(x), \\ \Psi(x)=e^{\kappa\sigma(x)}-e^{2\kappa\|\sigma\|_{\infty}}, \end{gathered} \end{equation} where $\Theta$ is given by \eqref{weight-functions}, $\kappa>0$, and $\sigma$ is the function given by \begin{equation}\label{sigma-2} \begin{gathered} \sigma \in C^2([0,1]),\quad \sigma(x)>0 \quad \text{in } (0,1), \quad \sigma(0)=\sigma(1)=0, \\ \sigma_{x}(x)\neq0 \quad \text{in } [0,1]\backslash \omega_0, \end{gathered} \end{equation} where $ \omega_0\Subset\omega $ is an open subset. The existence of the function $\sigma$ is proved in \cite{Fursikov}. On the other hand by the definition of $\varphi$ \eqref{weight-functions} and taking \begin{equation}\label{c-1-restr-0} c_1\geq \max\Big(\frac{k(1)(2-\gamma)(e^{2\kappa\|\sigma\|_{\infty}}-1)} {c_{2}k(1)(2-\gamma)-(1-x_0)^2}, \frac{k(0)(2-\gamma)(e^{2\kappa\|\sigma\|_{\infty}}-1)} {c_{2}k(0)(2-\gamma)-x_0^2}\Big), \end{equation} one can prove that \begin{equation}\label{c-1-restr} \varphi\leq \Phi. \end{equation} Our main theorem is stated as follows. \begin{theorem}\label{Carle-estimate} Assume that assumptions \eqref{hyp-k}, \eqref{hyp-beta-mu} and \eqref{H-P} hold. Let $A>0$ and $T>0$ be given. Then, there exist positive constants $C$ and $s_0$ such that for all $s\geq s_0$, every solution $w$ of \eqref{inter-adjoint sys-2} satisfies \begin{equation} \label{Carle-esti} \begin{aligned} &\int_{Q}(s\Theta k w_{x}^2+s^{3}\Theta^{3}\frac{(x-x_0)^2}{k} w^2)e^{2s\varphi} \,dt \,da\,dx \\ &\leq C\left(\int_{Q}h^2e^{2s\Phi} \,dt\,da\,dx + \int_{q}s^{3}\Theta^{3}w^2e^{2s\Phi}\,dt\,da\,dx\right), \end{aligned} \end{equation} \end{theorem} To prove this theorem, we need the following result which represents the Carleman estimate of nondegenerate population dynamics systems. The inequality is stated as follows. \begin{proposition}\label{Carl-nondegenerate} Let us consider the system \begin{equation}\label{564} \begin{gathered} \frac{\partial z}{\partial t} + \frac{\partial z}{\partial a}+(k(x)z_{x})_{x}-c(t, a, x)z = h \quad\text{in } Q_{b},\\ z(t, a, b_1)=z(t, a, b_{2})=0 \quad\text{on } (0,T)\times(0,A), \end{gathered} \end{equation} where $Q_{b}:=(0,T)\times(0,A)\times(b_1,b_{2})$, $(b_1,b_{2})\subset[0,x_0)$, or $(b_1,b_{2})\subset(x_0,1]$, $h \in L^2(Q_{b})$, $k \in C^{1}([0,1])$ is a strictly positive function and $c\in L^{\infty}(Q_{b})$. Then, there exist two positive constants $C$ and $s_0$, such that for any $s\geq s_0$, $z$ satisfies \begin{equation}\label{570} \begin{aligned} &\int_{Q_{b}}(s^{3}\phi^{3}z^2+s\phi z_{x}^2)e^{2s\Phi}\,dt\,da\,dx \\ &\leq C \Big(\int_{Q_{b}}h^2e^{2s\Phi}\,dt\,da\,dx+\int_{\omega} \int_0^{A}\int_0^{T}s^{3}\phi^{3}z^2e^{2s\Phi}\,dt\,da\,dx\Big), \end{aligned} \end{equation} where \begin{equation}\label{phi} \phi(t,a,x)=\Theta(t,a)e^{\kappa\sigma(x)}, \end{equation} $\Theta$ and $\Phi$ are defined by \eqref{poids}, and $\sigma$ by \eqref{sigma-2}. \end{proposition} Before giving the proof of Theorem \ref{Carle-estimate}, we note that a similar result was demonstrated in \cite[Lemma 2.1]{Ain3} in the case when $k$ is a positive constant, for any dimension $n$ without the source term $h$ and with the weight function $\Theta(t,a)=\frac{1}{t(T-t)a}$. By careful computations, the same proof can be adapted to \eqref{570} where $k$ is a positive general nondegenerate coefficient, with our weight function $\Theta(t,a)=\frac{1}{t^{4}(T-t)^{4}a^{4}}$ and the source term $h$. \begin{proof}[Proof of Theorem \ref{Carle-estimate}] Let us introduce the smooth cut-off function $\xi:\mathbb{R}\to \mathbb{R}$ defined by \begin{equation}\label{cut-off func-1} \begin{gathered} 0\leq\xi(x)\leq1, \quad x \in [0,1],\\ \xi(x)=1, \quad x \in [\lambda_1, \lambda_{2}],\\ \xi(x)=0, \quad x \in [0,1]\backslash \omega, \end{gathered} \end{equation} where $\lambda_1=\frac{x_1+2x_0}{3}$ and $\lambda_{2}=\frac{x_0+2x_{2}}{3}$. Let $w$ be the solution of \eqref{inter-adjoint sys-2} and define $v:=\xi w$. Then, $v$ satisfies the system \begin{equation}\label{inter-adjoint sys-3} \begin{gathered} \frac{\partial v}{\partial t} + \frac{\partial v}{\partial a} +(k(x)v_{x})_{x}-\mu(t, a, x)v = \overline{h},\\ v(t, a, 1)=v(t, a, 0)=0, \\ v(T, a, x)=\xi w_{T}(a, x),\\ v(t, A, x)=0, \end{gathered} \end{equation} where $\overline{h}:=\xi h+(k(x)\xi_{x}w)_{x}+k(x)w_{x}\xi_{x}$. Using Carleman estimate \eqref{Inter-Carl} and the definition of $\xi$, one has \begin{equation} \label{Carle-esti-1} \int_{Q}(s\Theta k v_{x}^2+s^{3}\Theta^{3}\frac{(x-x_0)^2}{k} v^2)e^{2s\varphi} \,dt\,da\,dx \leq C\int_{Q}\overline{h}^2e^{2s\varphi} \,dt\,da\,dx. \end{equation} On the other hand, using again the definition of $\xi$ we can check readily that \begin{equation}\label{equ-1} \begin{aligned} &\int_{\lambda_1}^{\lambda_{2}}\int_0^{A}\int_0^{T} (s\Theta k v_{x}^2+s^{3}\Theta^{3}\frac{(x-x_0)^2}{k} v^2)e^{2s\varphi} \,dt\,da\,dx\\ &=\int_{\lambda_1}^{\lambda_{2}}\int_0^{A} \int_0^{T}(s\Theta k w_{x}^2+s^{3}\Theta^{3}\frac{(x-x_0)^2}{k} w^2)e^{2s\varphi} \,dt\,da\,dx. \end{aligned} \end{equation} Therefore, combining \eqref{Carle-esti-1} and \eqref{equ-1} we have \begin{equation}\label{ineq-1} \begin{aligned} &\int_{\lambda_1}^{\lambda_{2}}\int_0^{A} \int_0^{T}(s\Theta k w_{x}^2+s^{3}\Theta^{3}\frac{(x-x_0)^2}{k} w^2)e^{2s\varphi} \,dt\,da\,dx \\ &\leq C\int_{Q}\overline{h}^2e^{2s\varphi} \,dt\,da\,dx. \end{aligned} \end{equation} Hence by Caccioppoli's inequality \eqref{Caccio} and \eqref{ineq-1}, we conclude that \begin{equation}\label{ineq-2} \begin{aligned} &\int_{\lambda_1}^{\lambda_{2}}\int_0^{A} \int_0^{T}(s\Theta k w_{x}^2+s^{3}\Theta^{3} \frac{(x-x_0)^2}{k} w^2)e^{2s\varphi} \,dt\,da\,dx\\ &\leq C\Big(\int_{Q}h^2e^{2s\varphi}\,dt\,da\,dx +\int_{q} s^2\Theta^2w^2e^{2s\varphi} \,dt\,da\,dx\Big), \end{aligned} \end{equation} where $\omega'$ of Lemma \ref{Caccio-1} here is exactly $(x_1, \lambda_1)\cup(\lambda_{2}, x_{2})$. Now, let $z:=\eta w$, with $\eta$ is the smooth cut-off function defined by \begin{equation}\label{cut-off func-2} \begin{gathered} 0\leq\eta(x)\leq1, \quad x \in [0,1],\\ \eta(x)=0, \quad x \in [0, \frac{\lambda_{3}+2\lambda_{2}}{3}],\\ \eta(x)=1, \quad x \in [\lambda_{2}, 1], \end{gathered} \end{equation} where $\lambda_{3}=\frac{x_{2}+2x_0}{3}$. We can observe easily that $\lambda_{3}<\frac{\lambda_{3}+2\lambda_{2}}{3}<\lambda_{2}$. Then, $z$ satisfies the population dynamics equation \begin{equation}\label{inter-adjoint sys-4} \begin{gathered} \frac{\partial z}{\partial t} + \frac{\partial z}{\partial a}+k(x)z_{xx}+k'(x)z_{x} -\mu(t, a, x)z = \widetilde{h}, \quad \text{in } (\lambda_{3}, 1)\\ z(t, a, 1)=z(t, a, \lambda_{3})=0, \quad \text{in }, (0,T)\times(0,A), \end{gathered} \end{equation} where $\widetilde{h}:=\eta h+(k(x)\eta_{x}w)_{x}+k(x)w_{x}\eta_{x}$. By assumption on $k$, we have $k(x)>0, \hspace{0.25cm} x\in(\lambda_{3}, 1)$. Hence, \eqref{inter-adjoint sys-4} is a non-degenerate