\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 189, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/189\hfil Backward uniqueness] {Backward uniqueness for heat equations with coefficients of bounded variation in time} \author[S. Tarama \hfil EJDE-2013/189\hfilneg] {Shigeo Tarama} \address{Shigeo Tarama \newline Lab. of Applied Mathematics, Osaka City University, Osaka 558-8585, Japan} \email{starama@mech.eng.osaka-cu.ac.jp} \thanks{Submitted February 27, 2013. Published August 28, 2013.} \subjclass[2000]{35K05, 16D10} \keywords{Heat equation; backward Cauchy problem; uniqueness} \begin{abstract} Uniqueness of solutions to the backward Cauchy problem for heat equations with coefficients of bounded variation in time is shown through the Carleman estimate. \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} We consider a heat operator in the time backward form $$\label{heat} Lu=\partial_t+\sum_{j,k=1}^d\partial_{x_j}(a_{jk}(x,t)\partial_{x_j}u),$$ with real bounded and measurable coefficients $a_{jk}(x,t)$ on $\mathbb{R}^d\times [0,T]$, for some $T>0$, satisfying $a_{jk}(x,t)=a_{kj}(x,t)$ ($j,k=1,2,\dots,d$) and $$\label{elliptic} \sum_{j,k=1}^d a_{jk}(x,t)\xi_j\xi_k \ge D_0|\xi|^2$$ for any $\xi\in \mathbb{R}^d$ with some positive $D_0$. The Cauchy problem for $Lu=f$ on $\mathbb{R}^d\times [0,T]$ with Cauchy data on $t=0$ is not well-posed. But the uniqueness of solutions to the Cauchy problem is valid under some conditions on the coefficients. Since the work of Mizohata \cite{mz}, there are many works on this problem. See for example the survey paper of Vessella \cite{vess} and the papers cited therein. But it seems that the backward uniqueness for heat operators with discontinuous coefficients is not well studied. We consider an operator whose coefficients $a_{jk}(x,t)$ ($j,k=1,2,\dots,d$) are of bounded variation in $t$ uniformly with respect to $x\in \mathbb{R}^d$. That is, there exists a constant $M\ge 0$ such that we have $$\label{hyp-2} \sum_{l=1}^L\sup_{x \in \mathbb{R}^d}|a_{jk}(x,t_l)-a_{jk}(x,t_{l-1})|\le M$$ for any partition of $[0,T]$, $t_0=01$. Then $\phi_nT_a u-T_a \phi_n u$ is equal to $\sum_{l;|l-n|\le 2}\phi_na_{l-3}\phi_l u-a_{l-3}\phi_l \phi_n u,$ which, using the symbol of commutator, is equal to $\sum_{l;|l-n|\le 2}[\phi_n,a_{l-3}]\phi_l u .$ Since $a_l(x)=\int_{\mathbb{R}^d}a(y)2^{ld}P(2^l(x-y))\,dy$ where $P(x)$ is the inverse Fourier transform of $\phi_0(\xi)$, then we have $|a_l(x)-a_l(y)|\le C\|a\|_{W^{1,\infty}} |x-y|$. Note that $[\phi_n,a_{l-3}]u$ is equal to $2^{nd}\int_{\mathbb{R}^d}Q(2^n(x-y))(a_l(y)-a_l(x))u(y)\,dy$ where $Q(x)=P(x)-2^{-d}P(x/2)$ if $n\ge 1$ and $Q(x)=P(x)$ if $n=0$. Then $|[\phi_n,a_{l-3}]\phi_l u(x)|\le C\|a\|_{W^{1,\infty}}2^{-n}2^{nd} \int_{\mathbb{R}^d}P_1(2^n(x-y))|u(y)|\,dy$ where $P_1(x)=|P(x)||x|$. Since $P_1(x)$ is integrable, we have $\|[\phi_n,a_{l-3}]\phi_l u\|\le C\|a\|_{W^{1,\infty}}2^{-n}\|u\|.