\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 149, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/149\hfil Lipschitz stability] {Lipschitz stability for degenerate \\ parabolic systems} \author[I. Boutaayamou, A. Hajjaj, L. Maniar \hfil EJDE-2014/149\hfilneg] {Idriss Boutaayamou, Abdelkarim Hajjaj, Lahcen Maniar} % in alphabetical order \address{Idriss Boutaayamou \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences Semlalia\\ LMDP, UMMISCO (IRD-UPMC), Universit\'e Cadi Ayyad, Marrakech 40000, B.P. 2390, Morocco} \email{dsboutaayamou@gmail.com} \address{Abdelkarim Hajjaj \newline Facult\'e des Sciences et Techniques, Universit\'e Hassan 1er, Laboratoire MISI, B.P. 577, Settat 26000, Morocco} \email{abdelkarim.hajjaj@uhp.ac.ma} \address{Lahcen Maniar \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences Semlalia\\ LMDP, UMMISCO (IRD-UPMC), Universit\'e Cadi Ayyad, Marrakech 40000, B.P. 2390, Morocco} \email{maniar@uca.ma} \thanks{Submitted October 4, 2013. Published June 26, 2014.} \subjclass[2000]{35K65} \keywords{Degenerate parabolic system; Carleman estimate; Lipschitz stability; \hfill\break\indent inverse problem} \begin{abstract} In this article, we study an inverse problem for weakly degenerate coupled parabolic systems with one force. We establish Lipschitz stability for the source term from measurements of one component of the solution at a positive time and on a subset of the space domain. The key ingredient is the derivation of a Carleman-type estimate. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} The null controllability and inverse problems of parabolic equations and parabolic coupled systems have attracted much interest in the previous years; see \cite{Ammar2,Ammar4, Ben, BCrGT,BukhgeimKlibanov,CrGR,CrGRY,FG,FGPY,Gu,Yam, LR,RqCr1,RqCr2,Rus}. The main result in these article is the development of adequate Carleman estimates, which is a crucial tool to obtain observability inequalities and Lipschitz stability for term sources, initial data, potentials and diffusion coefficients. The above systems are considered to be nondegenerate. In other words, the diffusion coefficients are uniformly coercive. The case of degenerate coefficients at the boundary is also considered in several papers by developing adequate Carleman estimates. The null controllability of degenerate parabolic equations is studied in \cite{Can1,Can2,Can3,21}, and the null controllability of coupled degenerate parabolic systems in \cite{bahm,hajjaj,CanTer,Liu}. While, the inverse problem for degenerate parabolic equations is studied in \cite{Tort,Tort2,Tort1,Tort3}. In this article, we consider the Lipschitz stability of an inverse problem for the linear coupled degenerate parabolic systems with two different diffusion coefficients. \begin{equation}\label{sys} \begin{gathered} u_t-(x^{\alpha_1}u_x)_x+b_{11}(x)u+b_{12}(x)v=F, \quad (t,x)\in Q,\\ v_t-(x^{\alpha_2}v_x)_x+b_{21}(x)u+b_{22}(x)v=0, \quad (t,x)\in Q,\\ u(t,0)=u(t,1)=v(t,0)=v(t,1)=0, \quad t \in (0,T),\\ u(0,x)=u_0(x), \, v(0,x)=v_0(x), \quad x\in (0,1), \end{gathered} \end{equation} where $u_0,\, v_0 \in L^2 (0,1) $, $\alpha_1,\alpha_2\in (0,1)$, $T> 0$ fixed, $Q:=(0,T)\times (0,1)$ and $b_{ij}\in L^\infty(0,1)$, $i,j=1,2$. For $t_0\in (0,T)$ given, let $Q_{t_0}^T=(t_0,T)\times (0,1)$ and $T':=\frac{T+t_0}{2}$. For a given $C_0>0$, we denote by $S(C_0)$ the space \[ S(C_0):=\{F\in H^1(0,T;L^2(0,1)) : |F_t(t,x)| \leqslant C_0|F(T',x)|, \; (t,x)\in Q\}. \] Our purpose is to determine $F$, belonging to the space $S(C_0)$, from the measurements $(x^{\alpha_1}u_x)_x(T',\cdot)$ and additional observations of the component $u$. The main ingredient to obtain Lipschitz stability is Carleman estimates. We prove a Carleman estimates for the coupled system \eqref{sys}, similar to the one obtained in \cite{Tort} but with different weight functions, that are necessary for the case of different exponents $\alpha_1$ and $\alpha_2$. Having the Carleman estimates in hand, we follow the method developed in \cite{BukhgeimKlibanov, Ben, Yam} to obtain the Lipschitz stability results. If we restrict ourselves to the particular case $F\in \{ rf : f\in L^2(0,1)\}$ for some given function $r$, uniqueness results can be shown for the system \eqref{sys} as an immediate consequence of our Lipschitz stability results, see \cite{BukhgeimKlibanov, Yam,RqCr2}. To prove our Carleman estimates, we use the following Hardy-Poincar\'e inequality proved in \cite{hajjaj} $$ \int_0^1x^{\kappa-2}v^2dx \leqslant \frac{4}{(1-\kappa)^2}\int_0^1x^{\kappa}v_x^2dx, $$ for $\kappa<1$ and $v$ locally absolutely continuous on $[0,1]$, continuous at $0$ and satisfying $v(0)=0$ and $\int_0^1x^{\kappa}v_x^2dx<\infty$. This paper is organised as follows: in Section 2, we discuss the well-posedness of the problem \eqref{sys}. Then, we establish different Carleman estimates for parabolic equations