model. In this case, applying Proposition \ref{Carl-nondegenerate} to the function $\widetilde{h}$ with $b_1=\lambda_{3}$ and $b_{2}=1$ and using again Caccioppoli's inequality \eqref{Caccio}, we infer that \begin{equation}\label{431} \begin{aligned} &\int_{\lambda_{3}}^{1}\int_0^{A}\int_0^{T}(s^{3}\phi^{3}z^2 +s\phi z_{x}^2)e^{2s\Phi}\,dt\,da\,dx\\ &\leq C\Big(\int_{\lambda_{3}}^{1}\int_0^{A}\int_0^{T}(\eta h +(k\eta_{x}w)_{x}+k\eta_{x}w_{x})^2e^{2s\Phi} \,dt\,da\,dx \\ &\quad + \int_{\omega}\int_0^{A}\int_0^{T} s^{3}\Theta^{3}w^2e^{2s\Phi} \,dt\,da\,dx\Big)\\ &\leq \widetilde{C}\Big(\int_{Q}h^2e^{2s\Phi}+((k\eta_{x}w)_{x} +k\eta_{x}w_{x})^2e^{2s\Phi} \,dt\,da\,dx \\ &\quad + \int_{\omega}\int_0^{A}\int_0^{T} s^{3}\Theta^{3}w^2e^{2s\Phi} \,dt\,da\,dx\Big) \\ &\leq \widetilde{C}(\int_{Q}h^2e^{2s\Phi}\,dt\,da\,dx +\int_{\omega'}\int_0^{A}\int_0^{T}(8(k\eta_{x})^2w_{x}^2 +2((k\eta_{x})_{x})^2w^2)e^{2s\Phi} \,dt\,da\,dx\\ &\quad + \int_{\omega}\int_0^{A}\int_0^{T}s^{3}\Theta^{3} w^2e^{2s\Phi} \,dt\,da\,dx)\\ & \leq \widetilde{C}_1\Big(\int_{Q}h^2e^{2s\Phi}\,dt\,da\,dx +\int_{\omega'}\int_0^{A}\int_0^{T}(w_{x}^2+w^2)e^{2s\Phi} \,dt\,da\,dx\\ &\quad + \int_{\omega}\int_0^{A}\int_0^{T}s^{3}\Theta^{3} w^2e^{2s\Phi} \,dt\,da\,dx\Big)\\ &\leq \widetilde{C}_{2}\Big(\int_{Q}h^2e^{2s\Phi}\,dt\,da\,dx + \int_{\omega}\int_0^{A}\int_0^{T} s^{3}\Theta^{3}w^2e^{2s\Phi} \,dt\,da\,dx\Big), \end{aligned} \end{equation} with $\omega':=(\frac{\lambda_{3}+2\lambda_{2}}{3}, \lambda_{2})$. By the restriction \eqref{c-1-restr} there exists $c_{3}>0$ such that, for $ (t,a,x)\in [0,T]\times[0,A]\times[\lambda_{3},1]$, we have \begin{equation*} \Theta k(x)e^{2s\varphi}\leq c_{3}\phi e^{2s\Phi},\quad \Theta^{3}\frac{(x-x_0)^2}{k(x)}e^{2s\varphi}\leq c_{3}\phi^{3}e^{2s\Phi}. \end{equation*} Then \begin{equation}\label{ineq-z-1} \begin{aligned} &\int_{\lambda_{3}}^{1}\int_0^{A}\int_0^{T}(s\Theta k z_{x}^2+s^{3}\Theta^{3} \frac{(x-x_0)^2}{k} z^2)e^{2s\varphi} \,dt\,da\,dx\\ &\leq c_{3}\int_{\lambda_{3}}^{1}\int_0^{A}\int_0^{T}(s^{3}\phi^{3}z^2 +s\phi z_{x}^2)e^{2s\Phi}\,dt\,da\,dx. \end{aligned} \end{equation} This inequality and \eqref{431} lead to \begin{equation}\label{ineq-z-2} \begin{aligned} &\int_{\lambda_{3}}^{1}\int_0^{A}\int_0^{T}(s\Theta k z_{x}^2+s^{3} \Theta^{3}\frac{(x-x_0)^2}{k} z^2)e^{2s\varphi} \,dt\,da\,dx\\ &\leq \widetilde{c_{3}}\Big(\int_{Q}h^2e^{2s\Phi}\,dt\,da\,dx + \int_{\omega}\int_0^{A}\int_0^{T} s^{3}\Theta^{3}w^2e^{2s\Phi} \,dt\,da\,dx\Big). \end{aligned} \end{equation} Taking into account the definition of $\eta$ \eqref{cut-off func-2}, we can say that \begin{equation}\label{z-3} \begin{aligned} &\int_{\lambda_{2}}^{1}\int_0^{A}\int_0^{T}(s\Theta k z_{x}^2+s^{3} \Theta^{3}\frac{(x-x_0)^2}{k} z^2)e^{2s\varphi} \,dt\,da\,dx\\ &=\int_{\lambda_{2}}^{1}\int_0^{A}\int_0^{T}(s\Theta k w_{x}^2+s^{3}\Theta^{3} \frac{(x-x_0)^2}{k} w^2)e^{2s\varphi} \,dt\,da\,dx. \end{aligned} \end{equation} Hence, \begin{equation}\label{w-3} \begin{aligned} &\int_{\lambda_{2}}^{1}\int_0^{A}\int_0^{T}(s\Theta k w_{x}^2+s^{3} \Theta^{3}\frac{(x-x_0)^2}{k} w^2)e^{2s\varphi} \,dt\,da\,dx\\ & \leq \widetilde{c_{3}}\Big(\int_{Q}h^2e^{2s\Phi}\,dt\,da\,dx + \int_{\omega}\int_0^{A}\int_0^{T} s^{3}\Theta^{3}w^2e^{2s\Phi} \,dt\,da\,dx\Big), \end{aligned} \end{equation} as a consequence of \eqref{ineq-z-2} and \eqref{z-3}. Arguing in the same way for $(0, \lambda_1)$, one can show that \begin{equation}\label{w-4} \begin{aligned} &\int_0^{\lambda_1}\int_0^{A}\int_0^{T}(s\Theta k w_{x}^2+s^{3} \Theta^{3}\frac{(x-x_0)^2}{k} w^2)e^{2s\varphi} \,dt\,da\,dx\\ & \leq \widetilde{c_{3}}\Big(\int_{Q}h^2e^{2s\Phi}\,dt\,da\,dx + \int_{\omega}\int_0^{A}\int_0^{T} s^{3}\Theta^{3}w^2e^{2s\Phi} \,dt\,da\,dx\Big), \end{aligned} \end{equation} Finally, summing the inequalities \eqref{ineq-2}, \eqref{w-3} and \eqref{w-4} side by side, taking $s$ large and using again the restriction on $c_1$, \eqref{c-1-restr}, we arrive at \begin{equation}\label{w-5} \begin{aligned} &\int_0^{1}\int_0^{A}\int_0^{T}(s\Theta k w_{x}^2 +s^{3}\Theta^{3}\frac{(x-x_0)^2}{k} w^2)e^{2s\varphi} \,dt\,da\,dx\\ &\leq \widetilde{c_{3}}\Big(\int_{Q}h^2e^{2s\Phi}\,dt\,da\,dx + \int_{\omega}\int_0^{A}\int_0^{T} s^{3}\Theta^{3}w^2e^{2s\Phi} \,dt\,da\,dx\Big), \end{aligned} \end{equation} and this is exactly the desired estimate \eqref{Carle-esti}. \end{proof} Before providing the main Carleman estimate, we make the following remarks. \begin{remark} \rm 1. The proof of our distributed-Carleman estimate \eqref{Carle-esti} is based on the cut-off functions and given by two different weighted functions $\varphi$ and $\Phi$, in addition by \eqref{c-1-restr} there is no positive constant $C$ such that $$ e^{2s\Phi}\leq Ce^{2s\varphi}. $$ 2. Our proof is not based on the reflection method used for the proof of \cite[Lemma 4.1]{genni} which is needed to eliminate the boundary term arising in the classical Carleman estimate for nondegenerate heat equation. \end{remark} By the Carleman estimate \eqref{Carle-esti}, we are able to show the following $\omega$-Carleman estimate for the full adjoint system \eqref{adj-sys}. \begin{theorem} \label{thm2.7} Assume that \eqref{hyp-k}, \eqref{hyp-beta-mu} and \eqref{H-P} hold. Let $A>0$ and $T>0$ be given such that $0t+(A-T)\\[4pt] S(T-t)w_{T}(T+(a-t),\cdot)+\int_{t}^{T}S(l-t)\beta(l,a,\cdot)w(l,0,\cdot)dl \\ \quad \text{if } a\leq t+(A-T), \end{cases} \end{equation} where $(S(t))_{t\geq 0}$ is the semi-group generated by the operator $A_{2}w=(kw_{x})_{x}-\mu w$. Thus, \begin{equation}\label{w} w(t,0,\cdot)=S(T-t)w_{T}(T-t,\cdot), \end{equation} using the last hypothesis in \eqref{hyp-beta-mu} on $\beta$. Inserting this formula in \eqref{434} and using the fact $(S(t))_{t\geq 0}$ is a bounded semi-group, we obtain \begin{equation} \label{Carlesti} \begin{aligned} &s^{3}\int_{Q} \Theta^{3}\frac{(x-x_0)^2}{k(x)}w^2e^{2s\varphi} \,dt\,da\,dx + s\int_{Q} \Theta k(x)w_{x}^2e^{2s\varphi} \,dt\,da\,dx \\ &\leq \widehat{C}\Big(\int_0^{1}\int_0^{T} w_{T}^2(T-t,x)\,dt\,dx +\int_{q}s^{3}\Theta^{3}w^2e^{2s\Phi}\,dt\,da\,dx\Big)\\ &\leq \widehat{C}\Big(\int_0^{1}\int_0^{T} w_{T}^2(m,x)\,dm\,dx +\int_{q}s^{3}\Theta^{3}w^2e^{2s\Phi}\,dt\,da\,dx\Big)\\ &\leq \widehat{C}\Big(\int_0^{1}\int_0^{\delta} w_{T}^2(m,x)\,dm\,dx +\int_{q}s^{3}\Theta^{3}w^2e^{2s\Phi}\,dt\,da\,dx\Big), \end{aligned} \end{equation} since $T\in(0, \delta)$ with $\delta \in (0,A)$ small enough and this achieves the proof of \eqref{main-Carle}. \end{proof} \section{Observability inequality and null controllability}\label{third-section} \subsection{Observability inequality} The objective of this paragraph is to reach the observability inequality of the adjoint system \eqref{adj-sys}. To attain this purpose, we combine the Carleman estimate \eqref{main-Carle} with the Hardy-Poincar\'e inequality stated in \cite[Proposition 2.3]{genni} and arguing