$ If $|l-n|\le 2$, the spectrum of $[\phi_n,a_{l-3}]\phi_l u$ is contained in $|\xi|\le2^{ n+2}$, then $\|[\phi_n,a_{l-3}]\phi_l u\|_1\le (2^{n+4}+1)\|[\phi_n,a_{l-3}]\phi_l u\|.$ Hence we obtain $\|\sum_{l; |l-n|\le 2}[\phi_n,a_{l-3}]\phi_l u\|_1\le \sum_{l; |l-n|\le 2}C\|a\|_{W^{1,\infty}}\|\phi_l u\|.$ Since $\phi_nT_a u-T_a \phi_n u=\sum_{l; |l-n|\le 2}[\phi_n,a_{l-3}]\phi_l u$, we have $\|\phi_nT_a u-T_a \phi_n u\|_1\le \sum_{l; |l-n|\le 2}C\|a\|_{W^{1,\infty}}\|\phi_l u\|,$ from which we obtain \eqref{commutator-1}. \end{proof} \subsection{Bounded variation} Next we recall the properties of functions with bounded variation. (See, for example, the appendix of \cite{Br} for the detail.) Let $X$ be a Banach space with a norm $\|\cdot\|_X$ and let $f(t)$ be a $X$-valued function on $[0,T]$ with bounded variation; that is, whose total variation $V(f,[0,T])$ given by $V(f,[0,T])=\sup_{\substack{\text{any partition of [0,T] }\\ t_0=00 there exists a positive T_{\varepsilon} such that we have \[ \int_0^{T_{\varepsilon}}\|f(t+h)-f(t)\|_X\,dt\le h\varepsilon$ for any $0\le h\le T_{\varepsilon}$. \begin{remark}\label{rem-bv} \rm It follows from the argument above and assumption \eqref{hyp-2} that the right limit $\lim_{h\searrow 0}a_{jk}(x,t+h)$ converges uniformly on $\mathbb{R}^d_x$ for any $t\in [0,T)$. Then we see that \eqref{elliptic} and \eqref{hyp-3} still hold for $a_{jk}(x,t+0)=\lim_{h\searrow 0}a_{jk}(x,t+h)$. Furthermore, we have $a_{jk}(x,t+0)=a_{jk}(x,t)$ except for at most countably many $t$. Then, in Theorem \ref{thm1.1}, we may assume that $a_{jk}(x,t+0)=a_{jk}(x,t)$ on $[0,T]$ uniformly with respect to $x\in \mathbb{R}^d$. \end{remark} \section{Carleman estimate} Noting Remark \ref{rem-bv}, we may assume that for any positive $\varepsilon$, there exist $T_{\varepsilon}>0$ such that we have $$\label{3-1} \int_0^{T_{\varepsilon}}\sum_{j,k=1}^d\|a_{jk}(\cdot,t+h)-a_{jk} (\cdot,t)\|_{L^\infty}\,dt\le \varepsilon h$$ for any $h\in [0,T_{\varepsilon}]$. Here we may assume that $T_{\varepsilon}\le \varepsilon$. We define $\psi_{\gamma}(t)$ and $\psi_{1,\gamma}(t)$ with $\gamma \ge 1/T_{\varepsilon}$ by \begin{gather*} \psi_{\gamma}(t)=\int_t^{T_{\varepsilon}}\Bigl(\frac{1}{\varepsilon} +\frac{1}{\varepsilon}\sum_{j,k=1}^d\gamma \int_0^1\|a_{jk}(x,\tau+\frac{s}{\gamma})-a_{jk}(x,\tau)\|_{L^\infty}\,ds\Bigr) \,d\tau, \\ \psi_{1,\gamma}(t)=\gamma\int_t^{T_{\varepsilon}} e^{\psi_{\gamma}(\tau)}\,d\tau. \end{gather*} We note that, since $\sum_{j,k=1}^d\int_0^{T_{}\varepsilon} \|a_{jk}(x,t+\frac{s}{\gamma})-a_{jk}(x,t)\|_{L^\infty}\,dt\le \frac{\varepsilon s}{\gamma}$ for $s\in[0,1]$ and $\gamma\ge 1/T_\varepsilon$, we have, on $[0,T_{\varepsilon}]$, $$\label{est-phi} 0\le \psi_{\gamma}(t) \le \frac{3}{2}.