\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 149, pp. 1--23.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/149\hfil Weakly coupled systems of damped waves] {Final-value problem for a weakly-coupled system of structurally damped waves} \author[N. H. Tuan, V. V. Au, N. H. Can, M. Kirane \hfil EJDE-2018/149\hfilneg] {Nguyen Huy Tuan, Vo Van Au, Nguyen Huu Can, Mokhtar Kirane} \address{Nguyen Huy Tuan \newline Applied Analysis Research Group Faculty of Mathematics and Statistics Ton Duc Thang University Ho Chi Minh City, Vietnam} \email{nguyenhuytuan@tdt.edu.vn} \address{Vo Van Au \newline Faculty of General Sciences, Can Tho University of Technology, Can Tho City, Vietnam} \email{vvau@ctuet.edu.vn} \address{Nguyen Huu Can \newline Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, (VNU-HCMC), Ho Chi Minh City, Vietnam} \email{nguyenhuucan@gmail.com} \address{Mokhtar Kirane \newline LaSIE, Facult\'e des Sciences, P\^ole Sciences et Technologies, Universit\'e de La Rochelle, Avenue M. Crepeau, 17042 La Rochelle Cedex, France.\newline NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia} \email{mkirane@univ-lr.fr} \dedicatory{Communicated by Vicentiu D. Radulescu} \thanks{Submitted February 22, 2018. Published August 7, 2018.} \subjclass[2010]{35K05, 35K99, 47J06, 47H10} \keywords{Ill-posed problems; regularization; systems of wave equations; \hfill\break\indent error estimate} \begin{abstract} We consider the final-value problem of a system of strongly-damped wave equations. First of all, we find a solution of the system, then by an example we show the problem is ill-posed. Next, by using a filter method, we propose stable approximate (regularized) solutions. The existence, uniqueness of the corresponding regularized solutions are obtained. Furthermore, we show that the corresponding regularized solutions converge to the exact solutions in $L^2$ uniformly with respect to the space coordinate under some a priori assumptions on the solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $T$ be a positive number and $\Omega \subset \mathbb{R}^n, n \ge 1$, be an open bounded domain with a smooth boundary $\Gamma$. Set $D_T=\Omega \times (0, T)$, $\Sigma = \Gamma \times (0, T)$. In this article, we consider the question of finding a couple of functions $(u,v)(x,t)$, $(x,t) \in \overline{\Omega} \times [0,T]$, satisfying the Cauchy problem for the weakly-coupled system of nonlinear structurally damped wave equations \begin{equation} \label{system} \begin{gathered} u_{tt}-\Delta u +2a (-\Delta)^{\gamma} u_t = {F} ( u, v),\quad \text{in } D_T,\\ v_{tt}-\Delta v +2a (-\Delta)^{\gamma } v_t= {G} ( u, v), \quad \text{in } D_T,\\ u = v = 0, \quad \text{on } \Sigma, \end{gathered} \end{equation} subject to the final observation \begin{equation} \label{T} \begin{gathered} u(x,T) = u_T (x),\, u_t(x,T) = \widetilde u_T (x),\quad \text{in }\Omega, \\ v(x,T) = v_T(x),\; v_t(x,T) = \widetilde v_T(x), \quad \text{in } \Omega, \end{gathered} \end{equation} where $\gamma > 1/2$ and $a>0$ is a damping constant, the functions $u_T, \widetilde{u}_T, v_T , \widetilde v_T $ are given in $L^2(\Omega)$. The source functions ${F}$ and ${G}$ will be defined later. The damped wave equations and systems occur in a wide range of applications modelling the motion of viscoelastic materials. Some more physical applications of strongly damped waves can be found in \cite{Pata}. The initial-value problem for damped wave equations (or pseudo-hyperbolic equations) have been widely studied, see for example Pata et al \cite{pata2005strongly,Pata}, Thomee et al \cite{thomee2004maximum}, Liu et al \cite{Liu}, Guo \cite{Guo}, Zelik et al \cite{Ka}, Yang et al \cite{Yang}. However, studies of the initial-value problem for strongly damped wave systems are limited. Recently, Hayashi et al \cite{Ha} studied the existence of small global solutions to the initial-value problem for system \eqref{system} in $\mathbb R^n$, $n \ge 4$, assuming that \[ a= \frac{1}{2},\quad \gamma=1,\quad {F} ( u, v)= {F} ( v),\quad {G} ( u, v)= {F} ( u). \] D'Abbicco \cite{Abb} studied the system of structurally damped waves \eqref{system} in $\mathbb R^n$ assuming that \[ \gamma= \frac{1}{2},\quad {F} ( u, v)= | v|^p,\quad {G} ( u, v)= | u|^q, \] where $p,q$ are chosen suitably. To the best of our knowledge, the final-value (backward) problem for the system \eqref{system} has not been studied yet. The final-value problem for systems of partial differential equations play an important role in engineering areas, which aims to obtain the previous data of a physical field from a given state. The first work on the regularization result for the strongly damped wave equation seems the one by Lesnic et al \cite{Lesnic}. In practice, the exact data $ u_T, \widetilde u_T, v_T , \widetilde v_T $ can only be measured with errors, and we thus would have as data some function $ u_T^\delta, \widetilde u_T^\delta, v_T^\delta , \widetilde v_T^\delta $ that belong to $L^2(\Omega)$, for which \[ \|u_T^\delta- u_T\|_{L^2(\Omega)} + \|\widetilde u_T^\delta- \widetilde u_T\|_{L^2(\Omega)} +\|v_T^\delta- v_T\|_{L^2(\Omega)} + \|\widetilde v_T^\delta- \widetilde v_T\|_{L^2(\Omega)} \le \delta, \] where the constant $\delta>0$ represents a bound on the measurement errors. It is well-known that problem \eqref{system}-\eqref{T} is ill-posed in the sense of Hadamard for data $ u_T^\delta, \widetilde u_T^\delta, v_T^\delta , \widetilde v_T^\delta$ in any reasonable topology (see \cite{Ha}). More details of ill-posedness of the solution are given in Section 2.2. In general, no solution which satisfies the system with final data and the boundary conditions exists. Even if a solution exists, it does not depend continuously on the final data and any small perturbation in the given data may cause large change to the solution. So we need some regularization methods to deal with this problem. This article is organized as follows. In Section 2, we present the mild solution and the ill-posedness of system \eqref{system}-\eqref{T}. In Section 3, we establish a regularized solution in the case of a global Lipschitz source function $F, G$. In Section 4, we extend Section 3 to the situation of the locally Lipschitz sources. Furthermore, we also obtain the convergence rate between the regularized solution and the exact solution in $L^2$ norm. \section{Solution of the initial inverse problem \eqref{system}-\eqref{T}} We begin by introducing some notation needed for our analysis throughout this paper. \subsection{Notation} We denote by $\langle \cdot,\cdot\rangle_{L^2(\Omega)}$ the inner product in ${L^2(\Omega)}$. \begin{itemize} \item For $w \in C([0,T];L^2(\Omega))$, we define \[ \| w\|_{C([0,T];{L^2(\Omega)})}=\sup_{0\le t \le T}\| w (t)\|_{{{L^2(\Omega)}}}. \] \item Let $\mathcal X, \mathcal Y$ be Banach spaces; $\mathcal X \times \mathcal Y$ is also a Banach space and its norm is defined as \[ \|(w_1,w_2)\|_{\mathcal X \times \mathcal Y} = \|w_1\|_{\mathcal X} + \|w_2\|_{\mathcal Y}, \] for