and parabolic systems \eqref{sys} and the last section treats the Lipschitz stability of $F$. \section{Well-Posedness of system} To study the well-posedness of \eqref{sys}, we introduce the following weighted spaces, for $0<\alpha<1$: \begin{align*} H_{\alpha}^1(0, 1) :=\Big\{& u \in L^2(0, 1) : u \text{ abs. cont. in } [0, 1],\\ & x^{\alpha/2} u_x \in L^2(0, 1),\; u(0)=u(1) = 0\Big\} \end{align*} with the norm $\|u\|^2_{ H^1_{\alpha}} := \|u\|^2_{L^2(0,1)} + \|x^{\alpha/2}u_x\|^2_{ L^2(0,1)}$ and $$ H^2_{\alpha} (0, 1):=\big\{ u \in H^1_{\alpha}(0, 1) : x^{\alpha}u_x \in H^1(0, 1)\big\}, \; \|u\|^2_{H^2_{\alpha}} := \|u\|^2 _{H^1_{\alpha}} + \|(x^{\alpha}u_x)_x\|^2_{ L^2(0,1)}. $$ We recall from \cite {cmp, Can1} that for $i=1,2,$ the operator $(A_i,D(A_i))$ defined by $A_iu := (x^{\alpha_i}u_x)_x$, $u \in D(A_i) = H^2_{\alpha_i}(0, 1)$ is closed self-adjoint, negative with dense domain in $L^2(0, 1)$. In the Hilbert space $\mathbb{H}:= L^2(0,1) \times L^2(0, 1) $, system \eqref{sys} can be transformed into the Cauchy problem \begin{gather*} X'(t) =\mathcal{A}X(t) -BX(t)+h(t), \quad t\in (0,T),\\ X(0)=\begin{pmatrix} u_0\\ v_0 \end{pmatrix}, \end{gather*} where $X(t)=\begin{pmatrix} u(t)\\ v(t) \end{pmatrix}$, $\mathcal{A}=\begin{pmatrix} A_1&0\\ 0&A_2 \end{pmatrix}$, $D(\mathcal{A})=D(A_1)\times D(A_2)$, $h(t)=\begin{pmatrix} F(t)\\ 0 \end{pmatrix}$ and $ B=\begin{pmatrix} b_{11}&b_{12}\\ b_{21}&b_{22} \end{pmatrix}$. Since $\mathcal{A}$ is a diagonal operator and $B$ is a bounded perturbation of $\mathcal{A}$, the following wellposedness and regularity results hold. \begin{proposition}\label{estimsemigroup} (i) The operator $\mathcal{A}$ generates a contraction strongly continuous semigroup $(T(t))_{t\geqslant0}$. (ii) For all $(u_0,v_0)\in H^2_{\alpha_1}\times H^2_{\alpha_2}$ and $F\in H^1(0,T;L^2(0,1))$, problem \eqref{sys} has a unique solution $(u,v)\in C([0,T],H^2_{\alpha_1}\times H^2_{\alpha_2}) \cap C^1(0,T;\mathbb{H})$. (iii) For all $F\in L^2(Q),$ $u_0,v_0\in L^2(0,1)$, and $\varepsilon\in (0,T)$, there exists a unique mild solution $(u,v)\in X_T:=H^1( [\varepsilon,T],\mathbb{H}) \cap L^2(\varepsilon,T;H^2_{\alpha_1}\times H^2_{\alpha_2})$ of \eqref{sys} satisfying \[ \|(u,v)\|_{X_T}\leqslant C_T\Big(\|(u_0,v_0)\|_{\mathbb{H}}^2 +\|(F,G)\|_{\mathbb{H}}^2\Big). \] Moreover, for $F\in H^1(0,T;L^2(0,1))$ and $\varepsilon\in (0,T)$, $$ (u,v)\in C\big([\varepsilon,T],H^2_{\alpha_1}\times H^2_{\alpha_2}\big) \cap C^1\big(\varepsilon,T;\mathbb{H}\big). $$ \end{proposition} \section{Carleman estimates} The main goal of this section is to establish a Carleman estimate for a degenerate parabolic single equation with a boundary observation on the right hand side. Then, we will deduce the one for the degenerate system \eqref{sys} with locally distributed observations of $(u,v)$. Some of these estimates were obtained in \cite{hajjaj} for a null controllability purpose. In the forthcoming theorems we will prove additional estimates on $u$ and $u_t$, that are crucial to prove Lipschitz stability results. As in \cite{hajjaj}, we introduce the following weight functions which will be used throughout the paper \begin{equation}\label{weightfunc} \begin{gathered} \varphi(t,x):=\theta(t)p(x),\quad \theta(t):=\frac{1}{(t-t_0)^4(T-t)^4}, \\ p(x):=\lambda\left(x^{2-\beta}-d\right), \quad \eta(t):=T+t_0-2t, \end{gathered} \end{equation} where $t_0>0$ is a fixed initial time, $T>0$ is a final time, $d>1$ and $\beta$ is a constant that will be chosen later. \subsection{Carleman estimates for parabolic equations} Consider the parabolic equation \begin{equation} \label{equparab} \begin{gathered} y_t-(x^{\alpha}y_x)_x+b(x)y=f(t,x), \quad (t,x)\in Q,\\ y(t,0)=y(t,1)=0, \quad t \in (0,T),\\ y(0,x)=y_0(x), \quad x\in (0,1) \end{gathered} \end{equation} We assume that $\alpha\in (0,1)$, $b\in L^\infty(0,1)$ and $f\in L^2(Q)$, and state the first Carleman estimate for smooth initial data. \begin{theorem}\label{dataregular} For all $T>0$ and $\beta\in [\alpha,1)$, there exist two positive constants $C$ and $s_0$ such that for all $s\geqslant s_0$, the solution $y$ of \eqref{equparab} with $y_0\in H_\alpha^1(0,1)$ satisfies \begin{equation} \label{carineq} \begin{aligned} &\int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+2\alpha-3\beta}y^2 +s\theta x^{2\alpha-\beta}y_x^2 +\frac{1}{s\theta}y_t^2 + s\theta^{3/2}|\eta p|y^2 \Big) e^{2s\varphi (t,x)} \,dt\,dx \\ &\leqslant C\Big(\int_{Q_{t_0}^T} f^2(t,x) e^{2s\varphi (t,x)} \,dt\,dx + \int_{t_0}^T s\theta(t)y_x^2(t,1) e^{2s\varphi (t,1)} d t\Big). \end{aligned} \end{equation} \end{theorem} \begin{proof} For $s>0$ and a solution $y$ of \eqref{equparab}, the function $w:= e^{s\varphi}y$ satisfies \[ \underbrace{-(x^{\alpha}w_x)_x-s\varphi_tw-s^2x^{\alpha}\varphi_x^2w}_{P_s^+w} + \underbrace{w_t+2sx^{\alpha}\varphi_xw_x+s(x^{\alpha}\varphi_x)_xw}_{P_s^-w} =\underbrace{f e^{s\varphi}-bw}_{f_s}. \] By \cite[Theorem 3.2]{hajjaj}, we have \begin{equation} \label{Eomega} \begin{aligned} & \|P_s^+w\|^2+\|P_s^-w\|^2+\int_{Q_{t_0}^T}(s\theta x^{2\alpha-\beta}w_x^2 + s^3\theta^3x^{2+2\alpha-3\beta}w^2) \,dt\,dx\\ &\leqslant C\Big(\int_{Q_{t_0}^T}f^2 e^{2s\varphi} \,dt\,dx +\int_{t_0}^T s\theta y_x^2(t,1) e^{2s\varphi(t,1)} \,dt\Big). \end{aligned} \end{equation} On one hand we have \begin{align*} \operatorname{sgn}(\eta)\theta^{1/4} w P_s^+w &= 4s\theta^{3/2}|\eta|pw^2- s^2\lambda^2(2-\beta)^2\operatorname{sgn}(\eta) \theta^{9/4}x^{2+\alpha-2\beta}w^2\\ &\quad -\operatorname{sgn}(\eta)\theta^{1/4}w(x^{\alpha}w_x)_x, \end{align*} where $\operatorname{sgn}(\eta)$ denotes the sign function of $\eta$. So, integrating by parts and using Young and Hardy-Poincar\'e inequalities and $\beta\in [\alpha,1)$, for s large, we obtain the following inequalities \begin{equation} \label{3.10} \begin{aligned} & \int_{Q_{t_0}^T}s\theta^{3/2}|\eta p|w^2\\ &\leqslant \frac{1}{4}\int_{Q_{t_0}^T}\theta^{1/4}w P_s^+w +\frac{\lambda^2(2-\beta)^2}{4} \int_{Q_{t_0}^T}s^2\theta^{9/4}x^{2+\alpha-2\beta}w^2 + \frac{1}{4}\int_{Q_{t_0}^T}\theta^{1/4}x^{\alpha}w_x^2\\ &\leqslant \frac{1}{8}\|P_s^+w\|_{L^2(Q_{t_0}^T)}^2 +\frac{1}{8}\int_{Q_{t_0}^T}\theta^{1/2}w^2 + \int_{Q_{t_0}^T}(s^3\theta^3x^{2+2\alpha-3\beta}w^2+s\theta x^{2\alpha-\beta}w_x^2), \end{aligned} \end{equation} and \begin{equation} \label{3.11} \begin{aligned} \int_{Q_{t_0}^T}\theta^{1/2}w^2 \,dt\,dx &=\int_{Q_{t_0}^T}(\theta^{1/4}x^{\alpha-\frac{\beta}{2}-1}w) (\theta^{1/4}x^{1-\alpha+\frac{\beta}{2}}w) \,dt\,dx \\ &\leqslant \frac{1}{2}\int_{Q_{t_0}^T} (\theta^{1/2}x^{2\alpha-\beta-2}w^2 +\theta^{1/2}x^{2-2\alpha+\beta}w^2) \,dt\,dx \\ &\leqslant \int_{Q_{t_0}^T}(s\theta x^{2\alpha-\beta}w_x^2 +s^3\theta^3x^{2+2\alpha-3\beta}w^2) \,dt\,dx. \end{aligned} \end{equation} On the other hand we have $$ \frac{1}{\sqrt{s\theta}}P_s^-w=\frac{1}{\sqrt{s\theta}}w_t +2\lambda(2-\beta)\sqrt{s\theta} x^{1+\alpha-\beta}w_x +\lambda(2-\beta)(1+\alpha-\beta)\sqrt{s\theta}x^{\alpha-\beta}w. $$ Therefore, using Hardy-Poincar\'e inequality and $\beta\in [\alpha,1)$, we obtain \begin{equation} \begin{aligned} \int_{Q_{t_0}^T}\frac{1}{s\theta}w_t^2 \,dt\,dx & \leqslant C\Big(\|P_s^-w\|^2+\int_{Q_{t_0}^T}(s\theta x^{2+2\alpha-2\beta}w_x^2 +s\theta x^{2\alpha-2\beta}w^2) \,dt\,dx\Big) \\ & \leqslant C\Big(\|P_s^-w\|^2+\int_{Q_{t_0}^T}(s\theta x^{2\alpha-\beta}w_x^2 +s\theta x^{2\alpha-\beta-2}w^2) \,dt\,dx\Big)\\ &\leqslant C\Big(\|P_s^-w\|^2+\int_{Q_{t_0}^T}s\theta x^{2\alpha-\beta}w_x^2 \,dt\,dx\Big). \end{aligned} \label{157673} \end{equation} Hence, combining \eqref{Eomega}-\eqref{157673}, it follows that \begin{align*} & \int_{Q_{t_0}^T}\Big(s^3 \theta^3x^{2+2\alpha-3\beta}{w}^2 + s\theta x^{2\alpha-\beta}w_x^2 + \frac{1}{s\theta}w_t^2 + s\theta^{3/2}|\eta p| w^2 \Big) \,dt\,dx \\ &\leqslant C\Big(\int_{Q_{t_0}^T} f^2 e^{2s\varphi (t,x)}\, \,dt\,dx + \int_{t_0}^T s\theta(t)y_x^2(t,1) e^{2s\varphi (t,1)} d t\Big). \end{align*} Finally, the definition of $w$ yields \[ w = y e^{s\varphi}, \quad y_x^2 e^{2s\varphi}\leqslant 2w_x^2 + cs^2\theta^2 x^{2-2\beta}w^2, \quad y_t^2 e^{2s\varphi}\leqslant 2w_t^2 + cs\theta^{5/4} |\eta p| w^2, \] and thus the estimate \eqref{carineq} can be deduced. \end{proof} The estimate \eqref{carineq} is obtained for regular initial data, by density we deduce the following result for general initial data. \begin{proposition}\label{datageneral} For all $T>0$ and $\beta\in [\alpha,1)$, there exist two positive constants $C$ and $s_0$ such that for every $y_0\in L^2(0,1)$ and all $s\geqslant s_0$, the solution $y$ of \eqref{equparab} satisfies \begin{align*} & \int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+\alpha-2\beta}y^2 +s\theta x^{\alpha}y_x^2 +\frac{1}{s\theta}y_t^2 + s\theta^{3/2}|\eta p|y^2 \Big) e^{2s\varphi (t,x)} \,dt\,dx \\ & \leqslant C\Big(\int_{Q_{t_0}^T} f^2(t,x) e^{2s\varphi (t,x)}\, \,dt\,dx+ \int_{t_0}^T s\theta(t)y_x^2(t,1) e^{2s\varphi (t,1)} d t\Big). \end{align*} \end{proposition} \begin{proof} Using the density of $H_\alpha^1(0,1)$ in $L^2(0,1)$, there exists a sequence $(y_0^n)_n\subset H_\alpha^1(0,1)$ converging to $y_0$. Let $y^n$ be the unique solution in the space $Z_T:= L^2(t_0,T;H_\alpha^2)\cap H^1(t_0,T;L^2(0,1))$ of the equation \eqref{equparab} associated to the initial data $y_0^n$. The sequence $(y^n)$ satisfies \begin{equation} \label{regularite} \begin{aligned} \|y^m-y^n\|_{Z_T}^2 &:=\int_{t_0}^T\|x^{\alpha/2}(y^m-y^n)_x\|_{L^2(0,1)}^2 +\|(x^{\alpha}(y^m-y^n)_x)_x\|_{L^2(0,1)}^2 d t\\ &\quad +\int_{t_0}^T\|y^m-y^n\|_{L^2(0,1)}^2+\|y^m_t-y^n_t\|_{L^2(0,1)}^2 d t\\ &\leqslant C_T\|y_0^m-y_0^n\|_{L^2(0,1)}^2, \end{aligned} \end{equation} hence, it has a limit $y$ in the Banach space $Z_T$. Using classical argument of semigroup theory, it is easy to show that $y$ is the solution of \eqref{equparab} associated to the initial