$$ In this section we show the following Carleman estimate. \begin{proposition} \label{calreman-prop} There exists a positive constant $\varepsilon_0$ so that, for any $\varepsilon\in(0,\varepsilon_0)$ we have, with a positive $\gamma_{\varepsilon}$, \label{calreman} \begin{aligned} &\frac{\gamma}{\varepsilon}\int_0^{T_{\varepsilon}} e^{2\psi_{1,\gamma}(t)}\|u(\cdot,t)\|^2\,dt+ \frac{1}{\varepsilon}\int_0^{T_{\varepsilon}} e^{2\psi_{1,\gamma}(t)}\|\nabla u(\cdot,t)\|^2\,dt\\ &\le C\int_0^{T_{\varepsilon}}e^{2\psi_{1,\gamma}(t)}\|Lu(\cdot,t)\|^2\,dt \end{aligned} for any $\gamma\ge \gamma_{\varepsilon}$ and any $u(x,t)\in L^2(\mathbb{R}^d \times [0,T_{\varepsilon}])$ satisfying $\partial_{x_j} u(x,t)\in L^2(\mathbb{R}^d \times [0,T_{\varepsilon}])$ ($j=1,2,\dots,d$) and $Lu\in L^2(\mathbb{R}^d \times [0,T_{\varepsilon}])$, $u(x,0)=0$ and $u(x,T_{\varepsilon})=0$. Here the constant $C$ is independent of $\varepsilon$ and of $\gamma$. \end{proposition} We define the operator $L_\gamma$ by $L_\gamma=e^{\psi_{1,\gamma}(t)}L e^{-\psi_{1,\gamma}(t)}$; that is, $L_\gamma u=\partial_tu+\gamma e^{\psi_{\gamma}(t)}u +\sum_{j,k=1}^d\partial_{x_j}( a_{jk}(x,t)\partial_{x_j}u).$ Then, by replacing $u$ by $e^{\psi_{1,\gamma}(t)}u$, \eqref{calreman} is equivalent to $$\label{calreman-1} \frac{\gamma}{\varepsilon}\int_0^{T_{\varepsilon}}\|u(\cdot,t)\|^2\,dt+ \frac{1}{\varepsilon}\int_0^{T_{\varepsilon}}\|\nabla u(\cdot,t)\|^2\,dt \le C\int_0^{T_{\varepsilon}}\|L_\gamma u(\cdot,t)\|^2\,dt.$$ We remark that from \eqref{commutator-1} it follows that $\sum_{n=0}^\infty \|\phi_n \partial_{x_j}(a_{jk}\partial_ku) -\partial_{x_j}(a_{jk}\partial_k\phi_n u)\|^2\le C\|\partial_{x_{k}}u\| ^2,$ from which we obtain $\sum_{n=0}^\infty \|\phi_n L_{\gamma}u-L_{\gamma}\phi_n u\|^2\le C\|u\|_1 ^2.$ Then, by \eqref{l-2} we get $$\label{sum} \sum_{n=0}^\infty \|L_{\gamma}\phi_n u\|^2 \le C( \| L_{\gamma}u\|^2 +\|u\|_1 ^2).$$ Therefore, we consider the estimate of $\|L_{\gamma}\phi_n u\|$. Note that \eqref{elliptic} implies $$\label{Garding} \sum_{j,k=1}^d(a_{jk}(x,t)\partial_{x_{k}}v,\partial_{x_{j}}v)\geq D_0\|\nabla v\|^2,$$ from which and from \eqref{est-phi} we obtain the following: for $u(x,t)$ satisfying $u(x,0)=0$ and $u(x,T_{\varepsilon_0})=0$, $- \int_0^{T_\varepsilon}(L_{\gamma}\phi_n u,\phi_n u) \ge D_0\int_0^{T_\varepsilon}\|\nabla (\phi_n u)\|^2\,dt -\gamma e^{3/2}\int_0^{T_\varepsilon}\|\phi_n u\|^2\,dt.$ When $\frac{D_0}{4}2^{2(n-1)}\ge \gamma e^{3/2}$ and $n\ge 1$, we see, noting $\|\nabla (\phi_n u)\|^2\ge 2^{2(n-1)}\|\phi_n u\|^2$, that $- \int_0^{T_\varepsilon}(L_{\gamma}\phi_n u,\phi_n u) \ge \frac{D_0}{2}\int_0^{T_\varepsilon}\|\nabla (\phi_n u)\|^2\,dt +\gamma e^{3/2}\int_0^{T_\varepsilon}\|\phi_n u\|^2\,dt.