in a similar way as in \cite{Ain3}. Our observability inequality is given by the following proposition. \begin{proposition}\label{prop-obser-ineq} Assume that \eqref{hyp-k}, \eqref{hyp-beta-mu} and \eqref{H-P} hold. Let $A>0$ and $T>0$ be given such that $00$ to be defined later, let $\widetilde{w}=e^{\kappa t}w$ be a solution of \begin{equation} \begin{gathered} \frac{\partial \widetilde{w}}{\partial t} + \frac{\partial \widetilde{w}}{\partial a} +(k(x)\widetilde{w}_{x})_{x}-(\mu(t,a, x)+\kappa)\widetilde{w} =-\beta \widetilde{w}(t,0,x), \\ \widetilde{w}(t,a, 1)=\widetilde{w}(t,a, 0)=0,\label{obse-1-3}\\ \widetilde{w}(T,a, x)=e^{\kappa T}w_{T}(a, x), \\ \widetilde{w}(t,A, x)=0. \end{gathered} \end{equation} We point out that the parameter $\kappa$ considered here is not the same as in \eqref{poids}. Multiplying the first equation of \eqref{obse-1-3} by $\widetilde{w}$ and integrating by parts on \\$Q_{t}=(0,t)\times(0,A)\times(0,1)$. Then, one obtains \begin{equation}\label{obs-1-4} \begin{aligned} &-\frac{1}{2}\int_{Q_{A}}\widetilde{w}^2(t,a,x)\,dx\,dx +\frac{1}{2}\int_{Q_{A}}w^2(0,a,x)\,dx\,dx +\frac{1}{2}\int_0^{1}\int_0^{t}\widetilde{w}^2(\tau,0,x)d\tau dx\\ &+\kappa\int_0^{1}\int_0^{A}\int_0^{t}\widetilde{w}^2(\tau,a,x)d\tau \,dx\,dx \\ &\leq \frac{\|\beta\|_{\infty}^2}{4\epsilon'}\int_0^{1}\int_0^{A} \int_0^{t}\widetilde{w}^2(\tau,a,x)d\tau \,dx\,dx +\epsilon'A\int_0^{1}\int_0^{t}\widetilde{w}^2(\tau,0,x)d\tau dx. \end{aligned} \end{equation} Thus, for $\kappa=\frac{\|\beta\|_{\infty}^2}{4\epsilon'}$ and $\epsilon'<\frac{1}{2A}$, integrating over $(\frac{T}{4}, \frac{3T}{4})$ we obtain \begin{equation}\label{obs-1-5} \int_{Q_{A}}w^2(0,a,x)\,dx\,dx \leq C_{12}e^{2\kappa T}\int_{Q_{A}}\int_{T/4}^{3T/4}w^2(t,a,x)\,dx\,dx. \end{equation} On the other hand, let us prove that there exists a positive constant $C_{\delta}$ such that \begin{equation}\label{obs1} \int_0^{1}\int_0^{\delta-\frac{3T}{4}}\int_{T/4}^{3T/4} w^2(t,a,x)\,dt\,da\,dx \leq C_{\delta}\int_0^{1}\int_0^{\delta} w_{T}^2(a,x)\,dx\,dx. \end{equation} For this purpose, we use the implicit formula of $w$ defined by \eqref{453} and we discuss the two cases: when $a>t+(A-T)$ and when $a\leq t+(A-T)$. In fact, if $a>t+(A-T)$ one has \begin{equation}\label{W} \begin{aligned} w(t,a,\cdot) &=\int_0^{A-a}S(A-a-l)\beta(t,A-l,\cdot)w(t,0,\cdot)dl\\ &= \int_0^{A-a}S(A-a-l)\beta(t,A-l,\cdot)S(T-t)w_{T}(T-t,\cdot)dl, \end{aligned} \end{equation} using \eqref{w}. Since $(S(t))_{t\geq 0}$ is a bounded semi-group and $\beta \in L^{\infty}(Q)$, one can see that for $T\in(0, \delta)$ \begin{equation}\label{W1} \begin{aligned} \int_0^{1}\int_0^{\delta-\frac{3T}{4}}\int_{T/4}^{3T/4} w^2(t,a,x)\,dt\,da\,dx &\leq \widetilde{C}_{10}\int_0^{1}\int_{T/4}^{3T/4}w_{T}^2(T-t,x)\,dt\,dx\\ &\leq \widetilde{C}_{10}\int_0^{1}\int_0^{\delta}w_{T}^2(m,x)\,dm\,dx, \end{aligned} \end{equation} Now, if $a\leq t+(A-T)$ one has \begin{equation} \label{W2} \begin{aligned} w(t,a,\cdot) &=S(T-t)w_{T}(T+(a-t),\cdot)+\int_{t}^{T}S(l-t)\beta(l,a,\cdot)w(l,0,\cdot)dl\\ &=S(T-t)w_{T}(T+(a-t),\cdot) \\ &\quad +\int_{t}^{T}S(l-t)\beta(l,a,\cdot) S(T-l)w_{T}(T-l,\cdot)dl. \end{aligned} \end{equation} Thanks to the same argument employed to get \eqref{W1}, we conclude that \begin{equation}\label{W3} \begin{aligned} &\int_0^{1}\int_0^{\delta-\frac{3T}{4}}\int_{T/4}^{3T/4} w^2(t,a,x)\,dt\,da\,dx\\ &\leq 2\widetilde{C}_{11}(\int_0^{1}\int_0^{\delta-\frac{3T}{4}} \int_{T/4}^{3T/4}w_{T}^2(T+(a-t),x)\,dt\,da\,dx \\ &\quad +\int_0^{1}\int_{t}^{T}w_{T}^2(T-l,x)\,dl\,dx). \end{aligned} \end{equation} On the one hand, we can check that \begin{equation}\label{W4} \begin{aligned} &\int_0^{1}\int_0^{\delta-\frac{3T}{4}}\int_{T/4}^{3T/4}w_{T}^2(T+(a-t),x) \,dt\,da\,dx\\ & \leq \int_0^{1}\int_0^{\delta-\frac{3T}{4}} \int_{T/4}^{3T/4}w_{T}^2(a+m,x)dm\,dx\,dx\\ & \leq \int_0^{1}\int_0^{\delta-\frac{3T}{4}} \int_{a+\frac{T}{4}}^{a+\frac{3T}{4}}w_{T}^2(z,x)dz\,dx\,dx\\ & \leq \int_0^{1}\int_0^{\delta-\frac{3T}{4}}\int_0^{\delta}w_{T}^2(z,x)dz\,dx\,dx\\ & \leq \delta\int_0^{1}\int_0^{\delta}w_{T}^2(z,x)\,dz\,dx. \end{aligned} \end{equation} On the other hand, we have the inequality \begin{equation}\label{W5} \begin{aligned} \int_0^{1}\int_{t}^{T}w_{T}^2(T-l,x)\,dl\,dx &=\int_0^{1}\int_0^{T-t}w_{T}^2(z,x)\,dz\,dx \\ & \leq\int_0^{1}\int_0^{\delta}w_{T}^2(z,x)\,dz\,dx. \end{aligned} \end{equation} Combining inequalities \eqref{W3}, \eqref{W4} and \eqref{W5} we obtain \begin{equation}\label{W6} \int_0^{1}\int_0^{\delta-\frac{3T}{4}}\int_{T/4}^{3T/4} w^2(t,a,x)\,dt\,da\,dx \leq\widetilde{C}_{12}\int_0^{1}\int_0^{\delta}w_{T}^2(z,x)\,dz\,dx. \end{equation} Subsequently, \eqref{obs1} occurs in both studied cases. Therefore, in light of inequality \eqref{obs-1-5} we conclude that \begin{equation}\label{obs-1-6} \begin{aligned} &\int_{Q_{A}}w^2(0,a,x)\,dx\,dx \\ &\leq \widetilde{C}_{13}\int_0^{1}\int_0^{\delta}w_{T}^2(a,x)\,dx\,dx +\frac{2e^{2\kappa T}}{T}\int_0^{1}\int_{\delta-\frac{3T}{4}}^{A} \int_{T/4}^{3T/4}w^2(t,a,x)\,dt\,da\,dx. \end{aligned} \end{equation} Now, let $p$ defined by \eqref{p}. Then, using the hypotheses \eqref{hyp-k} on $k$ the function $x\mapsto\frac{(x-x_0)^2}{p(x)}$ is non-increasing at the left of $x_0$ and nondecreasing at the right of $x_0$. Hence, applying Hardy-Poincar\'e inequality (see \cite[Proposition 2.3]{genni}) and taking into account the definition of $\varphi$ stated in \eqref{weight-functions} we have %\label{obs-1-7} \begin{align*} \int_{Q_{A}}w^2(0,a,x)\,dx\,dx &\leq \widetilde{C}_{13}\int_0^{1}\int_0^{\delta}w_{T}^2(a,x)\,dx\,dx \\ &\quad +C_{\delta}^{13}\int_0^{1}\int_{\delta-\frac{3T}{4}}^{A} \int_{T/4}^{3T/4}s\Theta k(x) w_{x}^2(t,a,x)e^{2s\varphi}\,dt\,da\,dx. \end{align*} Therefore, using Carleman estimate \eqref{main-Carle} we infer that \[ %\label{obs-1-8} \int_{Q_{A}}w^2(0,a,x)\,dx\,dx \leq \widetilde{C}_{\delta}^{15}\Big(\int_{q}s^{3}\Theta^{3}w^2e^{2s\Phi}\,dt\,da\,dx +\int_0^{1}\int_0^{\delta}w_{T}^2(a,x)\,dx\,dx\Big), \] and then the proof is complete using the fact that $\sup_{(t,a,x)\in Q}s^{d}\Theta^{d}e^{2s\Phi}<+\infty$ for all $d\ in\mathbb{R}$. \end{proof} \subsection{Null controllability} In the previous paragraph, we obtained the observability inequality of system \eqref{adj-sys}. Such a tool will be very useful to prove the null controllability of the model \eqref{405} in the case where $T\in(0,\delta)$ as we emphasized in the introduction. Our main result is provided in the following theorem. \begin{theorem}\label{null-contr-result-1} Assume that the dispersion coefficient $k$ satisfies \eqref{hyp-k} and the natural rates $\beta$ and $\mu$ verify \eqref{hyp-beta-mu}. Let $A, T>0$ be given such that $00$ and consider the cost function $$ J_{\varepsilon}(\vartheta)=\frac{1}{2\varepsilon} \int_0^{1}\int_{\delta}^{A}y^2(T,a, x)\,dx\,dx +\frac{1}{2}\int_{q}\vartheta^2(t,a,x) \,dt\,da\,dx. $$ We can prove that $J_{\varepsilon}$ is continuous, convex and coercive. Then, it admits at least one minimizer $\vartheta_{\varepsilon}$ and we have \begin{equation}\label{438} \vartheta_{\varepsilon}=-w_{\varepsilon}(t,a,x)\chi_{\omega}(x) \quad \text{ in } Q, \end{equation} with $w_{\varepsilon}$ is the solution of the system \begin{equation}\label{439} \begin{gathered} \frac{\partial w_{\varepsilon}}{\partial t} + \frac{\partial w_{\varepsilon}}{\partial a} +(k(x)(w_{\varepsilon})_{x})_{x}-\mu(t,a, x)w_{\varepsilon} =-\beta w_{\varepsilon}(t,0,x) \quad\text{in } Q,\\ w_{\varepsilon}(t,a, 1)=w_{\varepsilon}(t,a, 0)=0 \quad\text{on }(0,T)\times (0,A),\\ w_{\varepsilon}(T,a, x)=\frac{1}{\varepsilon}y_{\varepsilon}(T,a, x) \chi_{(\delta, A)}(a) \quad\text{in }Q_{A},\\ w_{\varepsilon}(t,A, x)=0 \quad\text{in } Q_{T}, \end{gathered} \end{equation} and $y_{\varepsilon}$ is the solution of the system \eqref{405} associated to the control $\vartheta_{\varepsilon}$. Multiplying \eqref{439} by $y_{\varepsilon}$, integrating over $Q$, using \eqref{438} and the Young inequality we obtain \begin{equation}\label{440} \begin{aligned} &\frac{1}{\varepsilon}\int_0^{1}\int_{\delta}^{A}y_{\varepsilon}^2(T,a, x)\,dx\,dx +\int_{q}\vartheta^2_{\varepsilon}(t,a,x) \,dt\,da\,dx\\ &=\int_{Q_{A}}y_0(a,x)w_{\varepsilon}( 0, a, x)\,dx\,dx \\ & \leq \frac{1}{4C_{\delta}}\int_{Q_{A}}w_{\varepsilon}^2( 0, a, x)\,dx\,dx +C_{\delta}\int_{Q_{A}}y_0^2(a,x)\,dx\,dx, \end{aligned} \end{equation} with $C_{\delta}$ is the constant of the observability inequality \eqref{obser-ineq}. This again leads to \begin{equation}\label{441} \begin{aligned} &\frac{1}{\varepsilon}\int_0^{1}\int_{\delta}^{A}y_{\varepsilon}^2(T,a, x)\,dx\,dx +\int_{q}\vartheta^2_{\varepsilon}(t,a,x) \,dt\,da\,dx \\ &\leq \frac{1}{4}\int_{q}w^2\,dt\,da\,dx +C_{\delta}\int_{Q_{A}}y_0^2(a,x)\,dx\,dx. \end{aligned} \end{equation} Keeping in the mind \eqref{438}, we conclude that \[ \frac{1}{\varepsilon}\int_0^{1}\int_{\delta}^{A}y_{\varepsilon}^2(T,a, x)\,dx\,dx +\frac{3}{4}\int_{q}\vartheta^2_{\varepsilon}(t,a,x) \,dt\,da\,dx \leq C_{\delta}\int_{Q_{A}}y_0^2(a,x)\,dx\,dx. \] Hence, it follows that \begin{equation}\label{442} \begin{gathered} \int_0^{1}\int_{\delta}^{A}y_{\varepsilon}^2(T,a, x)\,dx\,dx \leq \varepsilon C_{\delta}\int_{Q_{A}}y_0^2(a,x)\,dx\,dx\\ \int_{q}\vartheta^2_{\varepsilon}(t,a,x) \,dt\,da\,dx \leq \frac{4C_{\delta}}{3}\int_{Q_{A}}y_0^2(a,x)\,dx\,dx. \end{gathered} \end{equation} Then, we can extract two subsequences of $y_{\varepsilon}$ and $\vartheta_{\varepsilon}$ denoted also by $\vartheta_{\varepsilon}$ and $y_{\varepsilon}$ that converge weakly towards $\vartheta$ and $y$ in $L^2(q)$ and $L^2((0, T)\times(0, A); H^{1}_{k}(0, 1))$ respectively. Now, by a variational technic, we prove that $y$ is a solution of \eqref{405} corresponding to the control $\vartheta$ and, by the first estimate of \eqref{442}, $y$ satisfies \eqref{406} for $T\in(0,\delta)$ and this shows our claim. \end{proof} \section{Appendix}\label{fourth-section} As we said in the introduction, this Appendix concerns a result which plays an important role to show the $\omega$-Carleman estimate associated to the full adjoint system \eqref{adj-sys} namely the Caccioppoli's inequality which is stated in the following lemma. \begin{lemma}\label{Caccio-1} Let $\omega'\subset\subset\omega$ and $w$ be the solution of \eqref{inter-adjoint sys-2}. Suppose