data $y_0\in L^2(0,1)$. Note that for all $t\in(t_0,T)$ we have \[ s\theta e^{2s\varphi(t,1)}\leqslant L:= \max_{y\geqslant 0}(y e^{-2\lambda(d-1)y}). \] Hence, using the Sobolev trace theorem, $\alpha\in (0,1)$ and \eqref{regularite} with $m\to\infty$, one has \begin{align*} &\int_{t_0}^T s\theta(t)|(y_x^n-y_x)(t,1)|^2 e^{2s\varphi (t,1)} \,dt\\ &\leqslant C\Big(\int_{t_0}^T\|x^{\alpha/2}(y^n-y)_x\|_{L^2(0,1)}^2\, d t +\int_{t_0}^T\|(x^{\alpha}(y^n-y)_x)_x\|_{L^2(0,1)}^2\, d t\Big)\\ &\leqslant C_T\|y_0^n-y_0\|_{L^2(0,1)}^2. \end{align*} On the other hand since $x^\alpha\leqslant x^{2\alpha-\beta}$ and $x^{2+\alpha-2\beta}\leqslant x^{2+2\alpha-3\beta}$, inequality \eqref{carineq} provides \begin{align*} &\int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+\alpha-2\beta}|y^n|^2 +s\theta x^{\alpha}|y_x^n|^2 +\frac{1}{s\theta}|y_t^n|^2 + s\theta^{3/2}|\eta p||y^n|^2 \Big) e^{2s\varphi (t,x)} \,dt\,dx \\ &\leqslant C\Big(\int_{Q_{t_0}^T} f^2(t,x) e^{2s\varphi (t,x)} \,dt\,dx + \int_{t_0}^T s\theta(t)|y_x^n|^2(t,1) e^{2s\varphi (t,1)} d t\Big). \end{align*} Consequently, since the functions $s^3\theta^3 e^{2s\varphi}$, $\frac{1}{s\theta} e^{2s\varphi}$ and $x^\alpha$ are bounded, passing to the limit, we obtain the claim. \end{proof} \subsection{$\omega$-Carleman estimates for the system \eqref{sys}} In the present subsection, we shall derive an internal Carleman inequality. As in \cite{Fursikov}, let \begin{equation} \label{rho} \Phi(t,x)=\Psi(x)\theta(t), \quad \Psi (x):=\big( e^{\rho \sigma(x)}- e^{2\rho||\sigma||_\infty}\big), \end{equation} with $\theta$ defined in \eqref{weightfunc} and $\sigma$ is a function in $C^2([0,1])$ satisfying $\sigma(x)>0$ in $(a,1)$, $\sigma(a)=\sigma(1)=0$ and $|\sigma_x(x)|>0$ in $[0,1]\backslash\omega_0$ for some open $\omega_0\subset\subset \omega:=(a,b)$. We choose the parameters $d$, $\lambda$ and $\rho$, such that $d \geqslant 5$, $\rho > \frac{4ln 2}{||\sigma||_\infty}$ and $\frac{ e^{2\rho||\sigma||_\infty}}{d-1} < \lambda < \frac{4}{3d}( e^{2\rho||\sigma||_\infty}- e^{\rho||\sigma||_\infty})$. Thus, one has $\frac43\Phi<\varphi<\Phi$. Let $\xi,\zeta\in C^\infty([0,1])$ such that $\zeta=1-\xi$, $0\leqslant\xi(x)\leqslant1$, $\xi(x)=1$ for $x\in(0,a'')$ and $\xi(x)=0$ for $x\in(b'',1)$, where $0 < a < a''< b''< b<1$. Set also $\omega':=(a',b')$ and $\omega'':=(a'',b'')$ where $0 < a 0$ and $\beta\in[\alpha,1)$, there exist two constants $C$ and $s_0$ such that, for every $y_0\in L^2(0,1)$ and all $s\geqslant s_0$, the solution $y$ of \eqref{equparab} satisfies \begin{align*}%\label{Carineq21} &\int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+\alpha-2\beta}\xi^2y^2 +s\theta x^{\alpha}\xi^2y_x^2 +\frac{1}{s\theta}\xi^2y_t^2 + s\theta^{3/2}|\eta p|\xi^2y^2 \Big)e^{2s\varphi (t,x)}\,dt\,dx \\ & \leqslant C\Big(\int_{Q_{t_0}^T} \xi^2f^2(t,x) e^{2s\varphi (t,x)}\,\,dt\,dx+ \int_{t_0}^T\!\!\!\int_{\omega'} (f^2+s^2\theta^2 y^2)e^{2s\varphi}\,dx\,dt\Big). \end{align*} \end{proposition} \begin{proof} The function $z:=\xi y$ satisfies the parabolic equation \begin{gather*} z_t-(x^{\alpha}z_x)_x+bz =\xi f-\xi_x x^\alpha y_x-(x^\alpha\xi_x y)_x, \quad x\in (0,1),\; t\in (0,T),\\ z(t,0)=z(t,1)=0, \quad t \in (0,T),\\ z(0,x)=\xi(x)y_0(x), \quad x\in (0,1). \end{gather*} Using Proposition \ref{datageneral}, $z$ satisfies the estimate \begin{equation} \label{carineq22} \begin{aligned} & \int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+\alpha-2\beta}z^2 +s\theta x^{\alpha}z_x^2 +\frac{1}{s\theta}z_t^2 + s\theta^{3/2}|\eta p|z^2 \Big)e^{2s\varphi (t,x)}\,dt\,dx \\ & \leqslant C\int_{Q_{t_0}^T} ( \xi^2f^2+(\xi_x x^\alpha y_x+(x^\alpha\xi_x y)_x)^2) e^{2s\varphi (t,x)}\,dt\,dx. \end{aligned} \end{equation} So using $\operatorname{supp}(\xi_x)=\omega''$ and the Caccioppoli inequality \eqref{Caccioppoli-ineq} applying for $\mu_1=\mu_2=p$, we obtain \begin{equation} \label{31} \begin{aligned} \int_{Q_{t_0}^T} (\xi_x x^\alpha y_x+(x^\alpha\xi_x y)_x)^2 e^{2s\varphi}\,dt\,dx & \leqslant C\int_{t_0}^T\!\!\!\int_{\omega''}[y^2+y_x^2]e^{2s\varphi}\,dx\,dt\\ &\leqslant C\int_{t_0}^T\!\!\!\int_{\omega'}(f^2+s^2\theta^2y^2)e^{2s\varphi}\,dx\,dt. \end{aligned} \end{equation} By the definition of $z$ and $\xi$, we get \begin{equation} \label{32} \int_{Q_{t_0}^T}s\theta x^\alpha\xi^2y_x^2e^{2s \varphi}\,dt\,dx\leqslant 2\int_{Q_{t_0}^T}s\theta x^\alpha z_x^2e^{2s\varphi}\,dt\,dx+2\int_{t_0}^T\!\!\!