any $(w_1,w_2) \in \mathcal X \times \mathcal Y$. \end{itemize} $\diamond$ When $\Omega$ is bounded, the system \eqref{system} can also be solved by a decomposition in a Hilbert basis of $L^2(\Omega)$. \begin{itemize} \item For this purpose, it is very convenient to choose a basis $\{\xi_p\}_{p \in \mathbb N^*}$ of $L^2(\Omega)$ composed of eigenfunctions of $-\Delta$ (with zero Dirichlet condition), i.e., \begin{equation} \begin{gathered} -\Delta \xi_p(x) = \lambda_p \xi_p(x), \quad \text{in } \Omega,\\ \xi_p(x) = 0, \quad \text{on } \Gamma, \end{gathered} \end{equation} which admits a family of eigenvalues $0< \lambda_1 \leq \lambda_2 \leq \lambda_3 \leq \cdots \leq \lambda_p \cdots$ and $\lambda_p \to \infty$ as $p\to \infty$, see \cite[p. 335]{Evan}. \item Via the spectral decomposition of $w \in L^2(\Omega)$, for each $\gamma > \frac{1}{2}$, we define the fractional Laplacian using the spectral theorem as follows \begin{equation} \label{dfL} (-\Delta )^{\gamma}w = \sum_{p=1}^{\infty} \lambda_p^{\gamma} \big_{L^2(\Omega)} \xi_p(x). \end{equation} More details on this fractional Laplacian can be found in \cite{Koba}. \end{itemize} $\diamond$ In addition, we introduce the abstract Gevrey class of functions of index $m,n>0$, see e.g., \cite{Cao}, defined by \[ \mathbb{G}_{m,n}^\gamma=\Big\{w \in L^2(\Omega):\sum_{p=1}^{\infty} (\lambda_p^{\gamma})^{2m} \exp(2n \lambda_{p}^\gamma) \langle w,\xi_{p}\rangle^2_{L^2(\Omega)} < \infty\Big\} , \] for $ 2\gamma > 1$, which is a Hilbert space equipped with the inner product \begin{align*} &\langle w_1,w_2\rangle_{ \mathbb{G}_{m,n}^\gamma}\\ &:=\big\langle ((-\Delta)^{\gamma/2})^m \exp\big(n(-\Delta)^{\gamma/2}\big) w_1, ((-\Delta)^{\gamma/2})^m \exp\big(n(-\Delta)^{\gamma/2}\big)w_2 \big\rangle_{L^2(\Omega)}\,, \end{align*} for all $w_1, w_2 \in \mathbb{G}_{m,n}^\gamma$, and its corresponding norm $$ \| w\|_{\mathbb{G}_{m,n}^\gamma}^2 =\sum_{p=1}^{\infty} (\lambda_p^\gamma)^{2m} \exp(2n \lambda_p^\gamma)\langle w,\xi_{p}\rangle^2_{L^2(\Omega)} < \infty. $$ \subsection{Mild solution of \eqref{system}-\eqref{T}} We look for a solution of problem \eqref{system}-\eqref{T} of the form \begin{equation}\label{Fsol} \begin{gathered} u(x,t) = \sum_{p = 1} ^\infty u_p(t) \xi_p(x), \quad v(x,t) = \sum_{p=1}^\infty v_p(t)\xi_p(x), \end{gathered} \end{equation} where $u_p(t)=\langle u(x,t), \xi_p(x)\rangle_{L^2(\Omega)}$, $v_p(t)=\langle v(x,t), \xi_p(x)\rangle_{L^2(\Omega)}$. Put $F_p( u, v)(t)=\langle F( u, v), \xi_p(x)\rangle_{L^2(\Omega)}$. We consider the problem of finding a function $u_p(t)$ satisfying \begin{equation}\label{systemu} \begin{gathered} \frac{\mathrm d^2}{\mathrm d t^2} u_{p}(t) + 2a \lambda_p^{\gamma } \frac{\mathrm d}{\mathrm dt} u_p(t) +\lambda_p u_p(t) = F_p( u, v)(t), \quad t \in (0,T), \\ u_{p}(T)=\langle u(x,T), \xi_p(x)\rangle_{L^2(\Omega)},\quad \frac{\mathrm d}{\mathrm dt } u_{p}(T) = \langle \widetilde{ u}(x,T), \xi_p(x)\rangle_{L^2(\Omega)}. \end{gathered} \end{equation} The quadratic characteristic polynomial of \eqref{systemu} is \begin{equation*} Z^2+2a\lambda_p^\gamma Z +\lambda_p = 0. \end{equation*} With the notation $\alpha_p = a^2\lambda_p^{2\gamma} - \lambda_p$, for any $a >0$ and $\gamma >1/2$, we consider three cases \smallskip \noindent\textbf{Case 1.} $p \in \mathbb{N}_1 = \{p \in \mathbb{N}^* : \lambda_p > a^\frac{2}{1-2\gamma}\}$. We put $\mu_{j} = \mu_{j,p} := a\lambda_p^\gamma + (-1)^j\sqrt{\alpha_p}$, $j = 1,2$. Multiplying the first equation in \eqref{systemu} by $\frac{\exp\big(\mu_{1}(s-t)\big) - \exp\big(\mu_{2}(s-t)\big)} {2\sqrt{\alpha_p} }$ and integrating both sides from $t$ to $T$, we obtain \begin{equation} \label{u1} \begin{aligned} u_p(t) =& \frac{\mu_{2} \exp\big(\mu_{1}(T-t)\big) - \mu_{1} \exp\big(\mu_{2}(T-t)\big)} {2\sqrt{\alpha_p } } \langle u(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &+ \frac{\exp\big(\mu_{1}(T-t)\big) - \exp\big(\mu_{2}(T-t)\big)} {2\sqrt{\alpha_p } } \langle \widetilde u(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &+ \int_t^T \frac{\exp\big(\mu_{1}(s-t)\big) - \exp\big(\mu_{2}(s-t)\big)} {2\sqrt{ \alpha_p } } F_p( u, v)(s) \mathrm d s. \end{aligned} \end{equation} \noindent\textbf{Case 2.} $p \in \mathbb{N}_2 = \{p \in \mathbb{N}^*: \lambda_p = a^\frac{2}{1-2\gamma} \}$. Multiplying the first equation in \eqref{systemu} by $(s-t)\exp\big(a^\frac{1}{1-2\gamma}(s-t)\big)$, and integrating both sides from $t$ to $T$, we have \begin{equation}\label{u2} \begin{aligned} u_p(t) = & \exp\big(a \lambda_p^\gamma (T-t)\big)(1-a \lambda_p^\gamma (T-t)) \langle u(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &- \exp\big(a \lambda_p^\gamma (T-t)\big)(T-t) \langle \widetilde u(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &- \int_t^T(s-t)\exp\big(a \lambda_p^\gamma (s-t)\big) F_{p}( u, v)(s)\mathrm d s. \end{aligned} \end{equation} \noindent\textbf{Case 3.} $p \in \mathbb{N}_3 = \{p \in \mathbb{N}^* : \lambda_p < a^\frac{2}{1-2\gamma}\}$. Multiplying the first equation in \eqref{systemu} by $$ \frac{\exp\big(a\lambda_p^\gamma\big)}{\sqrt{-\alpha_p}} \sin (\sqrt{\alpha_p} (s-t)), $$ and integrating both sides from $t$ to $T$, we have \begin{equation} \label{u3} \begin{aligned} u_p(t) &= \frac{\exp\big(a\lambda_p^\gamma(T-t)\big)}{\sqrt{-\alpha_p}} \big[a\lambda_p^\gamma \sin (\sqrt{-\alpha_p} (T-t)) - \cos ( \sqrt{-\alpha_p} (T-t) ) \big] \\ &\quad\times \langle u(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &\quad + \frac{\exp\big(a\lambda_p^\gamma(T-t)\big)}{\sqrt{-\alpha_p}} \sin ( \sqrt{-\alpha_p} (T-t)) \langle \widetilde u(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &\quad - \int_t^T \frac{\exp\big(a\lambda_p^\gamma\big)}{\sqrt{-\alpha_p}} \sin (\sqrt{\alpha_p} (s-t)) F_{p}( u, v)(s) \mathrm d s. \end{aligned} \end{equation} Similar considerations apply to $v_p(t)$ that satisfies \begin{equation}\label{systemv} \begin{gathered} \frac{\mathrm d^2}{\mathrm d t^2}v_{p}(t) + 2a \lambda_p^{\gamma } \frac{\mathrm d}{\mathrm d t} v_p(t) +\lambda_p v_p(t) = G_p( u, v)(t), \quad t \in (0,T), \\ v_{p}(T)=\langle v(x,T), \xi_p(x)\rangle_{L^2(\Omega)} ,\quad \frac{\mathrm d}{\mathrm d t } v_{p}(T) = \langle \widetilde v(x,T), \xi_p(x)\rangle_{L^2(\Omega)}, \end{gathered} \end{equation} where $v_p=\langle v, \xi_p\rangle_{L^2(\Omega)}$, $G_p( u, v)=\langle G( u, v), \xi_p\rangle_{L^2(\Omega)}$. We also have three cases. \smallskip \noindent\textbf{Case 1.} $p \in \mathbb{N}_1$. We obtain \begin{equation} \label{v1} \begin{aligned} v_p(t) &= \frac{\mu_{2} \exp\big(\mu_{1}(T-t)\big) - \mu_{1} \exp\big(\mu_{2}(T-t)\big)} {2\sqrt{\alpha_p } } \langle v(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &\quad + \frac{\exp\big(\mu_{1}(T-t)\big) - \exp\big(\mu_{2}(T-t)\big)} {2\sqrt{\alpha_p } } \langle \widetilde v(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &\quad + \int_t^T \frac{\exp\big(\mu_{1}(s-t)\big) - \exp\big(\mu_{2}(s-t)\big)} {2\sqrt{ \alpha_p } } G_p( u, v)(s) \mathrm d s. \end{aligned} \end{equation} \noindent\textbf{Case 2.