$ Hence, by $|(L_{\gamma}\phi_n u,\phi_n u)| \le \frac{\varepsilon}{2}\|L_{\gamma}u\|^2+\frac{1}{2\varepsilon}\|u\|^2$ we get $$\label{carleman-00} \varepsilon\int_0^{T_\varepsilon}\|L_{\gamma}u\|^2\,dt\ge D_0\int_0^{T_\varepsilon}\|\nabla (\phi_n u)\|^2\,dt +\int_0^{T_\varepsilon}( 2\gamma-\frac{1}{\varepsilon})\|\phi_n u\|^2\,dt.$$ For the case where $\frac{D_0}{4}2^{2(n-1)}\le \gamma e^{3/2}$ with $\gamma\ge 1/T_{\varepsilon}$, we have the following lemma. \begin{lemma}\label{carleman-lemma} There exists a positive $\varepsilon_0$ such that under the condition that \eqref{3-1} is valid for $0<\varepsilon<\varepsilon_0$, we have the following estimates. When $0<\varepsilon<\varepsilon_0$ and $\frac{D_0}{4}2^{2(n-1)}\le \gamma e^{3/2}$ with $\gamma\ge 1/T_{\varepsilon}$, we have $$\label{carleman-01} C\int_0^{T_{\varepsilon}}\| L_{\gamma}\phi_n u\|^2\, dt \geq \frac{1}{\varepsilon}\int_0^{T_{\varepsilon}}\|\nabla\phi_n u\|^2\,dt+ \frac{\gamma}{\varepsilon}\int_0^{T_{\varepsilon}}\|\phi_n u\|^2\,dt$$ for any $u(x,t)$ satisfying $u(x,0)=0$ and $u(x,T_{\varepsilon_0})=0$ \end{lemma} \begin{proof} Note that \begin{align*} \| L_{\gamma}\phi_n u\|^2 &=\|\partial_t(\phi_n u)\|^2+ \|\gamma e^{\psi_{\gamma}}\phi_n u+\sum_{j,k=1}^d\partial_{x_j}(a_{jk} \partial_{x_k}\phi_n u)\|^2 \\ &\quad + 2\Re (\partial_t(\phi_n u),\gamma e^{\psi_{\gamma}}\phi_n u) + \sum_{j,k=1}^d 2\Re (\partial_t(\phi_n u),\partial_{x_j}(a_{jk} \partial_{x_k}\phi_n u)). \end{align*} Let $\chi(s)\in C^{\infty}(\mathbb{R})$ satisfy $\chi(s)\ge 0$ on $\mathbb{R}$, $\chi(s)=0$ on $(-\infty,0]\cup [1,\infty)$ and $\int_{-\infty}^{\infty}\chi(s)\,ds=1$. Set $D_1=\sup_{s\in\mathbb{R}}|\chi(s)|+|\chi'(s)|$. We define the regularization of $a_{jk}$, $a^{\gamma}_{jk}(x,t)$, by $a_{jk}^{\gamma}(x,t)=\gamma\int_{-\infty}^{\infty}\chi(\gamma (s-t)) a_{jk}(x,s)\,ds.$ We see that $a_{jk}^{\gamma}(x,t)=\int_{-\infty}^{\infty}\chi(s)a_{jk}(x,t+s/\gamma)\,ds$ from which and from $\int_{-\infty}^{\infty}\chi(s)\,ds=1$ we see $a_{jk}^{\gamma}(x,t)-a_{jk}(x,t)=\int_{-\infty}^{\infty}\chi(s)(a_{jk} (x,t+s/\gamma) -a_{jk}(x,t))\,ds,$ while from $\partial_t a^{\gamma}(x,t)=-\gamma\int_{-\infty}^{\infty}\chi'(s)a_{jk} (x,t+s/\gamma)\,ds$ and $\int_{-\infty}^{\infty}\chi'(s)\,ds=0$, it follows that $\partial_ta_{jk}^{\gamma}(x,t)=-\gamma\int_{-\infty}^{\infty}\chi'(s) (a_{jk}(x,t+s/\gamma) -a_{jk}(x,t))\,ds.$ Then we have \begin{gather*} | a_{jk}^{\gamma}(x,t)-a_{jk}(x,t)|\le D_1\int_0^1|a_{jk}(x,t+s/\gamma) -a_{jk}(x,t)|\,ds,\\ | \partial_ta_{jk}^{\gamma}(x,t)|\le D_1\gamma\int_0^1|a_{jk}(x,t+s/\gamma) -a_{jk}(x,t)|\,ds. \end{gather*} Furthermore we note that $| a_{jk}^{\gamma}(x,t)|\le \|a_{jk}(\cdot,t)\|_{L^\infty}$ implies $| a_{jk}^{\gamma}(x,t)-a_{jk}(x,t)|\le 2\|a_{jk}(\cdot,t)\|_{L^\infty}.$ Then we have $| a_{jk}^{\gamma}(x,t)-a_{jk}(x,t)| \le \sqrt{2\|a_{jk}(\cdot,t)\|_{L^\infty}} \Big(D_1\gamma\int_0^1|a_{jk}(x,t+s/\gamma) -a_{jk}(x,t)|\,ds\Big)^{1/2}.