\int_{\omega'}s\theta y^2e^{2s\varphi}\,dt\,dx. \end{equation} Thus, from \eqref{carineq22}-\eqref{32} and the definition of $\xi$ we deduce the desired estimate. \end{proof} Proposition \ref{Carxi} gave a Carleman estimate in $(0,a')$. For the rest of the interval, we have the following Proposition. Its proof is similar to the previous result using \cite[Lemma 1.2]{Fursikov} and Cacciopoli inequality \eqref{Caccioppoli-ineq} applying for $\mu_1=p$, and $\mu_2=\Psi$. \begin{proposition}\label{Car1-xi} For all $T>0$ and $\beta\in[\alpha,1)$, there exist two constants $C$ and $s_0$ such that, for every $y_0\in L^2(0,1)$ and all $s\geqslant s_0$, the solution $y$ of \eqref{equparab} satisfies \begin{align*}%\label{Carineq22} &\int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+\alpha-2\beta}y^2+s\theta x^{\alpha}y_x^2 +\frac{1}{s\theta}y_t^2+ s\theta^{3/2}|\eta p|y^2 \Big)\zeta^2e^{2s\Phi (t,x)}\,dt\,dx \\ & \leqslant C\Big(\int_{Q_{t_0}^T} \zeta^2f^2(t,x) e^{2s\Phi (t,x)}\,\,dt\,dx+ \int_{t_0}^T\!\!\!\int_{\omega'} \Big(f^2e^{2s\varphi}+s^3\theta^3y^2e^{2s(2\Phi-\varphi)}\Big)\,dx\,dt\Big). \end{align*} And if $f=0$, \begin{align*}%\label{Carineq22} &\int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+\alpha-2\beta}y^2 +s\theta x^{\alpha}y_x^2 +\frac{1}{s\theta}y_t^2 + s\theta^{3/2}|\eta p|y^2 \Big)\zeta^2e^{2s\Phi (t,x)}\,dt\,dx\\ &\leqslant C\int_{t_0}^T\!\!\!\int_{\omega'} s^3\theta^3y^2e^{2s\Phi}\,dx\,dt. \end{align*} \end{proposition} Using the above propositions, we show a Carleman estimate for our coupled system with locally distributed measurements. \begin{theorem}\label{Carl-system} Let $T>0$ and $\beta=\max(\alpha_1,\alpha_2)$. There exist two constants $ C$, $s_0>0$ such that, for every $(u_0,v_0)\in (L^2(0,1))^2$ and all $s\geqslant s_0$, the solution $(u,v)$ of \eqref{sys} satisfies % \label{uvxi} \begin{align*} I(\xi,u,v) &:= \int_{Q_{t_0}^T}\Big( s^3 \theta^3(x^{2+\alpha_1-2\beta}u^2 + x^{2+\alpha_2-2\beta}v^2)+s\theta ( x^{\alpha_1}u_x^2+ x^{\alpha_2}v_x^2) \\ & + \frac{1}{s\theta}(u_t^2+v_t^2)+ s\theta^{3/2}|\eta p|(u^2+v^2) \Big)\xi^2e^{2s\varphi (t,x)}\,dt\,dx \\ &\leqslant C\Big( \int_{Q_{t_0}^T} \xi^2F^2 e^{2s\varphi (t,x)}\,\,dt\,dx +\int_{t_0}^T \!\!\!\!\int_{\omega'} (F^2+s^2\theta^2(u^2+v^2)) e^{2s\varphi}\,dx\,dt\Big), \end{align*} and \begin{equation} \label{uvzeta} \begin{aligned} &I(\zeta,u,v)\\ &:= \int_{Q_{t_0}^T}\Big( s^3 \theta^3(x^{2+\alpha_1-2\beta}u^2 + x^{2+\alpha_2-2\beta}v^2)+s\theta ( x^{\alpha_1}u_x^2+ x^{\alpha_2}v_x^2) \\ &\quad +\frac{1}{s\theta}(u_t^2+v_t^2)+ s\theta^{3/2}|\eta p|(u^2+v^2) \Big) \zeta^2e^{2s\Phi (t,x)}\,dt\,dx \\ &\leqslant C\Big( \int_{Q_{t_0}^T} \zeta^2F^2 e^{2s\Phi (t,x)}\,\,dt\,dx +\int_{t_0}^T\!\!\!\int_{\omega'} (s^3\theta^3v^2e^{2s\Phi} +F^2e^{2s\varphi}+u^2)\,dx\,dt\Big). \end{aligned} \end{equation} \end{theorem} \begin{proof} The first component $u$ is the solution of the parabolic equation \eqref{equparab}. Applying Proposition \ref{Carxi}, for $s$ large enough, we obtain \begin{equation} \label{Carineq211} \begin{aligned} & \int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+\alpha_1-2\beta}u^2+s\theta x^{\alpha_1}u_x^2 +\frac{1}{s\theta}u_t^2+ s\theta^{3/2}|\eta p|u^2 \Big)\xi^2e^{2s\varphi (t,x)}\,dt\,dx \\ & \leqslant C_2\Big(\int_{Q_{t_0}^T} ( \xi^2F^2+\xi^2b_{12}^2v^2) e^{2s\varphi (t,x)}\,\,dt\,dx+ \int_{t_0}^T\!\!\!\int_{\omega'} (F^2+v^2+s^2\theta^2 u^2)e^{2s\varphi}\,dx\,dt\Big). \end{aligned} \end{equation} Proceeding as in \eqref{3.11}, for $s$ large enough, we have \begin{equation} \label{xxx} \begin{aligned} C_2 \int_{Q_{t_0}^T} \xi^2 b_{12}^2v^2 e^{2s\varphi}\,dt\,dx &\leqslant C \int_{Q_{t_0}^T}(x^{\frac{\alpha_2}{2}-1}\xi ve^{s\varphi}) (x^{1-\frac{\alpha_2}{2}}\xi ve^{s\varphi}) \,dt\,dx\\ &\leqslant \frac{1}{2}\int_{Q_{t_0}^T} (s\theta x^{\alpha_2}\xi^2v_x^2+s^3 \theta^3x^{2+\alpha_2-2\beta} \xi^2v^2)e^{2s\varphi}\,dt\,dx \\ &\quad + C\int_{t_0}^T\!\!\!\int_{\omega'} s^2\theta^2v^2e^{2s\varphi}\,dx\,dt. \end{aligned} \end{equation} Therefore, by \eqref{Carineq211} and \eqref{xxx} we deduce \begin{equation} \label{331} \begin{aligned} & \int_{Q_{t_0}^T}\Big(s^3 \theta^3x^{2+\alpha_1-2\beta} u^2+s\theta x^{\alpha_1}u_x^2 +\frac{1}{s\theta}u_t^2 + s\theta^{3/2}|\eta p|u^2 \Big)\xi^2e^{2s\varphi (t,x)}\,dt\,dx \\ & \leqslant C\Big(\int_{Q_{t_0}^T} \xi^2F^2 e^{2s\varphi (t,x)}\,\,dt\,dx + \int_{t_0}^T\!\!\!\int_{\omega'} (F^2+s^2\theta^2(v^2+u^2))e^{2s\varphi} \,dx\,dt\Big)\\ & +\frac{1}{2}\int_{Q_{t_0}^T} (s^3\theta^3x^{2+\alpha_2-2\beta}v^2+s\theta x^{\alpha_2}v_x^2)\xi^2e^{2s\varphi}\,dt\,dx. \end{aligned} \end{equation} The same can be done for $v$ and we obtain \begin{equation} \label{3321} \begin{aligned} & \int_{Q_{t_0}^T}\Big( s^3 \theta^3x^{2+\alpha_2-2\beta}v^2 +s\theta x^{\alpha_2}v_x^2 +\frac{1}{s\theta}v_t^2 + s\theta^{3/2}|\eta p|v^2 \Big)\xi^2e^{2s\varphi (t,x)}\,dt\,dx \\ &\leqslant C\Big(\int_{t_0}^T\!\!\!\int_{\omega'} s^2\theta^2(u^2+v^2) e^{2s\varphi}\,dx\,dt\Big)\\ &\quad +\frac{1}{2}\int_{Q_{t_0}^T} (s^3\theta^3x^{2+\alpha_1-2\beta}u^2+s\theta x^{\alpha_1}u_x^2)\xi^2e^{2s\varphi}\,dt\,dx. \end{aligned} \end{equation} Therefore, summing \eqref{331} and \eqref{3321} we obtain \begin{align*} & \int_{Q_{t_0}^T}\Big( s^3 \theta^3(x^{2+\alpha_1-2\beta}u^2 + x^{2+\alpha_2-2\beta}v^2)+s\theta ( x^{\alpha_1}u_x^2+ x^{\alpha_2}v_x^2) \\ & + \frac{1}{s\theta}(u_t^2+v_t^2)+ s\theta^{3/2}|\eta p|(u^2+v^2) \Big)\xi^2e^{2s\varphi (t,x)}\,dt\,dx \\ &\leqslant C\Big( \int_{Q_{t_0}^T} \xi^2F^2 e^{2s\varphi (t,x)}\,\,dt\,dx +\int_{t_0}^T\!