} $p \in \mathbb{N}_2$. We obtain \begin{equation}\label{v2} \begin{aligned} v_p(t) & = \exp\big(a \lambda_p^\gamma (T-t)\big)(1-a \lambda_p^\gamma (T-t)) \langle v(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ & \quad - \exp\big(a \lambda_p^\gamma (T-t)\big)(T-t) \langle \widetilde v(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &\quad - \int_t^T(s-t)\exp\big(a \lambda_p^\gamma (s-t)\big) G_{p}( u, v)(s)\mathrm d s. \end{aligned} \end{equation} \noindent\textbf{Case 3.} $p \in \mathbb{N}_3$. We have \begin{equation}\label{v3} \begin{aligned} v_p(t) &= \frac{\exp\big(a\lambda_p^\gamma(T-t)\big)}{\sqrt{-\alpha_p}} ( a\lambda_p^\gamma \sin (\sqrt{-\alpha_p} (T-t)) \\ &\quad - \cos ( \sqrt{-\alpha_p} (T-t) ) ) \langle v(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &\quad + \frac{\exp\big(a\lambda_p^\gamma(T-t)\big)}{\sqrt{-\alpha_p}} \sin ( \sqrt{-\alpha_p} (T-t)) \langle \widetilde v(x,T), \xi_p(x)\rangle_{L^2(\Omega)} \\ &\quad - \int_t^T \frac{\exp(a\lambda_p^\gamma)}{\sqrt{-\alpha_p}} \sin (\sqrt{\alpha_p} (s-t)) G_{p}( u, v)(s) \mathrm d s. \end{aligned} \end{equation} Hence, the solution of \eqref{system} is \begin{equation}\label{Soluv} \begin{gathered} u(x,t) = \sum_{p\in \mathbb{N}_1} u_p(t)\xi_p + \sum_{p\in \mathbb{N}_2} u_p(t)\xi_p + \sum_{p\in \mathbb{N}_3} u_p(t)\xi_p,\\ v(x,t) = \sum_{p\in \mathbb{N}_1} v_p(t)\xi_p + \sum_{p\in \mathbb{N}_2} v_p(t)\xi_p + \sum_{p\in \mathbb{N}_3} v_p(t)\xi_p. \end{gathered} \end{equation} Let $ z \in (0,T), w \in L^2(\Omega)$, $w_p = \langle w, \xi_p\rangle_{L^2(\Omega)} $, we define \begin{gather} \begin{aligned} \mathcal{A}(z) w & = \sum_{p \in \mathbb{N}_1}^\infty \frac{\mu_2 \exp(z\mu_1) -\mu_1 \exp(z\mu_2)}{2\sqrt{\alpha_p}} w_p \xi_p \\ &\quad + \sum_{p \in \mathbb{N}_2}^\infty \exp(a \lambda_p^\gamma z)(1-a \lambda_p^\gamma z) w_p \xi_p \\ &\quad + \sum_{p \in \mathbb{N}_3}^\infty \frac{\exp(a\lambda_p^\gamma z )}{\sqrt{-\alpha_p}} [a\lambda_p^\gamma \sin (\sqrt{-\alpha_p} z ) - \cos ( \sqrt{-\alpha_p} z ) ] w_p \xi_p , \end{aligned} \label{A}\\ \begin{aligned} \mathcal{B}(z) w &= \sum_{p \in \mathbb{N}_1}^\infty \frac{\exp(z\mu_1) - \exp(z\mu_2)}{2\sqrt{\alpha_p}} w_p \xi_p + \sum_{p \in \mathbb{N}_2}^\infty z \exp(a \lambda_p^\gamma z ) w_p \xi_p \\ &\quad + \sum_{p \in \mathbb{N}_3}^\infty \frac{\exp(a\lambda_p^\gamma z)}{\sqrt{-\alpha_p}} \sin ( \sqrt{-\alpha_p} z ) w_p \xi_p. \end{aligned}\label{B} \end{gather} Then, we can rewrite \eqref{Soluv} as \begin{equation}\label{Soluv1} \begin{gathered} u(x,t) = \mathcal{A}(T-t) u_T + \mathcal{B}(T-t) \widetilde u_T + \int_t^T \mathcal{B}(s-t) F( u, v)(s) \mathrm d s, \\ v(x,t) = \mathcal{A}(T-t) v_T + \mathcal{B}(T-t) \widetilde v_T + \int_t^T \mathcal{B}(s-t) G( u, v)(s) \mathrm d s. \end{gathered} \end{equation} We expressed the solution of problem \eqref{system} with the final observation in an integral formulation \eqref{Soluv1}. In the next section, we indicate the reasons which make the solution \eqref{Soluv1} ill-posed in the Hadamard sense. For clarity, we give an example to show that the regularization method is necessary. \subsection{Ill-posedness of the inverse problem for\eqref{system}-\eqref{T}} We first observe that if $p \in \mathbb{N}_2 \cup \mathbb{N}_3 $ then $\lambda_p \leq a^\frac{2}{1-2\gamma}$. It is obvious that the terms $\sum_{p\in \mathbb{N}_2 } u_p(t)\xi_p + \sum_{p\,\in \,\mathbb{N}_3 } u_p(t)\xi_p$ and $\sum_{p\in \mathbb{N}_2 } v_p(t)\xi_p + \sum_{p\in \mathbb{N}_3 } v_p(t)\xi_p$ are bounded and stable in $L^2$ norm. However, since $p \in \mathbb{N}_1$ implies that $\lambda_p > a^\frac{2}{1-2\gamma}$ then exponential functions in the right-hand sides of \eqref{u1} and \eqref{v1} tend to infinity as $p$ tends to infinity. Therefore, the terms $\sum_{p\in \mathbb{N}_1 } u_p(t)\xi_p$ and $\sum_{p\in \mathbb{N}_1 } v_p(t)\xi_p$ are unbounded. From the above arguments, we take $\mathbb{N}_2 = \mathbb{N}_3 = \emptyset$ by assuming that $a^2\lambda_1^{2\gamma-1} > 1$. Note that this also implies $a^2\lambda_p^{2\gamma}- \lambda_p> 0$ for all $p \in \mathbb{N}^*$, and hence the root of $\alpha_p$ are real and distinct. Next, we give an example which shows the the solution of problem \eqref{system} is not stable. Let $ a = \lambda_1^{1/2 - \gamma} + 1$, $\partial_t u^{(k)}(x,T) = \partial_t v^{(k)}(x,T) =\frac{1}{\sqrt{\lambda_k} }\xi_k(x):= \varPsi^{(k)}$, $u(x,T)=v(x,T)=0$, for any $k \in \mathbb N^*$. Let us define functions \begin{align*} F(w_1,w_2)(t) &= \sum_{p=1}^\infty \frac{\exp({-2(\lambda_1^{1/2 - \gamma} + 1) T \lambda_p^\gamma } ) }{2^3\sqrt{2} T^2} \\ &\quad\times \Big( \langle w_1 (t),\xi_p \rangle_{L^2(\Omega)} \xi_p + \langle w_2(t),\xi_p \rangle_{L^2(\Omega)} \xi_p \Big), \\ G (w_1, w_2) (t) &= \sum_{p=1}^\infty \frac{\exp({-2(\lambda_1^{1/2 - \gamma} + 1) T \lambda_p^\gamma } ) }{2^3\sqrt{2}T^2} \\ &\quad\times \Big( \langle w_1 (t) ,\xi_p \rangle_{L^2(\Omega)} \xi_p + \langle w_2 (t) ,\xi_p \rangle_{L^2(\Omega)} \xi_p \Big). \end{align*} Let $ u^{(k)}, v^{(k)}$ satisfy the system \begin{equation} \label{u_ex1} \begin{gathered} u^{(k)}(x,t)=\mathcal B(T-t)\varPsi^{(k)} + \int_t^T \mathcal B(s-t) F ( u^{(k)}(x,s), v^{(k)}(x,s)) \mathrm ds ,\\ v^{(k)}(x,t)= \mathcal B(T-t) \varPsi^{(k)} + \int_t^T \mathcal B(s-t) G ( u^{(k)}(x,s), v^{(k)}(x,s)) \mathrm ds, \end{gathered} \end{equation} with $a = \lambda_1^{1/2 - \gamma} + 1$; recalling that \begin{equation} \mathcal{B}(z) w = \sum_{p = 1}^\infty \frac{\exp(z\mu_1) - \exp(z\mu_2)} {2\sqrt{\alpha_p}} \langle w,\xi_p\rangle_{L^2(\Omega)} \xi_p, \label{R} \end{equation} and, for $j = 1,2$, \begin{equation} \alpha_p = (\lambda_1^{1/2 - \gamma } + 1 )^2\lambda_p^{2\gamma}-\lambda_p\,, \quad \mu_{j} = \mu_{j,p} := (\lambda_1^{1/2 - \gamma } + 1 ) \lambda_p^\gamma + (-1)^j\sqrt{\alpha_p}. \end{equation} \noindent\textbf{Step 1.} We show that \eqref{u_ex1} has a unique solution $( u^{(k)}, v^{(k)}) \in [C([0,T]; L^2(\Omega))]^2$. Indeed, we consider for $(r_1,r_2) \in [C([0,T];L^2(\Omega))]^2$ the function \[ \mathbb E(r_1,r_2)(t) = \Big(\mathcal E ( r_1, r_2)(t), \widetilde{\mathcal E}( r_1, r_2)(t)\Big), \] where \begin{gather*} \mathcal{E}( r_1, r_2)(t) = {\mathcal B} (T-t) \varPsi^{(k)} + \int_t^T {\mathcal B} (\tau-t) F (r_1(x,\tau), r_2(x,\tau)) \mathrm d\tau, \\ \widetilde{\mathcal E}( r_1, r_2)(t) = {\mathcal B} (T-t) \varPsi^{(k)} + \int_t^T {\mathcal B} (\tau-t) G ( r_1(x,\tau), r_2(x,\tau)) \mathrm d\tau. \end{gather*} Then for any $( r_1, r_2), ( s_1, s_2) \in [C([0,T]; L^2(\Omega))]^2$, we obtain \begin{equation} \begin{aligned} &\|\mathcal{E}( r_1, r_2) (t)- \mathcal{E}( s_1, s_2) (t) \|_{L^2(\Omega)} \\ &\le \int_t^T \| {\mathcal{B}}(\tau-t) [F ( r_1, r_2) - F ( s_1, s_2) ] \|_{L^2(\Omega)} \mathrm d\tau \\ &=\int_t^T \Big\{ \sum_{p=1}^\infty \Big[ \frac{\exp((\tau-t)\mu_2) -\exp((\tau-t)\mu_1)} {\sqrt{\alpha_p}} \Big]^2 \\ &\quad\times \langle F ( r_1, r_2) - F ( s_1, s_2),\xi_p \rangle^2_{L^2(\Omega)} \Big\}^{1/2} \mathrm d\tau \\ &\leq \int_t^T \Big\{ \sum_{p=1}^\infty \Big[ \frac{\exp((\tau-t)\mu_2)-\exp((\tau-t)\mu_1)} {\sqrt{\alpha_p}}\Big]^2 \frac{\exp({-4(\lambda_1^{1/2 - \gamma} + 1) T \lambda_p^\gamma } )}{2^7T^4} \\ &\quad \times 2\langle r_1(x,\tau)- s_1(x,\tau),\xi_p\rangle^2_{L^2(\Omega)} + 2\langle r_2(x,\tau)- s_2(x,\tau),\xi_p\rangle^2_{L^2(\Omega)} \Big\}^{1/2} \mathrm d\tau. \end{aligned} \end{equation} Moreover, using the the inequality $|\exp(-b)- \exp(-c)| \le |b-c| $ for $b, c>0$, we obtain the estimate \begin{align*} &\Big[ \frac{\exp((\tau-t)\mu_2)-\exp((\tau-t)\mu_1)} {\sqrt{\alpha_p}}\Big]^2 \frac{\exp({- 4 (\lambda_1^{1/2 - \gamma} + 1) T \lambda_p^\gamma} ) }{2^7T^4} \\ &= \Big[\exp\big((\tau-t)(\mu_1+\mu_2) \big) \frac{ \exp\big(-(\tau-t)\mu_1\big)- \exp\big(-(\tau-t)\mu_2\big)} {\sqrt{\alpha_p}} \Big]^2\\ &\quad\times \frac{\exp({-4(\lambda_1^{1/2 - \gamma} + 1) T \lambda_p^\gamma } ) }{2^7T^4} \\ &\le \exp\Big(4{(\lambda_1^{1/2 - \gamma} + 1) (\tau-t) \lambda_p^\gamma } \Big) \Big[\frac{(-2\sqrt{\alpha_p})(\tau-t)}{\sqrt{\alpha_p}}\Big]^2 \frac{\exp({-4 (\lambda_1^{1/2 - \gamma} + 1) T \lambda_p^\gamma } ) }{2^7T^4} \\ &\le \Big[\frac{(-2\sqrt{\alpha_p})(\tau-t)}{\sqrt{\alpha_p}}\Big]^2 \frac{1}{2^7 T^4} \\ &=4(\tau-t)^2 \frac{1}{2^7 T^4} \le \frac{1}{2^5T^2}, \end{align*} where we have used $\mu_1 + \mu_2 = 2a\lambda_p^\gamma = 2(\lambda_1^{1/2 - \gamma } + 1 )\lambda_p^\gamma$ and $\mu_2-\mu_1 = 2 \sqrt{\alpha_p}$. According to the above observations, we deduce that for all $t \in [0,T]$ \begin{align*} \|\mathcal{E}( r_1, r_2)(t)- \mathcal{E}( s_1, s_2)(t)\|_{L^2(\Omega)} &\le \frac{1}{4} \int_t^T \frac{1}{T} \|R(\cdot,\tau) -S(\cdot,\tau)\|_{[L^2(\Omega)]^2} \mathrm d\tau \\ &\le \frac{1}{4} \|R-S\|_{[C([0,T];L^2(\Omega))]^2}, \end{align*} where $R:=(r_1,r_2)$, $S:=(s_1,s_2) \in [C([0,T]; L^2(\Omega))]^2$. Whereupon \begin{equation} \label{E1} \|\mathcal{E}( r_1, r_2)- \mathcal{E}( s_1, s_2) \|_{L^2(\Omega)} \le \frac{1}{4} \|R-S\|_{[C([0,T];L^2(\Omega))]^2}. \end{equation} Similarly, \begin{equation} \label{E2} \|\widetilde{\mathcal E} ( r_1, r_2)- \widetilde{\mathcal E} ( s_1, s_2) \|_{L^2(\Omega)} \le \frac{1}{4} \|R-S\|_{[C([0,T];L^2(\Omega))]^2}. \end{equation} Combining \eqref{E1} and \eqref{E2}, we obtain \[ \|\mathbb E (R) - \mathbb E (S) \|_{[C([0,T];L^2(\Omega))]^2} \le \frac{1}{2} \|R-S\|_{[C([0,T];L^2(\Omega))]^2}. \] Hence $\mathbb{E}$ is a contraction. Using the Banach fixed-point theorem, we conclude that $\mathbb{E} (R)=R$ has a unique solution $( u^{(k)}, v^{(k)}) \in [C([0,T];L^2(\Omega))]^2 $. \smallskip \noindent\textbf{Step 2.} Problem \eqref{u_ex1} is ill-posed in the sense of Hadamard. We have \begin{equation}\label{u_k} \| u^{(k)}(t)\|_{L^2(\Omega)} \ge \|{\mathcal{B}}(T-t) \varPsi^{(k)} \|_{L^2(\Omega)} - \| \int_t^T {\mathcal{B}}(s-t) F(\mathbf w^{(k)})(s)\mathrm ds \|_{L^2(\Omega)}, \end{equation} where $\mathbf w^{(k)} = ( u^{(k)}, v^{(k)}) \in [C([0,T];L^2(\Omega))]^2$. Firstly, it is easy to see that (noting that $F(0,0)=0$ and using \eqref{E1}) \begin{equation}\label{tpB} \begin{aligned} \| \int_t^T {\mathcal{B}}(s-t) F(\mathbf w^{(k)}(s))ds \|_{L^2(\Omega)} &= \| \mathcal{E}( u^{(k)}, v^{(k)})(t)- \mathcal{E}(0,0)(t) \|_{L^2(\Omega)} \\ &\le \frac{1}{4} \|\mathbf w^{(k)}\|_{[C([0,T];L^2(\Omega))]^2}. \end{aligned} \end{equation} Hence \[ \| u^{(k)}(t)\|_{L^2(\Omega)} \ge \Big\| {\mathcal{B}}(T-t) \varPsi^{(k)} \Big\|_{L^2(\Omega)} - \frac{1}{4} \|\mathbf w^{(k)}\|_{[C([0,T];L^2(\Omega))]^2}. \] This leads to \begin{equation} \| u^{(k)}\|_{C([0,T];{L^2(\Omega)})} \ge \sup_{0 \le t \le T}\Big\| {\mathcal{B}}(T-t) \varPsi^{(k)} \Big\|_{L^2(\Omega)} - \frac{}{4} \|\mathbf w^{(k)}\|_{[C([0,T];L^2(\Omega))]^2}. \label{B1} \end{equation} By an argument analogous to the previous one. We get \begin{equation} \| v^{(k)}\|_{C([0,T];{L^2(\Omega)})} \ge \sup_{0 \le t \le T}\Big\| {\mathcal{B}}(T-t) \varPsi^{(k)} \Big\|_{L^2(\Omega)} - \frac{}{4} \|\mathbf w^{(k)}\|_{[C([0,T];L^2(\Omega))]^2}. \label{B11} \end{equation} Combining \eqref{B1} and \eqref{B11} yields \begin{equation} \|\mathbf w^{(k)}\|_{[C([0,T];L^2(\Omega))]^2} \ge \frac{4}{3}\sup_{0 \le t \le T}\Big\| {\mathcal{B}}(T-t) \varPsi^{(k)} \Big\|_{L^2(\Omega)} . \label{B111} \end{equation} Secondly, we have \begin{align*} \| {\mathcal{B}}(T-t) \varPsi^{(k)} \|_{L^2(\Omega)}^2 & =\Big[\frac{\exp\big((T-t)\mu_{2k}\big)-\exp\big((T-t)\mu_{1k}\big)} {2\sqrt{\alpha_k}} \Big]^2 \frac{1}{{\lambda_k}} \\ & =\frac{\exp\big(2(T-t) \mu_{2k}\big) \big( 1- \exp\big(-2(T-t) \sqrt{\alpha_k}\big) \big)^2 } { 4\lambda_k \alpha_k } \\ &\ge \frac{\exp\big(2(T-t) \mu_{2k}\big) \big( 1- \exp\big(-2(T-t) \sqrt{\alpha_k}\big) \big)^2} {4\lambda_k \alpha_k }, \end{align*} where we set $\mu_{j,k} := (\lambda_1^{1/2 - \gamma } + 1 )\lambda_k^\gamma + (-1)^j\sqrt{\alpha_k}$, $j = 1,2$, $k \in \mathbb N^*$. Since the function $\Theta(t)=\exp\big(2(T-t) \mu_{2k}\big) \big( 1- \exp\big(-2(T-t) \sqrt{\alpha_k}\big) \big)^2$ is a decreasing function with respect to the variable $t$, noting that $\mu_{2k} \approx 2(\lambda_1^{1/2 - \gamma } + 1 )\lambda_k^\gamma$, we deduce that \begin{equation} \begin{aligned} &\sup_{0 \le t \le T}\| {\mathcal{B}}(T-t) \varPsi^{(k)} \|_{L^2(\Omega)}^2 \\ &= \sup_{0 \le t \le T}\| {\mathcal{B}}(T-t) \varPsi^{(k)} \|_{L^2(\Omega)}^2 \\ & \geq \sup_{0\le t \le T} \frac{\exp\big(2(T-t) \mu_{2k}\big) \big( 1- \exp\big(-2(T-t) \sqrt{\alpha_k}\big) \big)^2 } {4\lambda_k \alpha_k } \\ &\geq \frac{\exp(2T \mu_{2k}) \big( 1- \exp\big(-2T \sqrt{\alpha_k}\big) \big)^2 } {4\lambda_k \alpha_k } \\ &\ge \frac{\exp(4T(\lambda_1^{1/2 - \gamma } + 1 ) \lambda_k^\gamma ) \big( 1- \exp\big(-2T \sqrt{\alpha_k}\big) \big)^2 } {4\lambda_k \alpha_k }. \end{aligned} \label{B2} \end{equation} Next we estimate the right-hand side of the latter inequality. Indeed, combining \eqref{B111} and \eqref{B2} yields \begin{equation} \|\mathbf w^{(k)}\|_{[C([0,T];L^2(\Omega))]^2} \ge \frac{2}{3}\frac{\exp\big(2T(\lambda_1^{1/2 - \gamma } + 1 ) \lambda_k^\gamma \big) \big( 1- \exp\big(-2T \sqrt{\alpha_k}\big) \big) } { \sqrt{\lambda_k \alpha_k} }. \end{equation} As $k \to +\infty$, we see that \begin{gather} \lim_{k \to \infty} \Big(\| u^{(k)}(x,T)\|_{L^2(\Omega)}+ \| v^{(k)}(x,T)\|_{L^2(\Omega)} \Big) = \lim_{k \to \infty} \frac{2}{ \sqrt{\lambda_k}}=0, \nonumber\\ \begin{aligned} &\lim_{k \to \infty} \|\mathbf w^{(k)}\|_{[C([0,T];L^2(\Omega))]^2} \\ &\ge \lim_{k \to \infty} \frac{2}{3} \frac{\exp\big(2T(\lambda_1^{1/2 - \gamma } + 1 ) \lambda_k \big) \big( 1- \exp\big(-2T \sqrt{\alpha_k}\big) \big) } {\sqrt{\lambda_k \alpha_k }}=\infty. \end{aligned} \end{gather} Thus, Problem \eqref{u_ex1} is ill-posed in the sense of Hadamard in $L^2$-norm. \section{Regularization and error estimate in for globally Lipschitz nonlinearities} Observe that when $p \to \infty$ the operators $\mathcal A, \mathcal B$ are unbounded; to establish a regularized solution, we need to find new operators which are bounded operators, more specifically, \begin{gather} \mathcal A^{\Lambda}(t) w:= \mathcal H^{\Lambda} \mathcal