$ Using the estimates above, we estimate the term $(\partial_t(\phi_n u),\partial_{x_j}(a_{jk}\partial_{x_k}\phi_n u))$. Note that \begin{aligned} &(\partial_t(\phi_n u),\partial_{x_j}(a_{jk}\partial_{x_k}\phi_n u))\\ &= (\partial_t(\phi_n u),\partial_{x_j}(a_{jk}^{\gamma}\partial_{x_k}\phi_n u)) +(\partial_t(\phi_n u),\partial_{x_j}((a_{jk}-a_{jk}^{\gamma}) \partial_{x_k}\phi_n u)). \end{aligned} Note that $|(\partial_t(\phi_n u),\partial_{x_j}((a_{jk}-a_{jk}^{\gamma})\partial_{x_k} \phi_n u))|= |(\partial_t\partial_{x_j}(\phi_n u),((a_{jk}-a_{jk}^{\gamma})\partial_{x_k} \phi_n u))|$ which is dominated by $2^{n+1}\|\partial_t(\phi_n u)\|\sqrt{2\|a_{jk}(\cdot,t)\|_{L^\infty}} \Big(D_1\int_0^1\|a_{jk}(\cdot,t+s/\gamma) -a_{jk}(\cdot,t)\|_{L^\infty}\,ds\Big)^{1/2}\|\nabla \phi_n u\|.$ Here we used $\phi_n(\xi)=0$ for $|\xi|\ge 2^{n+1}$. Setting $K=\sum_{j,k=1}^d\sup_{t\in[0,T_{\varepsilon}]} \|a_{jk}(\cdot,t)\|_{L^\infty}$, we obtain, from Schwarz's inequality, \begin{align*} &\sum_{j,k=1}^d|(\partial_t(\phi_n u),\partial_{x_j}((a_{jk}-a_{jk}^{\gamma}) \partial_{x_k}\phi_n u))| \\ &\leq 2^{n+1}\|\partial_t(\phi_n u)\|\sqrt{2K} \Big(D_1\sum_{j,k=1}^d \int_0^1\|a_{jk}(\cdot,t+s/\gamma) -a_{jk}(\cdot,t)\|_{L^\infty}\,ds\Big)^{1/2} \|\nabla \phi_n u\|. \end{align*} % 3.11 Then we get \begin{align*} &2\sum_{j,k=1}^d|(\partial_t(\phi_n u),\partial_{x_j}((a_{jk} -a_{jk}^{\gamma})\partial_{x_k}\phi_n u))|\\ &\le \|\partial_t(\phi_n u)\|^2+ 2^{2(n+1)+1}KD_1\sum_{j,k=1}^d \int_0^1\|a_{jk}(\cdot,t+s/\gamma)-a_{jk}(\cdot,t)\|_{L^\infty}\,ds \|\nabla \phi_n u\|^2. \end{align*} % 3.12 Hence \begin{align*} &\|\partial_t(\phi_n u)\|^2+2\sum_{j,k=1}^d \Re (\partial_t(\phi_n u), \partial_{x_j}((a_{jk}-a_{jk}^{\gamma})\partial_{x_k}\phi_n u)) \\ &\geq - 2^{2(n+1)+1}KD_1\sum_{j,k=1}^d\int_0^1\|a_{jk}(\cdot,t+s/\gamma) -a_{jk}(\cdot,t)\|_{L^\infty}\,ds \|\nabla \phi_n u\|^2. \end{align*} On the other hand, noting that \begin{align*} &\sum_{j,k=1}^d 2\Re (\partial_t(\phi_n u),\partial_{x_j}(a_{jk}^{\gamma} \partial_{x_k}\phi_n u))\\ &=\sum_{j,k=1}^d(\partial_{x_j}\phi_n u,(\partial_ta_{jk}^{\gamma}) \partial_{x_k}\phi_n u)- \sum_{j,k=1}^d\partial_t(\partial_{x_j}\phi_n u,a_{jk}^{\gamma} \partial_{x_k}\phi_n u), \end{align*} we see that, when $u(x,0)=0$ and $u(x,T_{\varepsilon_0})=0$ are satisfied, \begin{align*} &|\int_0^{T_{\varepsilon}}2\Re (\partial_t(\phi_n u), \partial_{x_j}(a^{\gamma}_{jk}\partial_{x_k}\phi_n u))\,dt|\\ &\leq \int_0^{T_{\varepsilon}}(\sum_{j,k=1}^d D_1\gamma\int_0^1\|a_{jk} (\cdot,t+s/\gamma)-a_{jk}(\cdot,t)\|_{L^\infty}\,ds)\|\nabla \phi_n u\|^2\,dt. \end{align*} Therefore, we see that, when $u(x,0)=0$ and $u(x,T_{\varepsilon_0})=0$ are satisfied, \begin{align*} & \int_0^{T_{\varepsilon}} \Bigl( \|\partial_t(\phi_n u)\|^2+ 2\Re (\partial_t(\phi_n u), \sum_{j,k=1}^d\partial_{x_j}(a_{jk}\partial_{x_k}\phi_n u)) \Bigr)\,dt\\ &\geq -( 2^{2(n+1)+1}K+\gamma)D_1\sum_{j,k=1}^d\int_0^{T_\varepsilon} \int_0^1\|a_{jk}(\cdot,t+s/\gamma)-a_{jk}(\cdot,t)\|_{L^\infty}\,ds \|\nabla \phi_n u\|^2\,dt. \end{align*} Similarly, we have $2\Re (\partial_t(\phi_n u),\gamma e^{\psi_{\gamma}}\phi_n u)= \partial_t(\phi_n u,\gamma e^{\psi_{\gamma}}\phi_n u)-\gamma (\frac{d}{dt}e^{\psi_{\gamma}})\|\phi_n u\|^2.