\!\! \int_{\omega'} (F^2+s^2\theta^2(u^2+v^2))e^{2s\varphi} \,dx\,dt\Big). \end{align*} Similarly, applying Proposition \ref{Car1-xi} to each equation of \eqref{sys}, we obtain the estimate \eqref{uvzeta}. \end{proof} \section{Inverse problem} In this section we establish a Lipschitz stability for the term $F$. More precisely, we show some inequalities estimating $F$ with an upper bound given by some measurements of the component $u$ only. For this aim, we start by giving adequate Carleman estimates for a system \eqref{sys}. The following result play a crucial role to absorb the observations on the component $v$. For the proof one can adapt a similar technique used in \cite[Lemma 3.4]{hajjaj} for the adjoint of degenerate parabolic systems. \begin{lemma}\label{absorption} Let $\omega_2\subset\subset\omega_1$. Moreover, assume that $b_{12}\geqslant \mu>0 \quad on\,\, \omega_1$. There is $ C>0$ such that the solution $(u,v)$ of \eqref{sys} satisfies \[ \int_{t_0}^T\int_{\omega_2}s^3\theta^3v^2e^{2s\Phi}\,dx\,dt \leq\varepsilon J(v)+C \int_{Q_{t_0}^T}F^2e^{2s\varphi}\,\,dx\,dt +C\int_{t_0}^T\int_{\omega} u^2\,dx\,dt, \] where $\varepsilon>0$ is small enough, $s$ is large enough and $$ J(v)=\int_{Q_{t_0}^T} \Big(s\theta x^{\alpha_2}v_x^2+s^3 \theta^3 x^{2+\alpha_2-2\beta}v^2\Big)e^{2s\varphi}\,\,dx\,dt. $$ \end{lemma} The following theorem is a consequence of Theorem \ref{Carl-system}, Lemma \ref{absorption} and the fact that \[ \int_{\omega}F^2e^{2s\varphi}\,dx\leqslant 2\int_{\omega}F^2(\xi^2+\zeta^2) e^{2s\varphi}\,dx \leqslant 2\int_0^1F^2(\xi^2e^{2s\varphi}+\zeta^2e^{2s\Phi})\,dx. \] \begin{theorem}\label{caroneforce} Let $T>0$ and $\beta=\max\{\alpha_1,\alpha_2\}$. Moreover, assume that \begin{equation} \label{hypb12} b_{12}\geqslant \mu>0 \quad \text{on } \omega'\Subset\omega. \end{equation} There exist two positive constants $C$ and $s_0$ such that, for every $(u_0,v_0)\in (L^2(0,1))^2$ and for all $s\geqslant s_0$, the solution $(u,v)$ of \eqref{sys} satisfies \begin{equation} \label{Carineq2oneforce} \begin{aligned} J_0(u,v) &:= I(\xi,u,v)+I(\zeta,u,v) \\ & \leqslant C\big\{ \int_{Q_{t_0}^T} F^2(\zeta^2 e^{2s\Phi (t,x)}+\xi^2 e^{2s\varphi (t,x)})\, \,dt\,dx+\int_{t_0}^T\!\!\!\int_{\omega}u^2 \,dt\,dx \big\}\\ &=:J_1(F,u). \end{aligned} \end{equation} \end{theorem} The main result of this article is as follows. \begin{theorem}\label{oneforce} Let $\alpha_1,\alpha_2\in (0,1)$ and $C_0>0$. There exists a positive constant $C=C(T,t_0,s_0,C_0,\alpha_1,\alpha_2)$ such that, for all $F \in S(C_0)$ and $(u_0,v_0)\in (L^2(0,1))^2$, we have \begin{equation}\label{stability} \begin{aligned} &\|F\|_{L^2(Q)}^2 \\ &\leqslant C\Big(\|u\|_{H^1(t_0,T;L^2(\omega))}^2 +\|(x^{\alpha_1}u_x)_x(T',\cdot)\|_{L^2(0,1)}^2+\|u(T',\cdot)\|_{L^2(0,1)}^2\Big). \end{aligned} \end{equation} \end{theorem} \begin{proof} The functions $y=u_t$ and $z=v_t$, where $(u,v)$ is the solution of \eqref{sys}, are solutions of the system \begin{gather*} y_t-(x^{\alpha_1}y_x)_x+b_{11}y+b_{12}z=F_t , \quad (t,x)\in Q,\\ z_t-(x^{\alpha_2}z_x)_x+b_{21}y+b_{22}z=0, \quad (t,x)\in Q,\\ y(t,0)=y(t,1)=z(t,0)=z(t,1)=0, \quad t \in (0,T). \end{gather*} When we apply Carleman estimate \eqref{Carineq2oneforce} to $(y,z)$, we obtain \begin{equation} \label{Carineq2oneforcet} \begin{aligned} J_0(y,z)&:= I(\xi,y,z)+I(\zeta,y,z)\\ &\leqslant C\Big\{ \int_{Q_{t_0}^T} F_t^2(\zeta^2 e^{2s\Phi (t,x)}+\xi^2 e^{2s\varphi (t,x)}) \,dt\,dx +\int_{t_0}^T\!\!\!\int_{\omega} y^2 \,dt\,dx \Big\}\\ &=:J_1(F_t,y). \end{aligned} \end{equation} The terms appearing in \eqref{stability} are well defined, indeed, by Proposition \ref{estimsemigroup}, we have $y\in L^2\left(t_0,T;H^2_{\alpha_1}\right) \cap H^1\left(t_0,T;L^2(0,1)\right)$. As in \cite{BukhgeimKlibanov}, we divide the proof into three steps. \smallskip \noindent\textbf{Step 1.} We show first that there exists a constant $C>0$ such that \begin{equation} \label{step1} \begin{aligned} J_1(F,u)+J_1(F_t,y) &\leqslant C\Big(\frac{1}{\sqrt{s}}\int_0^1F^2(T',x)(\zeta^2e^{2s\Phi(T',x)} +\xi^2e^{2s\varphi(T',x)})dx\\ &\quad + \|u\|_{L^2(\omega_{t_0}^T)}^2+\|u_t\|_{L^2(\omega_{t_0}^T)}^2\Big), \end{aligned} \end{equation} where, we used $T'= \frac{T+t_0}{2}$. To obtain \eqref{step1}, % complete the proof of this step, it remains to prove that \begin{align*} &\int_{Q_{t_0}^T} (F^2+F_t^2)(\zeta^2 e^{2s\Phi (t,x)} +\xi^2 e^{2s\varphi (t,x)})\,\,dx\,dt\\ &\leqslant \frac{C}{\sqrt{s}}\int_0^1F^2(T',x) (\zeta^2e^{2s\Phi(T',x)}+\xi^2e^{2s\varphi(T',x)})dx \end{align*} Since $\Phi_{t}(T')=\varphi_t(T')=0$, $\Phi_{tt}(t)\leqslant -\nu_0<0$ and $\varphi_{tt}(t)\leqslant -\nu_1<0$, then Taylor's formula provides $$ \Phi(t,x)\leqslant \Phi(T',x)-\frac{\nu_0 }{2}(t-T')^2,\quad \varphi(t,x)\leqslant \varphi(T',x)-\frac{\nu_1 }{2}(t-T')^2 $$ and then \begin{gather*} \int_{t_0}^Te^{2s\Phi(t,x)}dt\leqslant \frac{1}{\sqrt{\nu_0s}} e^{2s\Phi(T',x)}\int_{-\infty}^\infty e^{-l^2} dl \leqslant \frac{C}{\sqrt{s}}e^{2s\Phi(T',x)}, \\ \int_{t_0}^Te^{2s\varphi(t,x)}dt\leqslant \frac{1}{\sqrt{\nu_1s}} e^{2s\varphi(T',x)}\int_{-\infty}^\infty e^{-l^2} dl \leqslant \frac{C}{\sqrt{s}}e^{2s\varphi(T',x)}. \end{gather*} So, \begin{align*} &\int_{Q_{t_0}^T}F^2(T',x)(\zeta^2e^{2s\Phi(t,x)}+\xi^2e^{2s\varphi(t,x)})\,dx\,dt\\ &\leqslant \frac{C}{\sqrt{s}}\int_0^1F^2(T',x) (\zeta^2e^{2s\Phi(T',x)}+\xi^2e^{2s\varphi(T',x)})dx . \end{align*} For $F\in S(C_0)$, one has \begin{equation} \label{estimF} |F(t,x)| \leqslant |F(T',x)|+\int_{T'}^t|F_t(s,x)|ds \leqslant C|F(T',x)|. \end{equation} Thus \begin{align*} &\int_{Q_{t_0}^T}(F^2+F_t^2)(\zeta^2e^{2s\Phi(t,x)}+\xi^2e^{2s\varphi(t,x)})\,dx\,dt\\ &\leqslant \frac{C}{\sqrt{s}}\int_0^1F^2(T',x)(\zeta^2e^{2s\Phi(T',x)} +\xi^2e^{2s\varphi(T',x)})dx . \end{align*} The purpose of the first step is then achieved. \smallskip \noindent\textbf{Step 2.} Now, let us show that there exists a constant $C>0$ such that \begin{equation} \label{step2} \int_0^1(y(T',x)+b_{12}v(T',x))^2(\zeta^2e^{2s\Phi(T',x)} +\xi^2e^{2s\varphi(T',x)})dx\leqslant C(J_1(F,u)+J_1(F_t,y)). \end{equation} Since, for a.e. $x\in(0,1)$, $$ \lim_{t\to t_0} (y(t,x)+b_{12}v(t,x))^2(\zeta^2e^{2s\Phi(t,x)} +\xi^2e^{2s\varphi(t,x)})=0. $$ Hence \begin{equation} \label{step21} \begin{aligned} &\int_0^1(y(T',x)+b_{12}v(T',x))^2(\zeta^2e^{2s\Phi(T',x)} +\xi^2e^{2s\varphi(T',x)})dx \\ &= \int_0^1\!\!\!\int_{t_0}^{T'}\frac{\partial}{\partial t} \Big((y+b_{12}v)^2(\zeta^2e^{2s\Phi(t,x)}+\xi^2e^{2s\varphi(t,x)})\Big) \,\,dt\,dx \\ &= \int_{t_0}^{T'}\!\!\!\!\int_0^12(y+b_{12}v)(y_t+b_{12}z) (\zeta^2e^{2s\Phi(t,x)}+\xi^2e^{2s\varphi(t,x)})\,dx\,dt\\ &\quad + \int_{t_0}^{T'}\!\!\!\!\int_0^1(y+b_{12}v)^2(2s\Phi_t\zeta^2 e^{2s\Phi(t,x)}+2s\varphi_t\xi^2e^{2s\varphi(t,x)})\,dx\,dt. \end{aligned} \end{equation} Using the Young inequality, for $s$ large enough, we obtain \begin{equation} \label{step22} \begin{aligned} &\Big|\int_{t_0}^{T'}\!\!\!\!\int_0^1 2(y+b_{12}v) (y_t+b_{12}z)(\zeta^2e^{2s\Phi(t,x)}+\xi^2e^{2s\varphi(t,x)})\,dx\,dt\Big|\\ &\leqslant \int_{Q_{t_0}^T}\Big(s\theta y^2+s\theta z^2+s\theta v^2 +\frac{1}{s\theta}y_t^2\Big)\big(\zeta^2e^{2s\Phi(t,x)} +\xi^2e^{2s\varphi(t,x)}\big)\,dx\,dt. \end{aligned} \end{equation} On one hand, using Young and Hardy-Poincar\'e inequalities, we obtain \begin{equation} \label{step23} \begin{aligned} &\int_{Q_{t_0}^T}s\theta (y^2+z^2)\xi^2e^{2s\varphi(t,x)} \,dt\,dx \leqslant \int_{Q_{t_0}^T}s\theta (x^{\alpha_1-2}y^2+x^{\alpha_2-2}z^2)\xi^2 e^{2s\varphi} \,dt\,dx\\ &\leqslant \int_{Q_{t_0}^T}s\theta \Big\{x^{\alpha_1} [\xi y_x+\xi_xy+s\varphi_x\xi y]^2+x^{\alpha_2} [\xi z_x+\xi_xz+s\varphi_x\xi z]^2\Big\} e^{2s\varphi} \,dt\,dx \\ & \leqslant C\int_{Q_{t_0}^T} \big( s\theta x^{\alpha_1}y_x^2+s^3\theta^3x^{2+\alpha_1-2\beta}y^2+s\theta x^{\alpha_2}z_x^2+s^3\theta^3x^{2+\alpha_2-2\beta}z^2\big) \xi^2e^{2s\varphi} \,dt\,dx\\ &\quad +C\int_{t_0}^T\!\!\!\int_{\omega''}s\theta(y^2+z^2)e^{2s\varphi}\,dx\,dt. \end{aligned} \end{equation} By the definition of $\zeta$, we obtain \begin{equation} \label{step24} \begin{aligned} &\int_{Q_{t_0}^T}s\theta (y^2+z^2)\zeta^2e^{2s\Phi(t,x)} \,dt\,dx\\ &\leqslant C\int_{Q_{t_0}^T} \big( s^3\theta^3x^{2+\alpha_1-2\beta}y^2+ s^3\theta^3x^{2+\alpha_2-2\beta}z^2\big) \zeta^2e^{2s\Phi} \,dt\,dx. \end{aligned} \end{equation} Similarly \begin{align*} &\int_{Q_{t_0}^T}s\theta v^2(\xi^2e^{2s\varphi(t,x)} +\zeta^2e^{2s\Phi(t,x)}) \,dt\,dx \\ & \leqslant C\int_{Q_{t_0}^T} \big(s\theta x^{\alpha_2}v_x^2 +s^3\theta^3x^{2+\alpha_2-2\beta}v^2\big) (\xi^2e^{2s\varphi} +\zeta^2e^{2s\Phi}) \,dt\,dx\\ &\quad +C\int_{t_0}^T\!\!\!\int_{\omega''}s\theta v^2e^{2s\varphi}\,dx\,dt. \end{align*} On the other hand, since $|\theta_t|=|-4\eta\theta^{\frac{5}{4}}|\leqslant C |\eta|\theta ^{3/2}$ and $|\Psi|\leqslant |p|$, we have \begin{equation} \label{step25} \begin{aligned} &\int_{t_0}^{T'}\!\!\!\!\int_0^1(y+b_{12}v)^2 (2s\Phi_t\zeta^2e^{2s\Phi(t,x)}+2s\varphi_t\xi^2e^{2s\varphi(t,x)})\,dx\,dt\\ &\leqslant C \int_{Q_{t_0}^T} s\theta^{3/2}|\eta p|(y^2+v^2) (\xi^2e^{2s\varphi}+\zeta^2e^{2s\Phi}) \,dt\,dx. \end{aligned} \end{equation} Thus, Lemma \ref{absorption}, \eqref{Carineq2oneforce}, \eqref{Carineq2oneforcet} and \eqref{step21}-\eqref{step25} yield the estimate \eqref{step2}. \smallskip \noindent \textbf{Step 3.