A(t)w, \\ \mathcal B^{\Lambda}(t) w := \mathcal H^{\Lambda}\mathcal B(t)w, \\ \mathcal H^{\Lambda}w := \sum_{p=1}^\infty [1 + \Lambda C_pe^{C_pT}]^{-1}\langle w,\xi_p\rangle_{L^2(\Omega)}\xi_p, \end{gather} where $C_p=2a\lambda_p^\gamma$. Here $\Lambda:=\Lambda(\delta)>0$ is a parameter regularization which satisfies $\lim_{\delta \to 0^+} \Lambda=0$. The function $\mathcal H^{\Lambda}$ is called the \emph{filter function}. The regularized problem is \begin{equation} \label{reg-sys} \begin{gathered} \begin{aligned} U_\delta^{\Lambda}(x,t) &= \mathcal A^{\Lambda}(T-t) u_T^\delta(x) + \mathcal B^{\Lambda}(T-t) \widetilde u_T^\delta(x)\\ &\quad + \int_t^T \mathcal B^{\Lambda}(s-t) F ( U_\delta^\Lambda(x,s), V_\delta^\Lambda(x,s)) \mathrm ds, \end{aligned} \\ \begin{aligned} V_\delta^\Lambda(x,t) &= \mathcal A^{\Lambda}(T-t) v_T^\delta(x) + \mathcal B^{\Lambda}(T-t) \widetilde v_T^\delta(x) \\ &\quad + \int_t^T \mathcal B^{\Lambda}(s-t) G ( U_\delta^\Lambda(x,s), V_\delta^\Lambda(x,s)) \mathrm ds . \end{aligned} \end{gathered} \end{equation} The following technical lemma plays a key role in our analysis. \begin{lemma} \label{lem1} Let $t \in [0,T], \frac{1}{2} \leq \gamma \leq 1 $ and $a>0$. Then \begin{gather} \label{lem:bound} \|\mathcal A^{\Lambda}(t) \|_{\mathcal{L} (L^2(\Omega),L^2(\Omega))} \le T_a \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{t/T},\\ \|\mathcal B^{\Lambda}(t) \|_{\mathcal{L} (L^2(\Omega),L^2(\Omega))} \le T_a \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{t/T}, \label{lem:boundb} \end{gather} where $T_a:=\max \{2,T,1+ T a^{\frac{1}{1-2\gamma}}\}$. \end{lemma} \begin{proof} To show \eqref{lem:bound}, letting $w \in L^2(\Omega)$, we have \begin{equation} \label{Abeta} \begin{aligned} &\|\mathcal A^{\Lambda}(t)w \|_{L^2(\Omega)}^2 \\ &=\sum_{p \in \mathbb{N}_1} \Big[ \frac{\mu_{2} e^{\mu_1t} - \mu_{1} e^{\mu_2 t}} {2\sqrt{\alpha_p } [1 + C_p\Lambda e^{C_pT} ] } \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad+\sum_{p\in \mathbb{N}_2} \frac{e^{C_pt}(1-\frac{C_p}{2} t)^2} {[1 + C_p\Lambda e^{C_pT} ]^2}\langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad + \sum_{p \in \mathbb{N}_3} \frac{e^{C_p t}}{(-\alpha_p)[1 + C_p\Lambda e^{C_pT} ]^2}\\ &\quad\times \Big[\frac{C_p}{2} \sin (\sqrt{-\alpha_p} t) - \cos ( \sqrt{-\alpha_p} t ) \Big]^2\langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\le \sum_{p \in \mathbb{N}_1} \Big[ \frac{\mu_2 e^{-\mu_2t} - \mu_1 e^{-\mu_1 t}} {2\sqrt{\alpha_p } } \Big]^2 \Big[ \frac{1 } {e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad +\sum_{p\in \mathbb{N}_2} e^{-C_pt}(1-\frac{C_p}{2} t)^2 \Big[ \frac{1 } {e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &+ \sum_{p \in \mathbb{N}_3} \Big[ \frac{1 } {e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big]^2 e^{-C_pt} \frac{[\frac{ C_p}{2} \sin (\sqrt{-\alpha_p} t) - \cos ( \sqrt{-\alpha_p} t ) ]^2}{(-\alpha_p)} \\ &\quad\times \langle w,\xi_p\rangle_{L^2(\Omega)}^2. \end{aligned} \end{equation} Now, we continue estimating the terms in \eqref{Abeta}: First, we have \begin{equation} \begin{aligned} \frac{1 } {e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } & = \frac{e^{-C_p(T-t)}}{[C_p\Lambda + e^{-C_p T }]^{\frac{T-t}{T}} [C_p\Lambda + e^{-C_p T }]^{t/T}} \\ &\le \frac{1}{[C_p\Lambda + e^{-C_p T }]^{t/T} }. \end{aligned}\label{es3} \end{equation} On other hand, it is easy to see that $h(y)= \frac{1}{by+ e^{-yT}} \le \frac{T}{b \log (\frac{T}{b})}$ for $0 a^{\frac{2}{1-2\gamma}}$; using the inequalities $1-e^{-z} \leq z$ and $ze^{-z} \leq 1$ for $z>0$, we obtain (noting that $ \mu_2 - \mu_1= 2\sqrt{\alpha_p}$) \begin{equation} \begin{aligned} \big|\frac{\mu_2 e^{-t\mu_2} -\mu_1 e^{-t\mu_1}}{2\sqrt{ \alpha_p }} \big| &= e^{-t\mu_2 } + \mu_1 \frac{ | e^{-t\mu_2} - e^{-t\mu_1} | }{ 2 \sqrt{\alpha_p } } \\ &\le e^{-t\mu_2}+ \mu_1 e^{-t\mu_1} \frac{|1 - e^{-t(\mu_2 - \mu_1)}|}{2\sqrt{\alpha_p} } \\ &\le e^{-t\mu_2} + \mu_1 e^{-t\mu_1} \frac{t (\mu_2 - \mu_1)}{2\sqrt{\alpha_p}} \\ &\leq e^{-t\mu_2} + t \mu_1 e^{-t\mu_1} \le 2. \end{aligned}\label{es2} \end{equation} Combining \eqref{Abeta}, \eqref{es5}, \eqref{esN2}, \eqref{esN3}, \eqref{es2}, we conclude that \begin{equation} \|\mathcal A^{\Lambda}(t)w \|_{L^2(\Omega)}^2 \leq \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{2t/T} T_a^2 \|w\|_{L^2(\Omega)}^2. \end{equation} To show \eqref{lem:boundb}, letting $w \in L^2(\Omega)$, we have \begin{align*} % 3.14 &\|\mathcal B^{\Lambda}(t)w \|_{L^2(\Omega)}^2 \\ &= \sum_{p \in \mathbb{N}_1} \Big[e^{t(\mu_1+\mu_2)}\frac{e^{-\mu_2t} - e^{-\mu_1t}} {2\sqrt{\alpha_p } } \frac{1}{1 + C_p\Lambda e^{C_pT} } \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad + \sum_{p \in \mathbb{N}_2} \Big[e^{\frac{C_p}{2}t}t \frac{1}{1 + C_p\Lambda e^{C_pT}}\Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad + \sum_{p \in \mathbb{N}_3} \Big[\frac{1}{1 + C_p\Lambda e^{C_pT}} \frac{e^{\frac{C_p}{2} t}}{\sqrt{-\alpha_p}}\sin ( \sqrt{-\alpha_p} t) \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\leq \sum_{p \in \mathbb{N}_1} \Big[\frac{1}{e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big ]^2 \Big[\frac{e^{-\mu_2t} - e^{-\mu_1t}} {2\sqrt{\alpha_p } } \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad + \sum_{p \in \mathbb{N}_2} \Big[\frac{1}{e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big]^2 e^{-C_pt}t^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad + \sum_{p \in \mathbb{N}_3} \Big[\frac{1}{e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big]^2 \frac{e^{-C_pt}}{(-\alpha_p)}\sin^2 ( \sqrt{-\alpha_p} t) \langle w,\xi_p\rangle_{L^2(\Omega)}^2. \end{align*} As $|e^{-c}- e^{-d}| \le |c-d| $ for $c, d>0$, and noting that $e^{-C_pt} \leq 1$, for all $t \in [0,T]$ and $\sin z \leq z$, for $z>0$, we obtain \begin{align*} \|\mathcal B^{\Lambda}(t)w \|_{L^2(\Omega)}^2 &\leq T^2\sum_{p \in \mathbb{N}_1}\Big[\frac{1}{e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad + T^2 \sum_{p \in \mathbb{N}_2} \Big[\frac{1}{e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\quad + T^2 \sum_{p \in \mathbb{N}_3} \Big[\frac{1}{e^{-C_pt} + C_p\Lambda e^{C_p(T-t)} } \Big]^2 \langle w,\xi_p\rangle_{L^2(\Omega)}^2 \\ &\leq T_a^2 \Big[ \frac{T}{ \Lambda \log (\frac{T}{\Lambda })} \Big]^{2t/T} \|w\|_{L^2(\Omega)}^2. \end{align*} This completes the proof. \end{proof} Now we are ready to state and prove the main results of this paper. \subsection{Existence and uniqueness for problem \eqref{u_ex1}} \begin{theorem}\label{theorem1} The nonlinear integral system \eqref{u_ex1} has a solution $(U_\delta^\Lambda,V_\delta^\Lambda) \in [C([0,T];L^2(\Omega))]^2$. \end{theorem} \begin{proof} For any $w_1,w_2 \in [C([0,T];L^2(\Omega))]^2$, we define the function\\ $\mathbb L \colon [C([0,T];L^2(\Omega))]^2 \to [C([0,T];L^2(\Omega))]^2$ as $$ \mathbb L(w_1,w_2)(t):=(\mathbb X(w_1,w_2)(t),\mathbb Y(w_1,w_2)(t)),$$ where \begin{gather} \label{existregsol} \begin{aligned} \mathbb X(w_1,w_2)(t) &:=\mathcal A^{\Lambda}(T-t) u_T^\delta(x) + \mathcal B^{\Lambda}(T-t) \widetilde