$ We note that $-\gamma (\frac{d}{dt}e^{\psi_{\gamma}})=\gamma e^{\psi_{\gamma}}\times (\frac{1}{\varepsilon}+\frac{\gamma}{\varepsilon} \sum_{j,k=1}^d \int_0^1\|a_{jk}(\cdot,t+s/\gamma)-a_{jk}(\cdot,t)\|_{L^\infty}\,ds)$ from which and from $e^{\psi_{\gamma}(t)}\ge 1$, we obtain $-\gamma (\frac{d}{dt}e^{\psi_{\gamma}})\ge \frac{\gamma}{\varepsilon}(1+\gamma \sum_{j,k=1}^d \int_0^1\|a_{jk}(\cdot,t+s/\gamma)-a_{jk}(\cdot,t)\|_{L^\infty}\,ds)\,.$ Then we see that, when $u(x,0)=0$ and $u(x,T_{\varepsilon})=0$ are satisfied, \begin{align*} &\int_0^{T_{\varepsilon}}2\Re (\partial_t(\phi_n u),\gamma e^{\psi_{\gamma}} \phi_n u)\,dt \\ &\ge \int_0^{T_{\varepsilon}}\frac{\gamma}{\varepsilon}(1+\gamma \sum_{j,k=1}^d \int_0^1\|a_{jk}(\cdot,t+s/\gamma)-a_{jk}(\cdot,t)\|_{L^\infty}\,ds) \|\phi_n u\|^2\,dt \end{align*} Therefore, \begin{align*} \int_0^{T_{\varepsilon}}\| L_{\gamma}\phi_n u\|^2\,dt &\ge \int_0^{T_{\varepsilon}}\frac{\gamma}{\varepsilon}\|\phi_n u\|^2\,dt \\ &\quad +\int_0^{T_{\varepsilon}}(\frac{\gamma^2}{\varepsilon} \|\phi_n u\|^2 -(2^{2(n+1)+1}KD_1+D_1\gamma)\|\nabla \phi_n u\|^2 )\\ &\quad \times \Big(\sum_{j,k=1}^d \int_0^1\|a_{jk} (\cdot,t+s/\gamma)-a_{jk}(\cdot,t)\|_{L^\infty}\,ds \Big)\,dt. \end{align*} Since $\|\nabla \phi_n u\|^2\le 2^{2(n+1)}\|\phi_n u\|^2$, when $\frac{D_0}{4}2^{2(n-1)}\leq \gamma e^{3/2}$, we have \begin{gather*} (2^{2(n+1)+1}KD_1+D_1\gamma)\|\nabla \phi_n u\|^2 \le C \gamma^2 \| \phi_n u\|^2, \\ \|\nabla \phi_n u\|^2\le C\gamma \| \phi_n u\|^2. \end{gather*} Choosing $\varepsilon_0$ small, we have, for $0<\varepsilon<\varepsilon_0$, $(\frac{\gamma^2}{\varepsilon} \|\phi_n u\|^2 -(2^{2(n+1)+1}KD_1+D_1\gamma)\|\nabla \phi_n u\|^2)\ge 0.$ Hence $\int_0^{T_{\varepsilon}}\| L_{\gamma}\phi_n u\|^2\,dt \ge \int_0^{T_{\varepsilon}}\frac{\gamma}{\varepsilon}\|\phi_n u\|^2\,dt.$ Using $\gamma \| \phi_n u\|^2\ge \frac{1}{C}\|\nabla \phi_n u\|^2$, we obtain $\int_0^{T_{\varepsilon}}\| L_{\gamma}\phi_n u\|^2\,dt \ge \int_0^{T_{\varepsilon}}(\frac{\gamma}{2\varepsilon}\|\phi_n u\|^2+ \frac{1}{2C\varepsilon}\|\nabla\phi_n u\|^2) \,dt.