that $x_0\notin \overline{\omega'}$. Then, there exists a positive constant $C$ such that $w$ satisfies \begin{equation}\label{Caccio} \begin{aligned} &\int_{\omega'}\int_0^{A}\int_0^{T}w_{x}^2e^{2s\varphi} \,dt\,da\,dx \\ &\leq C\Big(\int_{q}s^2\Theta^2w^2e^{2s\varphi} \,dt\,da\,dx+\int_{q}h^2 e^{2s\varphi}\,dt\,da\,dx\Big). \end{aligned} \end{equation} \end{lemma} \begin{proof} Define the smooth cut-off function $\zeta:\mathbb{R}\to \mathbb{R}$ by \begin{equation}\label{81} \begin{gathered} 0\leq\zeta(x)\leq1, \quad x \in \mathbb{R},\\ \zeta(x)=0, \quad xx_2,\\ \zeta(x)=1, \quad x \in \omega'.\\ \end{gathered} \end{equation} For the solution $w$ of \eqref{inter-adjoint sys-2}, we have \begin{equation}\label{131} \begin{aligned} 0&=\int_0^{T}\frac{d}{dt}\Big[\int_0^{1}\int_0^{A} \zeta^2e^{2s\varphi}w^2\,dx\,dx\Big]dt\\ &=2s \int_0^{1}\int_0^{A}\int_0^{T}\zeta^2\varphi_{t}w^2e^{2s\varphi}\,dt\,da\,dx +2\int_0^{1}\int_0^{A}\int_0^{T}\zeta^2ww_{t}e^{2s\varphi}\,dt\,da\,dx\\ &=2s\int_0^{1}\int_0^{A}\int_0^{T}\zeta^2\varphi_{t}w^2 e^{2s\varphi}\,dt\,da\,dx\\ &\quad +2\int_0^{1}\int_0^{A}\int_0^{T}\zeta^2w(-(kw_{x})_{x}-w_{a} +h+\mu w)e^{2s\varphi}\,dt\,da\,dx. \end{aligned} \end{equation} Then, integrating by parts we obtain \begin{align*} %\label{82} &2\int_{Q}k\zeta^2e^{2s\varphi}w_{x}^2\,dt\,da\,dx\\ &=-2s\int_{Q}\zeta^2w^2\psi(\Theta_{a} +\Theta_{t})e^{2s\varphi}\,dt\,da\,dx-2\int_{Q}\zeta^2whe^{2s\varphi}\,dt\,da\,dx\\ &\quad -2\int_{Q}\zeta^2\mu w^2e^{2s\varphi}\,dt\,da\,dx +\int_{Q}(k(\zeta^2e^{2s\varphi})_{x})_{x}w^2\,dt\,da\,dx. \end{align*} On the other hand, by the definitions of $\zeta$, $\psi$ and $\Theta$, thanks to Young inequality, taking $s$ quite large and using the fact that $x_0\notin \overline{\omega'}$, one can prove the existence of a positive constant $c$ such that \begin{gather*} %\label{88} 2\int_{Q}k\zeta^2e^{2s\varphi}w_{x}^2\,dt\,da\,dx \geq 2\min_{x\in\omega'}k(x)\int_{\omega'}\int_0^{A}\int_0^{T}w_{x}^2e^{2s\varphi} \,dt\,da\,dx,\\ \int_{Q}(k(\zeta^2e^{2s\varphi})_{x})_{x}w^2\,dt\,da\,dx \leq c\int_{\omega}\int_0^{A}\int_0^{T}s^2\Theta^2w^2e^{2s\varphi}\,dt\,da\,dx,\\ -2s\int_{Q}\zeta^2w^2\psi(\Theta_{a}+\Theta_{t})e^{2s\varphi}\,dt\,da\,dx \leq c\int_{\omega}\int_0^{A}\int_0^{T}s^2\Theta^2w^2e^{2s\varphi}\,dt\,da\,dx,\\ \begin{aligned} &-2\int_{Q}\zeta^2whe^{2s\varphi}\,dt\,da\,dx \\ &\leq c\Big(\int_{\omega}\int_0^{A}\int_0^{T}s^2\Theta^2w^2e^{2s\varphi}\,dt\,da\,dx +\int_{\omega}\int_0^{A}\int_0^{T}h^2e^{2s\varphi}\,dt\,da\,dx\Big), \end{aligned} \\ -2\int_{Q}\zeta^2\mu w^2e^{2s\varphi}\,dt\,da\,dx \leq c\int_{\omega}\int_0^{A}\int_0^{T}s^2\Theta^2w^2e^{2s\varphi}\,dt\,da\,dx. \end{gather*} This implies that there is $ C>0$ such that \[ \int_{\omega'}\int_0^{A}\int_0^{T}w_{x}^2 e^{2s\varphi}\,dt\,da\,dx \leq C\Big(\int_{q}s^2\Theta^2w^2e^{2s\varphi} \,dt\,da\,dx+\int_{q}h^2 e^{2s\varphi}\,dt\,da\,dx\Big). \] Thus, the proof is complete. \end{proof} \begin{remark}\label{rem-cacc} \rm Lemma \ref{Caccio-1} remains valid for any function $\pi \in C([0,1], (-\infty, 0))\cap C^{1}([0,1]\backslash \{x_0\}, (-\infty, 0))$ satisfying \[ |\pi_{x}|\leq\frac{c}{\sqrt{k}}, \quad \text{for } x\in[0,1]\backslash\{x_0\}, \] where $c>0$. see \cite[Proposition 4.2]{genni} for more details. \end{remark} \subsection*{Acknowledgements} The authors would like to thank the anonymous referee and the Professors B. 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