u_T^\delta(x) \\ &\quad + \int_t^T \mathcal B^{\Lambda}(s-t) F (w_1(x,s),w_2(x,s)) \mathrm ds, \end{aligned} \\ \begin{aligned} \mathbb Y(w_1,w_2)(t) &:= \mathcal A^{\Lambda}(T-t) v_T^\delta(x) + \mathcal B^{\Lambda}(T-t) \widetilde v_T^\delta(x) \\ &\quad + \int_t^T \mathcal B^{\Lambda}(s-t) G (w_1(x,s),w_2(x,s)) \mathrm ds. \end{aligned} \end{gather} Then for any $W=( w_1, w_2), \overline{W} = (\overline w_1,\overline w_2) \in [C([0,T]; L^2(\Omega))]^2$, we obtain \begin{equation} \label{CM1} \begin{aligned} &\| \mathbb L(W)(t) - \mathbb L(\overline{W})(t) \|_{[L^2(\Omega)]^2}\\ &\le \| \mathbb X(W)(t) - \mathbb X(\overline{W})(t) \|_{L^2(\Omega)} + \| \mathbb Y(W)(t) - \mathbb Y(\overline{W})(t) \|_{L^2(\Omega)}. \end{aligned} \end{equation} Let the functions $ F, G \colon [L^2(\Omega)]^2 \to L^2(\Omega)$ satisfy the global Lipschitz condition \begin{gather} \| F(W) - F(\overline{W})\|_{L^2(\Omega)} \le K_F \|W-\overline{W}\|_{[L^2(\Omega)]^2}, \label{Lip-F}\\ \| G(W) - G(\overline{W})\|_{L^2(\Omega)} \le K_G \|W-\overline{W}\|_{[L^2(\Omega)]^2} \label{Lip-G}, \end{gather} where $K_F, K_G$ are constants which are independent of $W, \overline{W}$. We shall prove the estimate \begin{equation} \label{dpcm} \| \mathbb X^n(W)(t) - \mathbb X^n(\overline{W})(t) \|_{L^2(\Omega)} \le \frac{\mathbf{E}_{\Lambda,\delta}^n(t)}{n!} \| W - \overline{W} \|_{[C([0,T];L^2(\Omega))]^2} , \end{equation} where $\mathbf{E}_{\Lambda,\delta}^n(t) :=\big[ \frac{\mathbb{K}T_a (T-t)}{\Lambda} \big]^n, n\ge 1$, and $ \mathbb{K} = \max \left\lbrace K_F,K_G \right\rbrace $, by induction. $\bullet$ For $n=1$, using Lemma \eqref{lem1} and the global Lipschitz condition of the function $F$, we obtain \begin{align*} &\| \mathbb X(W)(t) - \mathbb X(\overline{W})(t) \|_{L^2(\Omega)} \\ &\le \int_t^T \| \mathcal{B}^\Lambda(s-t) (F (W)(s) - F (\overline{W})(s) ) \|_{L^2(\Omega)} \mathrm ds \\ &\le \int_t^T \| \mathcal{B}^\Lambda(s-t) \|_{\mathcal L (L^2(\Omega) \times L^2(\Omega))} \| F(W)(s) - F(\overline{W})(s)\|_{L^2(\Omega)} \mathrm ds \\ &\le \int_t^T T_a \Big[ \frac{T}{ \Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{s-t}{T}} K_F \| W(s) - \overline{W}(s)\|_{[L^2(\Omega)]^2} \mathrm d s \\ &\le \mathbb{K}T_a {\Lambda }^{-1} \int_t^T \| W(s) - \overline{W}(s)\|_{[L^2(\Omega)]^2} \mathrm d s \\ &\le \mathbb{K}T_a {\Lambda }^{-1} (T-t) \| W - \overline{W}\|_{[C([0,T];L^2(\Omega))]^2} \\ &= \mathbf{E}_{\Lambda,\delta}(t)\| W - \overline{W} \|_{[C([0,T];L^2(\Omega))]^2}, \end{align*} where $\mathbf{E}_{\Lambda,\delta}(t):= \frac{\mathbb{K}T_a (T-t)}{\Lambda} \cdot$ $\bullet$ Assume that \eqref{dpcm} holds for $n=k$. Then we obtain \begin{equation} \label{gtqn} \| \mathbb X^k(W)(t) - \mathbb X^k(\overline{W})(t) \|_{L^2(\Omega)} \le \frac{\mathbf{E}_{\Lambda,\delta}^k(t)}{k!} \| W - \overline{W} \|_{[C([0,T];L^2(\Omega))]^2}. \end{equation} $\bullet$ We show that \eqref{dpcm} holds for $n=k+1$. In fact, we have \begin{align*} &\| \mathbb X^{k+1}(W)(t) - \mathbb X^{k+1}(\overline{W})(t) \|_{L^2(\Omega)}\\ &= \Big\| \int_t^T \mathcal{B}^\Lambda(s-t) [F (\mathbb X^k(W)(s)) - F (\mathbb X^k(\overline W)(s)) ] \mathrm ds \Big\|_{L^2(\Omega)} \\ &\le \int_t^T T_a \Big[ \frac{T}{ \Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{s-t}{T}} K_F \| \mathbb X^k(W)(s) - \mathbb X^k(\overline{W})(s) \|_{L^2(\Omega) } \mathrm d s \\ &\le \mathbb{K}T_a \Lambda^{-1} \int_t^T \| \mathbb X^k(W)(s) - \mathbb X^k(\overline{W})(s) \|_{ L^2(\Omega)} \mathrm d s \\ &\le \mathbb{K}T_a \Lambda^{-1} \int_t^T \frac{\big[\mathbb{K}T_a \Lambda^{-1} (T-s)\big]^k}{ k!}\| W - \overline{W} \|_{[C([0,T];L^2(\Omega))]^2} \mathrm d s \\ &= \frac{(\mathbb{K}T_a \Lambda^{-1})^{k+1}}{k!} \| W - \overline{W} \|_{[C([0,T];L^2(\Omega))]^2} \int_t^T (T-s)^k \mathrm d s \\ &= \frac{\mathbf{E}_{\Lambda,\delta}^{k+1}(t)}{(k+1)!} \| W - \overline{W} \|_{[C([0,T];L^2(\Omega))]^2}. \end{align*} Therefore, \eqref{dpcm} holds for $n \ge 1$. Secondly, we estimate $\| \mathbb Y(W)(t) - \mathbb Y(\overline{W})(t) \|_{L^2(\Omega) }$. Using similar arguments, we infer that if $W:=(w_1,w_1), \overline{W}:=(\overline{w}_1, \overline{w}_2) \in [L^2(\Omega)]^2$ then \begin{equation} \label{dpcmY} \| \mathbb Y^n(W)(t) - \mathbb Y^n(\overline{W})(t) \|_{ L^2(\Omega)} \le \frac{\mathbf{E}_{\Lambda,\delta}^n (t)}{n!} \| W - \overline{W} \|_{[C([0,T];L^2(\Omega))]^2}, \end{equation} where $\mathbf{E}_{\Lambda,\delta}^n(t) :=\big[ \frac{\mathbb{K}T_a (T-t)}{\Lambda} \big]^n$. Combining \eqref{CM1}, \eqref{dpcm} and \eqref{dpcmY}, we obtain \begin{equation} \| \mathbb L^n(W)(t) - \mathbb L^n(\overline{W})(t) \|_{[L^2(\Omega)]^2} \le \frac{2\mathbf{E}_{\Lambda,\delta}^n(t) }{n!} \| W - \overline{W} \|_{[C([0,T];L^2(\Omega) ]^2} \end{equation} On the other hand, $$ \mathbf{E}_{\Lambda,\delta}^n(t) \le \mathbf{E}_{\Lambda,\delta}^n(0) = \Big( \frac{\mathbb{K}T_aT}{\Lambda}\Big)^n,\quad n \ge 1. $$ This implies \begin{gather*} \| \mathbb L^n(W) - \mathbb L^n(\overline{W}) \|_{[C([0,T];L^2(\Omega) ]^2} \le \frac{2\mathbf{E}_{\Lambda,\delta}^n(0) }{n!} \| W - \overline{W} \|_{[C([0,T];L^2(\Omega) ]^2} \,, \\ \lim_{n \to \infty} \frac{2 \mathbf{E}_{\Lambda,\delta}^n(0)}{n!} = 0. \end{gather*} There exits a positive integer $n_0$ such that $\mathbb{L}^{n_0}$ is a contraction. Thus, the existence and uniqueness arguments are obtained by the Banach fixed-point theorem, i.e, $\mathbb{L}(w_1,w_2) = (w_1,w_2)$ has a unique solution $(w_1,w_2) \in [C([0,T];L^2(\Omega) ]^2$. Hence, $\mathbb{X}(U_\delta^\Lambda,V_\delta^\Lambda) = U_\delta^\Lambda$ and $\mathbb{Y}(U_\delta^\Lambda,V_\delta^\Lambda) = V_\delta^\Lambda$. \end{proof} \subsection{Error estimate} Now, we shall state (and prove) some regularization results under some conditions on the exact solution $(u,v)$ of system \eqref{Soluv1}. \begin{theorem} \label{theo-main1} Let $\Lambda(\delta)$ be a regularization parameter such that \begin{equation} \lim_{\delta \to 0^+}\Lambda = \lim_{\delta \to 0^+} \delta/\Lambda = 0. \end{equation} For $m\geq 1, n \geq 2aT$, we assume that system \eqref{u_ex1} has a unique solution $$ \mathcal S := (u,v) \in [C([0,T];L^2(\Omega)) \cap L^\infty(0,T;\mathbb G_{m,n}^\gamma)]^2 $$ which satisfies \begin{equation} \label{priori} \|\mathcal S(t)\|_{[C([0,T];L^2(\Omega))]^2} + \|\mathcal S(t)\|_{[L^\infty(0,T;\mathbb G_{m,n}^\gamma)]^2}\le Q. \end{equation} Then we have the estimate \begin{equation} \label{estimate1} \| \mathcal S_\delta^\Lambda(t) - \mathcal S(t) \|_{[L^2(\Omega)]^2} \le \Big(2a Q + T_a\delta\Lambda^{-1} \Big) e^{2T_a \mathbb{K} (T-t)} \Big[ \frac{T}{ \log (\frac{T}{\Lambda })} \Big]^{1-\frac{t}{T}} \Lambda^{t/T}, \end{equation} where $\mathcal S_\delta^\Lambda := ( U_\delta^\Lambda, V_\delta^\Lambda ) \in [C([0,T];L^2(\Omega))]^2$ and $\mathbb{K}:=\max\{K_F,K_G\}$. \end{theorem} \begin{remark} \rm In \eqref{estimate1}, the error estimate is of order $\Lambda^{t/T} [ \frac{T}{ \log (\frac{T}{\Lambda })} ]^{1-\frac{t}{T}}$. If $t \approx T$, the first term $\Lambda^{t/T}$ tends to zero quickly, and if $t\approx 0$, the second term $[ \frac{T}{ \log (\frac{T}{\Lambda })} ]^{1-\frac{t}{T}}$ tends to zero as $\delta \to 0^+$. And if $t=0$, the error \eqref{estimate1} becomes \begin{equation}\label{remark} \| \mathcal S_\delta^\Lambda(t) - \mathcal S(t) \|_{[L^2(\Omega)]^2} \le C[\log (\frac{T}{\Lambda })]^{-1}. \end{equation} We also note that the right-hand side of \eqref{remark} tends to zero when $\delta \to 0^+$. \end{remark} \begin{proof} Let ${\mathcal S}^\Lambda := ( U^\Lambda, V^\Lambda ) \in [C([0,T];L^2(\Omega))]^2 $ satisfy the nonlinear integral equations \begin{equation} \label{reg-sys.