$ \end{proof} Now we complete the proof of Proposition \ref{calreman-prop}. We choose $\varepsilon_0$ and $\gamma_{\varepsilon}$ so that the assertion of Lemma \ref{carleman-lemma} is valid. Furthermore we choose $\gamma_{\varepsilon}$ so large that we have $\gamma_{\varepsilon}> 2/\varepsilon$. Then we obtain from \eqref{carleman-00} $C\int_0^{T_\varepsilon}\|L_{\gamma}\phi_nu\|^2\,dt\ge \frac{1}{\varepsilon}\int_0^{T_\varepsilon}\|\nabla \phi_nu\|^2\,dt+ \frac{\gamma}{\varepsilon}\int_0^{T_\varepsilon}\| \phi_nu\|^2\,dt$ for $n$ satisfying $\frac{D_0}{4}2^{2(n-1)}\geq \gamma e^{3/2}$ with $\gamma\ge \gamma_\varepsilon$. Hence it follows from the estimate above, \eqref{carleman-01} and \eqref{l-2} that $C\sum_{n=0}^{\infty}\int_0^{T_\varepsilon}\|L_{\gamma}\phi_nu\|^2\,dt\ge \frac{1}{\varepsilon}\int_0^{T_\varepsilon}\|\nabla u\|^2\,dt+ \frac{\gamma}{\varepsilon}\int_0^{T_\varepsilon}\| u\|^2\,dt.$ Noting \eqref{sum}, we have $C\int_0^{T_\varepsilon}(\|L_{\gamma}u\|^2+\|u\|_1^2)\,dt\ge \frac{1}{\varepsilon}\int_0^{T_\varepsilon}\|\nabla u\|^2\,dt+ \frac{\gamma}{\varepsilon}\int_0^{T_\varepsilon}\| u\|^2\,dt.$ Then, by choosing $\varepsilon_0$ so small, we obtain the desired estimate \eqref{calreman-1}. The proof of Proposition \ref{calreman-prop} is complete. \section{Proof of Theorem \ref{thm1.1}} First, we show time local uniqueness under the assumptions of Theorem \ref{thm1.1}. Then, using the well-known continuity argument, we show that the assertion of Theorem \ref{thm1.1} is valid. \begin{proposition}\label{local-u1} Under the assumptions of Theorem \ref{thm1.1}, there exists $t_0\in(0,T]$ such that we have $u(x,t)=0$ for $t\in[0,t_0]$. \end{proposition} \begin{proof} Set $f=Lu$. Then we see from the assumptions of Theorem \ref{thm1.1}, that $f\in L^2([0,T],L^2(\mathbb{R}^d ))$ and $\|f(\cdot,t)\|^2\le C_0(\|\nabla u(\cdot,t)\|^2+\| u(\cdot,t)\|^2)$ for almost all $t\in[0,T]$. Let the non-negative function $\chi_0(t)\in C^{\infty}(\mathbb{R})$ satisfy $\chi_0(t)= \begin{cases} 1& t<3/4\\ 0& t>7/8\,. \end{cases}$ Set $u_{\varepsilon}(x,t)=\chi_0(t/T_{\varepsilon})u(x,t)$. Here we use the notation of Proposition \ref{calreman-prop}. Then we see that $u_{\varepsilon}(x,0)=0$, $u_{\varepsilon}(x,T_\varepsilon)=0$ and that $Lu_{\varepsilon}=\chi_0(t/T_{\varepsilon})f+\frac{\chi_0'(t/T_{\varepsilon})}{T_\varepsilon}u.$ Then from \eqref{calreman} we obtain \begin{align*} &\int_0^{T_\varepsilon}(\frac{1}{\varepsilon}\|\nabla u_\varepsilon \|^2 +\frac{\gamma}{\varepsilon}\|u_\varepsilon \|^2)e^{2\psi_{1,\gamma}(t)}\,dt\\ &\le C \int_0^{T_\varepsilon}(\|\chi_0(t/T_{\varepsilon})f\|^2 +(\frac{\chi_0'(t/T_{\varepsilon})}{T_\varepsilon})^2\|u\|^2) e^{2\psi_{1,\gamma}(t)}\,dt. \end{align*} Since $\|\chi_0(t/T_{\varepsilon})f\|^2\le C_0(\|\nabla u_{\varepsilon}\|^2+ \|u_\varepsilon\|^2),$ by choosing $\varepsilon$ small, we have $$\label{carleman-03} \int_0^{T_\varepsilon}(\frac{1}{\varepsilon}\|\nabla u_{\varepsilon}\|^2 +\frac{\gamma}{\varepsilon}\|u_{\varepsilon}\|^2)e^{2\psi_{1,\gamma}(t)}\,dt \le C \int_0^{T_\varepsilon}(\frac{\chi_0'(t/T_{\varepsilon})}{T_\varepsilon})^2 \|u_{\varepsilon}\|^2e^{2\psi_{1,\gamma}(t)}\,dt$$ for any $\gamma\ge \gamma_{\varepsilon}$. Since $\chi_0'(t/T_{\varepsilon})=0$ for $t\le 3T_\varepsilon/4$ and $\psi_{1,\gamma}(t)$ is decreasing, we note that the right-hand side of \eqref{carleman-03} can be dominated by $C_{\varepsilon}e^{2\psi_{1,\gamma}(3T_{\varepsilon}/4)}.