} Combining \eqref{step1} and \eqref{step2}, we deduce \begin{equation}\label{step31} \begin{aligned} &\int_0^1(y(T',x)+b_{12}v(T',x))^2(\zeta^2e^{2s\Phi(T',x)} +\xi^2e^{2s\varphi(T',x)})dx\\ &\leqslant C\Big(\frac{1}{\sqrt{s}}\int_0^1F^2(T',x)(\zeta^2e^{2s\Phi(T',x)} +\xi^2e^{2s\varphi(T',x)})dx+\|u\|_{L^2(\omega_{t_0}^T)}^2 +\|u_t\|_{L^2(\omega_{t_0}^T)}^2\Big), \end{aligned} \end{equation} Since $y+b_{12}v$ satisfies $$ y(T',x)+b_{12}v(T',x)=(x^{\alpha_1}u_x)_x(T',x)-b_{11}u(T',x)+F(T',x), $$ it follows that \begin{equation} \label{step32} \begin{aligned} & \int_0^1F^2(T',x)(\zeta^2e^{2s\Phi(T',x)}+\xi^2e^{2s\varphi(T',x)})dx\\ &\leqslant C\Big(\|(x^{\alpha_1}u_x)_x(T')\|_{L^2(0,1)}^2+ \|u(T')\|_{L^2(0,1)}^2\\ &+\int_0^1(y(T',x)+b_{12}v(T',x))^2(\zeta^2e^{2s\Phi(T',x)}+\xi^2e^{2s\varphi(T',x)})dx\Big). \end{aligned} \end{equation} Hence, by \eqref{step31}-\eqref{step32}, for $s$ large enough, we obtain \begin{equation} \label{bis} \begin{aligned} &\int_0^1F^2(T',x)(\xi^2e^{2s\varphi(T',x)}+\zeta^2e^{2s\Phi(T',x)})dx\\ &\leqslant C\Big(\|u\|_{H^1(t_0,T;L^2(\omega)}^2 +\|u(T')\|_{L^2(0,1)}^2+\|(x^{\alpha_1}u_x)_x(T')\|_{L^2(0,1)}^2\Big). \end{aligned} \end{equation} Moreover, by $\frac12\leqslant \xi^2+\zeta^2$ and $e^{2s\varphi(T',x)}\leqslant e^{2s\Phi(T',x)}$, then \begin{equation} \begin{aligned}\label{conc} &\gamma\int_0^1F^2(T',x) \,dx\\ &\leqslant C\Big(\|u\|_{H^1(t_0,T;L^2(\omega))}^2 +\|(x^{\alpha_1}u_x)_x(T',\cdot)\|_{L^2(0,1)}^2 +\|u(T',\cdot)\|_{L^2(0,1)}^2\Big), \end{aligned} \end{equation} where $\gamma=\min_{x\in[0,1]}\{ e^{2s\varphi(T',x)}\}$. Thus \eqref{estimF} and \eqref{conc} gives the thesis. \end{proof} \section{Appendix: Cacciopoli inequality} As in \cite{bahm,hajjaj,CanTer}, we adapt the proof of the Caccioppoli inequality for nonhomogenuous degenerate parabolic equations. Let $\omega_1$ and $\omega_2$ two arbitrary non empty open subsets of $(0,1)$ such that $\overline{\omega}_2 \subset \omega_1$. Consider the equation \begin{equation}\label{equationCacc} U_t -(x^{\alpha} U_x)_x + b(x) U=F_1(t,x) ,\quad (t,x)\in \omega_1\times (t_0,T):=Q_{\omega_1}, \end{equation} where $F_1\in L^2(Q_{\omega_1})$ and $b\in L^\infty(Q_{\omega_1})$. \begin{lemma} Let $\mu_1, \mu_2\in C^{2}(\overline{\omega}_1,\mathbb{R}^-)$ such that $\frac{4}{3}\mu_2\leqslant\mu_1\leqslant \mu_2$. Then, there exists a constant $C>0$ such that for any solution $U$ of \eqref{equationCacc}, one has \begin{align}\label{Caccioppoli-ineq} \int_{t_0}^T\!\!\!\!\int_{\omega_2} U_x^2 e^{2s\theta(t)\mu_2(x)} \,dt\,dx \leqslant C\Big( \int_{t_0}^T\!\!\!\!\int_{\omega_1} U^2 \,dt\,dx +\int_{t_0}^T\!\!\!\!\int_{\omega_1} F_1^2 e^{2s\theta(t)\mu_1(x)} \,dt\,dx \Big), \end{align} where $\theta(t)=\frac{1}{t^{k}(T-t)^k}$, $k\geqslant 1$. \end{lemma} \begin{proof} Let $\chi\in\mathcal{C}^\infty(0,1)$ such that $\operatorname{supp}\chi \subset \omega_1$ and $\chi \equiv 1$ on $\omega_2$. We have \begin{align*} 0&=\int_{t_0}^T\frac{d}{dt} \Big[\int_0^1\chi^2 U^2 e^{2s\theta(t)\mu_2(x)}dx \Big] dt\\ &= -2\int_{Q_{t_0}^T} \chi^2 x^{\alpha}U_x^2 e^{2s\theta(t)\mu_2(x)} \,dx\,dt + \int_{Q_{t_0}^T} (x^{\alpha}(\chi^2 e^{2s\theta(t)\mu_2(x)})_x)_x U^2 \,dx\,dt \\ &\quad -2\int_{Q_{t_0}^T} b\chi^2 U^2 e^{2s\theta(t)\mu_2(x)} \,dx\,dt +2\int_{Q_{t_0}^T} s \varphi_t\chi^2U^2e^{2s\theta(t)\mu_2(x)} \,dx\,dt\\ &\quad +2\int_{Q_{t_0}^T} \chi^2 UF_1 e^{2s\theta(t)\mu_2(x)} \,dx\,dt. \end{align*} Then \begin{align*} &\int_{Q_{t_0}^T} \chi^2 x^{\alpha}U_x^2 e^{2s\theta(t)\mu_2(x)} \,dx\,dt\\ &= \int_{Q_{t_0}^T} (x^{\alpha}(\chi^2 e^{2s\theta(t)\mu_2(x)})_x)_x U^2 dx -\int_{Q_{t_0}^T} b\chi^2 U^2 e^{2s\theta(t)\mu_2(x)} \,dx\,dt\\ &\quad +\int_{Q_{t_0}^T} s \varphi_t\chi^2U^2e^{2s\theta(t)\mu_2(x)} \,dx\,dt +\int_{Q_{t_0}^T} \chi^2 UF_1 e^{2s\theta(t)\mu_2(x)} \,dx\,dt\\ & \leqslant \int_{Q_{t_0}^T} \Big\{(x^{\alpha}(\chi^2 e^{2s\theta(t)\mu_2(x)})_x)_x U^2 + \chi^2 U^2(b e^{2s\theta(t)\mu_2(x)} + s \varphi_te^{2s\theta(t)\mu_2(x)}\\ &\quad +\frac12 e^{2s\theta(t)(2\mu_2(x)-\mu_1(x))}) \Big\} \,dx\,dt +\frac12\int_{Q_{t_0}^T} F_1 e^{2s\theta(t)\mu_1(x)} \,dx\,dt. \end{align*} Therefore, since $\chi$ is supported in $\omega_1$, $\chi\equiv 1$ in $\omega_2$ and $\frac{4}{3}\mu_2\leqslant\mu_1\leqslant \mu_2$. Then, one obtains \begin{align*} &\int_{t_0}^T\!\!\!\!\int_{\omega_2} U_x^2 e^{2s\theta(t)\mu_2(x)} \,dx\,dt \\ &\leqslant C \int_{Q_{t_0}^T} \chi^2 x^{\alpha}U_x^2 e^{2s\theta(t)\mu_2(x)} \,dx\,dt\\ &\leqslant C\Big(\int_{t_0}^T\!\!\!\!\int_{\omega_1} s^2\theta^2 U^2 e^{2s\theta(t)(2\mu_2(x)-\mu_1(x))} \,dx\,dt +\int_{t_0}^T\!\!\!\!\int_{\omega_1} F_1^2 e^{2s\theta(t)\mu_1(x)} \,dx\,dt \Big) \\ &\leqslant C\Big(\int_{t_0}^T\!\!\!\!\int_{\omega_1} U^2 \,dx\,dt +\int_{t_0}^T\!\!\!\!\int_{\omega_1} F_1^2 e^{2s\theta(t)\mu_1(x)} \,dx\,dt \Big). \end{align*} This completes the proof. \end{proof} \begin{thebibliography}{99} \bibitem{bahm} E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj, L. Maniar; \emph{Null controllability of degenerate parabolic cascade systems Portugaliae Mathematica}, \textbf{68} (2011), 345-367. \bibitem{hajjaj} E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj, L. Maniar; \emph{Carleman estimate and null controllability of degenerate parabolic systems}, Evol. Equ. 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