} \begin{aligned} U^{\Lambda}(x,t) &= \mathcal A^{\Lambda}(T-t) u_T(x) + \mathcal B^{\Lambda}(T-t) \widetilde u_T(x) \\ &\quad + \int_t^T \mathcal B^{\Lambda}(s-t) F (U^\Lambda(x,s), V^\Lambda(x,s)) \mathrm ds, \\ V^\Lambda(x,t) &= \mathcal A^{\Lambda}(T-t) v_T(x) + \mathcal B^{\Lambda}(T-t) \widetilde v_T(x) \\ &\quad + \int_t^T \mathcal B^{\Lambda}(s-t) G (U^\Lambda(x,s), V^\Lambda(x,s)) \mathrm ds . \end{aligned} \end{equation} Using the triangle inequality \begin{equation} \| \mathcal S_\delta^\Lambda(t) - {\mathcal S}(t) \|_{[L^2(\Omega)]^2} \le \| \mathcal S_\delta^\Lambda(t) - \mathcal S^\Lambda(t) \|_{[L^2(\Omega)]^2} + \| \mathcal S^\Lambda(t) - {\mathcal S}(t) \|_{[L^2(\Omega)]^2}, \end{equation} the proof of \eqref{estimate1} can be completed in two steps. \smallskip \noindent\textbf{Step 1.} Estimate of $\| \mathcal S_\delta^\Lambda(t) - {\mathcal S}^\Lambda(t) \|_{[L^2(\Omega)]^2}$. From \eqref{reg-sys} and \eqref{reg-sys.}, we have \begin{align*} U_\delta^\Lambda(t) - U^\Lambda(t) &= \mathcal A^{\Lambda}(T-t) ( u_T^\delta - u_T ) + \mathcal B^{\Lambda}(T-t)( \widetilde u_T^\delta - \widetilde u_T ) \\ &\quad + \int_t^T \mathcal B^{\Lambda}(s-t) [ F ( \mathcal S_\delta^\Lambda(s)) - F (\mathcal S^\Lambda(s) ) ] \mathrm ds. \end{align*} Using Lemma \eqref{lem1} and that $F$ satisfies the global Lipschitz condition, we obtain \begin{equation} \begin{aligned} &\|U_\delta^\Lambda(t) - U^\Lambda(t) \|_{L^2(\Omega)} \\ &\le \| \mathcal A^{\Lambda}(T-t) ( u_T^\delta - u_T ) \|_{L^2(\Omega)} + \|\mathcal B^{\Lambda}(T-t)( \widetilde u_T^\delta - \widetilde u_T ) \|_{L^2(\Omega)} \\ &\quad + \| \int_t^T \mathcal B^{\Lambda}(s-t) [ F ( \mathcal S_\delta^\Lambda(s)) - F (\mathcal S^\Lambda(s) ) ] \mathrm ds\|_{L^2(\Omega)} \\ &\le \| \mathcal A^{\Lambda}(T-t)\|_{\mathcal L (L^2(\Omega),L^2(\Omega))} \| u_T^\delta - u_T \|_{L^2(\Omega)} \\ &\quad + \|\mathcal B^{\Lambda}(T-t)\|_{\mathcal L (L^2(\Omega),L^2(\Omega))} \| \widetilde u_T^\delta - \widetilde u_T \|_{L^2(\Omega)} \\ &\quad + \int_t^T \| \mathcal B^{\Lambda}(s-t) \|_{\mathcal L (L^2(\Omega), L^2(\Omega))} \| F ( \mathcal S_\delta^\Lambda(s)) - F (\mathcal S^\Lambda(s) ) \|_{L^2(\Omega)} \mathrm ds \\ &\le T_a \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{T-t}{T} } \| u_T^\delta - u_T \|_{L^2(\Omega)} + T_a \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{T-t}{T} } \| \widetilde u_T^\delta - \widetilde u_T \|_{L^2(\Omega)} \\ &\quad + \int_t^T TK_F \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{s-t}{T} } \| \mathcal S_\delta^\Lambda(s) - \mathcal S^\Lambda(s) \|_{[L^2(\Omega)]^2} \mathrm ds \\ &\le \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{T-t}{T} } T_a \Big( \| u_T^\delta - u_T \|_{L^2(\Omega)} + \| \widetilde u_T^\delta - \widetilde u_T \|_{L^2(\Omega)}\Big) \\ &\quad + T_aK_F \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{-t/T} \int_t^T \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{s/T} \| \mathcal S_\delta^\Lambda(s) - \mathcal S^\Lambda(s) \|_{[L^2(\Omega)]^2} \mathrm ds. \end{aligned}\label{U-U.} \end{equation} Similarly, one has \begin{equation} \begin{aligned} &\|V_\delta^\Lambda(t) - V^\Lambda(t) \|_{L^2(\Omega)} \\ &\le \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{T-t}{T} } T_a \Big( \| v_T^\delta - v_T \|_{L^2(\Omega)} + \| \widetilde v_T^\delta - \widetilde v_T \|_{L^2(\Omega)}\Big) \\ &\quad + T_a K_G \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{-t/T} \int_t^T \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{s/T} \| \mathcal S_\delta^\Lambda(s) - \mathcal S^\Lambda(s) \|_{[L^2(\Omega)]^2} \mathrm ds. \end{aligned} \label{V-V.} \end{equation} Combining \eqref{U-U.} and \eqref{V-V.} yields \begin{align*} &\| \mathcal S_\delta^\Lambda(t) - {\mathcal S}^\Lambda(t) \|_{[L^2(\Omega)]^2}\\ &\le \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{T-t}{T} } T_a \Big( \| u_T^\delta - u_T \|_{L^2(\Omega)} + \| \widetilde u_T^\delta - \widetilde u_T \|_{L^2(\Omega)}\\ &\quad + \| v_T^\delta - v_T \|_{L^2(\Omega)} + \| \widetilde v_T^\delta - \widetilde v_T \|_{L^2(\Omega)} ) \\ &\quad + T_a (K_F+K_G) \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{-t/T} \int_t^T \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{s/T} \| \mathcal S_\delta^\Lambda(s) - {\mathcal S}^\Lambda(s) \|_{[L^2(\Omega)]^2} \mathrm ds. \end{align*} Consequently, \begin{equation} \begin{aligned} \| \mathcal S_\delta^\Lambda(t) - {\mathcal S}^\Lambda(t) \|_{[L^2(\Omega)]^2} &\le \Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{\frac{T-t}{T} } T_a \delta + 2T_a \mathbb{K} \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{-t/T}\\ &\quad\times \int_t^T \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{s/T} \| \mathcal S_\delta^\Lambda(s) - {\mathcal S}^\Lambda(s) \|_{[L^2(\Omega)]^2} \mathrm ds, \end{aligned}\label{S-S.2} \end{equation} where $\mathbb{K}:=\max\{K_F,K_G\}$. Multiplying both sides of \eqref{S-S.2} by $\Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{t/T}$, we obtain \begin{align*} & \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{t/T} \| \mathcal S_\delta^\Lambda(t) - {\mathcal S}^\Lambda(t) \|_{[L^2(\Omega)]^2} \\ &\le \frac{T}{\Lambda \log (\frac{T}{\Lambda })} T_a \delta + 2T_a \mathbb{K} \int_t^T \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{s/T} \| \mathcal S_\delta^\Lambda(s) - {\mathcal S}^\Lambda(s) \|_{[L^2(\Omega)]^2} \mathrm ds. \end{align*} Then Gronwall's inequality yields \begin{align*} \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{t/T} \| \mathcal S_\delta^\Lambda(t) - {\mathcal S}^\Lambda(t) \|_{[L^2(\Omega)]^2} \le e^{2T_a \mathbb{K} (T-t)} \frac{T}{\Lambda \log (\frac{T}{\Lambda })} T_a \delta. \end{align*} From this we have \begin{equation} \| \mathcal S_\delta^\Lambda(t) - {\mathcal S}^\Lambda(t) \|_{[L^2(\Omega)]^2} \le e^{2T_a \mathbb{K} (T-t)} \Big[ \frac{T}{\Lambda \log (\frac{T}{\Lambda })} \Big]^{1-\frac{t}{T}} T_a \delta. \label{SDL-SL} \end{equation} \noindent\textbf{Step 2.