$ Since $\psi_{\gamma,1}=\gamma\int_t^{T_\varepsilon}e^{\psi_{\gamma}(\tau)}\,d\tau$ and $e^{\psi_{\gamma}(\tau)}\ge 1$ for $\tau\ge0$, we see that for $t\in[0,T_\varepsilon/4]$, $\psi_{\gamma,1}(t)\geq \psi_{\gamma,1}(T_\varepsilon/4) \geq \psi_{\gamma,1}(3T_\varepsilon/4)+\gamma T_\varepsilon/2.$ Then, noting that $u_{\varepsilon}(x,t)=u(x,t)$ on $[0,3T_{\varepsilon}/4]$, we see that $\frac{\gamma}{\varepsilon} e^{ 2\psi_{\gamma,1}(3T_\varepsilon/4) +\gamma T_\varepsilon}\int_0^{T_\varepsilon/4} \|u\|^2\,dt$ is not greater than the left hand side of \eqref{carleman-03}. Then we have $\int_0^{T_\varepsilon/4} \|u\|^2\,dt\le \frac{ \varepsilon C_{\varepsilon}}{\gamma} e^{ -\gamma T_\varepsilon}.$ As $\gamma$ tends to infinity, the right hand side converges to zero. Then we see that $\int_0^{T_\varepsilon/4} \|u\|^2\,dt=0$, which implies $u(x,t)=0$ on $[0,T_\varepsilon/4]$. \end{proof} Using the same argument we have the following proposition. \begin{proposition}\label{local-u2} We assume that the assumptions of Theorem \ref{thm1.1} except for $u(x,0)=0$ are satisfied. For any $t_0\in[0,T)$ there exists $t_1\in(t_0,T]$ such that, if $u(x,t_0)=0$, then we have $u(x,t)=0$ for $t\in[t_0,t_1]$. \end{proposition} Now we prove Theorem \ref{thm1.1}. Let $S$ be the subset of $(0,T)$ that consists of $t_0\in(0,T)$ satisfying $u(x,t)=0$ on $[0,t_0]$. From Proposition \ref{local-u1} and Proposition \ref{local-u2} we see that the set $S$ is not empty and open set. Since $u(x,t)\in C^0([0,T],L^2(\mathbb{R}^d))$, we see that $S$ is closed subset of $(0,T)$. Then the connectedness of $(0,T)$ implies $S=(0,T)$. Then we see that $u(x,t)=0$ on $[0,T)$. Hence $u(x,t)=0$ on $[0,T]$. The proof of Theorem \ref{thm1.1} is complete. \begin{thebibliography}{999} \bibitem{Br} H. Brezis; \emph{Op\'erateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,} North Holland, Amsterdam, 1973. \bibitem{del} D. Del Santo, M. Prizzi; \emph{Backward uniqueness for parabolic operators whose coefficients are non-Lipschitz continuous in time,} J. Math. Pures Appl., 84(2005) 471--491. \bibitem{dl} R. Dautray, J. L. Lions; \emph{Mathematical Analysis and Numerical Methods for Science and Technology Volume 5 Evolution Problems 1,} Springer-Verlag, Berlin, 2000. \bibitem{met} G. Metivier; \emph{Para-differential calculus and applications to the Cauchy problem for nonlinear systems,} Centro di Riccerca Matematica Ennio De Giorgi'' (CRM) Series 5, Edizioni della Normale, Pisa, 2008 \bibitem{mz} S. 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