} Estimate of $\| {\mathcal S}^\Lambda(t) - \mathcal S (t) \|_{[L^2(\Omega)]^2}$. First, we note that \begin{equation} \begin{aligned} \mathcal H^{\Lambda} u(x,t) &= \mathcal H^{\Lambda} \mathcal{A}(T-t) u_T - \mathcal H^{\Lambda} \mathcal{B}(T-t) \widetilde{u}_T - \int_t^T \mathcal H^{\Lambda}\mathcal{B}(s-t) F(\mathcal S)(s) \mathrm d s \\ &=\mathcal{A}^{\Lambda}(T-t) u_T - \mathcal{B}^{\Lambda}(T-t) \widetilde{u}_T - \int_t^T \mathcal{B}^{\Lambda}(s-t) F(\mathcal S)(s) \mathrm d s. \end{aligned} \end{equation} Thanks to the triangle inequality, we conclude that \begin{equation} \label{triangle} \|U^\Lambda(t) - u(t)\|_{L^2(\Omega)} \leq \underbrace{\|U^\Lambda(t) - \mathcal H^{\Lambda} u(t)\|_{L^2(\Omega)}} _{\mathbb I} + \underbrace{\| u(t) - \mathcal H^{\Lambda} u(t)\|_{L^2(\Omega)}}_{\mathbb{II}}. \end{equation} One has \begin{equation} \label{estI} \begin{aligned} |I| &\leq \int_t^T \| \mathcal{B}^{\Lambda}(s-t)\|_{\mathcal{L}(L^2(\Omega),L^2(\Omega))} \|[F ( \mathcal S^\Lambda)(s) - F (\mathcal S )(s)] \|_{L^2(\Omega)} \mathrm d s \\ &\leq K_FT_a \int_t^T \Big[ \frac{T}{ \Lambda \log (\frac{T}{\Lambda })} \Big]^{\frac{s-t}{T} } \|\mathcal S^\Lambda(s) - \mathcal S(s)\|_{L^2(\Omega)} \mathrm d s. \end{aligned} \end{equation} Now we have \begin{equation} \label{II} \begin{aligned} |II|^2 &= \|(1-\mathcal H^{\Lambda}) u(t)\|_{L^2(\Omega)}^2 \\ &= \sum_{p=1}^\infty \big[1-(1 + \Lambda C_pe^{C_pT})^{-1}\big]^2 \langle u(x,t),\xi_p\rangle_{L^2(\Omega)}^2 \\ & = \sum_{p=1}^\infty \Lambda^2 C_p^2 \Big[\frac{1}{e^{-C_pT}+\Lambda C_p }\Big]^2 \langle u(x,t),\xi_p\rangle_{L^2(\Omega)}^2 \\ & = \sum_{p=1}^\infty \Big[\frac{e^{-C_pt}}{e^{-C_pT}+\Lambda C_p }\Big]^2 \Lambda^2 C_p^2 e^{2C_pt}\langle u(x,t),\xi_p\rangle_{L^2(\Omega)}^2. \end{aligned} \end{equation} Similarly to \eqref{es3}, we infer that \begin{equation} \label{phu} \begin{aligned} \frac{e^{-C_pt}}{e^{-C_pT}+\Lambda C_p } & = \frac{e^{-C_pt}}{[e^{-C_pT}+\Lambda C_p]^{t/T}[e^{-C_pT} +\Lambda C_p]^{1-t/T} } \\ &\leq \frac{1}{[e^{-C_pT}+\Lambda C_p]^{1-t/T} } \\ &\leq \Big[\frac{T}{ \Lambda \log (\frac{T}{\Lambda})}\Big]^{1-t/T}. \end{aligned} \end{equation} Using \eqref{phu} in \eqref{II}, we obtain \begin{equation} \begin{aligned} |II|^2 &\leq \sum_{p=1}^\infty \Big[\frac{T}{ \Lambda \log (\frac{T}{\Lambda})}\Big]^{2-2t/T} \Lambda^2 C_p^2 e^{2C_pt}\langle u(x,t),\xi_p\rangle_{L^2(\Omega)}^2 \\ &\leq \Lambda^{2t/T}\Big[\frac{T}{\log(\frac{T}{\Lambda})}\Big]^{2-2t/T} \sum_{p=1}^\infty 4a^2\lambda_p^{2\gamma}e^{4a\lambda_p^{\gamma}} T \langle u(x,t),\xi_p\rangle_{L^2(\Omega)}^2 \\ &=4a^2\Lambda^{2t/T}\Big[\frac{T}{\log(\frac{T}{\Lambda})}\Big]^{2-2t/T} \| u(\cdot,t)\|_{\mathbb G_{2,2aT}^\gamma}^2. \end{aligned} \end{equation} Consequently, \begin{equation} \label{estII} |II| \leq 2a\Lambda^{t/T}\Big[\frac{T}{\log(\frac{T}{\Lambda})}\Big]^{1-t/T} \| u(\cdot,t)\|_{\mathbb G_{2,2aT}^\gamma}. \end{equation} Combining \eqref{triangle}, \eqref{estI} and \eqref{estII}, we deduce that \begin{equation} \begin{aligned} \|U^\Lambda(t) - u(t)\|_{L^2(\Omega)} &\leq K_F T_a \int_t^T \Big[\frac{T}{\Lambda \log (\frac{T}{\Lambda })}\Big]^{\frac{s-t}{T} } \|\mathcal S^\Lambda(s) - \mathcal S(s)\|_{[L^2(\Omega)]^2} \mathrm d s \\ &\quad + 2a\Lambda[\frac{T}{\Lambda\log(\frac{T}{\Lambda})}]^{1-t/T} \| u(\cdot,t)\|_{\mathbb G_{2,2aT}^\gamma}. \end{aligned}\label{Uu} \end{equation} Multiplying both sides of \eqref{Uu} by $\big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\big]^{t/T}$, we obtain \begin{equation} \begin{aligned} &\Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{t/T} \|U^\Lambda(t) - u(t)\|_{L^2(\Omega)} \\ &\leq \frac{2aT}{\log(\frac{T}{\Lambda})} \| u(\cdot,t)\|_{\mathbb G_{2,2aT}^\gamma} \\ &\quad + K_FT_a \int_t^T \Big[\frac{T}{\Lambda \log (\frac{T}{\Lambda })}\Big]^{s/T} \|\mathcal S^\Lambda(s) - \mathcal S(s) \|_{[L^2(\Omega)]^2} \mathrm d s. \end{aligned}\label{UL-u} \end{equation} Similarly, \begin{equation} \begin{aligned} &\Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{t/T} \|V^\Lambda(t) - v(t)\|_{L^2(\Omega)} \\ &\leq \frac{2aT}{\log(\frac{T}{\Lambda})} \| v(\cdot,t)\|_{\mathbb G_{2,2aT}^\gamma} \\ &\quad + K_GT_a \int_t^T \Big[\frac{T}{\Lambda \log (\frac{T}{\Lambda })}\Big]^{s/T} \|\mathcal S^\Lambda(s) - \mathcal S(s) \|_{[L^2(\Omega)]^2} \mathrm d s. \end{aligned} \label{VL-v} \end{equation} From \eqref{UL-u} and \eqref{VL-v}, we have \begin{align*} &\Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{t/T} \|\mathcal S^\Lambda (t) - \mathcal S(t) \|_{[L^2(\Omega)]^2} \\ &\leq \frac{2aT}{\log(\frac{T}{\Lambda})} \|\mathcal S(\cdot,t)\|_{[\mathbb G_{2,2aT}^\gamma]^2} + 2 \mathbb{K} T_a \int_t^T \Big[\frac{T}{\Lambda \log (\frac{T}{\Lambda })}\Big]^{s/T} \|\mathcal S^\Lambda(s) - \mathcal S(s)\|_{[L^2(\Omega)]^2} \mathrm d s . \end{align*} Applying Gronwall's inequality, we obtain \[ \Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{t/T} \|\mathcal S^\Lambda (t) - \mathcal S(t)\|_{[L^2(\Omega)]^2} \leq e^{2T_a \mathbb{K} (T-t)}~ \frac{2aT}{\log(\frac{T}{\Lambda})} \|\mathcal S(\cdot,t)\|_{[\mathbb G_{2,2aT}^\gamma ]^2}. \] It follows that \begin{equation} \|\mathcal S^\Lambda (t) - \mathcal S(t)\|_{[L^2(\Omega)]^2} \leq 2a \Lambda e^{2T_a \mathbb{K} (T-t)} \Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{1-\frac{t}{T}} \|\mathcal S(\cdot,t)\|_{[\mathbb G_{2,2aT}^\gamma]^2}. \label{SL-S} \end{equation} Combining \eqref{SDL-SL} and \eqref{SL-S}, we conclude that \begin{align*} &\|\mathcal S_\delta^\Lambda(t) - \mathcal S(t)\|_{[L^2(\Omega)]^2} \\ &\le 2a \Lambda e^{2T_a \mathbb{K} (T-t)} \Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{1-\frac{t}{T}} \|\mathcal S(\cdot,t)\|_{[\mathbb G_{2,2aT}^\gamma]^2} +e^{2T_a \mathbb{K} (T-t)} \Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })} \Big]^{1-\frac{t}{T}} T_a \delta. \end{align*} This completes the proof. \end{proof} \section{Locally Lipschitz source functions} In the rest of this paper, for solving System \eqref{system}, we concentrate on the case of locally Lipschitz functions. In many ways, the locally Lipschitz functions are more natural. For example, $h(u)=u^2, u^3, u\sin u$, etc, are locally Lipschitz functions but not globally ones. Results for the locally Lipschitz case are still very scarce. The local Lipschitz condition (coercive-type) \[ \|h(u_1,v_1)-h(u_2,v_2)\|_{L^2(\Omega)} \leq K(\mathcal R) (\|u_1-u_2\|_{L^2(\Omega)} + \|v_1-v_2\|_{L^2(\Omega)} ), \] for $\|u_1\|_{L^2(\Omega)},\|u_2\|_{L^2(\Omega)},\|v_1\|_{L^2(\Omega)}, \|u_2\|_{L^2(\Omega)} \leq \mathcal R$. The conditions hold for the following source. \subsection*{Example} Let \[ h_1(u,v) = u\|u\|_{L^2(\Omega)}^2 + v\|v\|_{L^2(\Omega)}^2. \] By direct computations, we obtain \begin{align*} &\|h_1(u_1,v_1)-h_1(u_2,v_2)\|_{L^2(\Omega)} \\ &= \|u_1\|u_1\|_{L^2(\Omega)}^2 - u_2\|u_2\|_{L^2(\Omega)}^2 + v_1\|v_1\|_{L^2(\Omega)}^2 - v_2\|v_2\|_{L^2(\Omega)}^2 \|_{L^2(\Omega)} \\ &\leq \|\|u_1\|_{L^2(\Omega)}^2(u_1-u_2) + u_2\Big(\|u_1\|_{L^2(\Omega)}^2 - \|u_2\|_{L^2(\Omega)}^2\Big) \|_{L^2(\Omega)} \\ &\quad + \|\|v_1\|_{L^2(\Omega)}^2(v_1-v_2) + v_2\Big(\|v_1\|_{L^2(\Omega)}^2 - \|v_2\|_{L^2(\Omega)}^2\Big) \|_{L^2(\Omega)} \\ &\leq \|u_1\|_{L^2(\Omega)}^2\|u_1-u_2\|_{L^2(\Omega)} + \|u_2\|_{L^2(\Omega)}\big(\|u_1\|_{L^2(\Omega)} + \|u_2\|_{L^2(\Omega)} \big)\\ &\quad\times \big|\|u_1\|_{L^2(\Omega)} - \|u_2\|_{L^2(\Omega)}\big| + \|v_1\|_{L^2(\Omega)}^2\|v_1-v_2\|_{L^2(\Omega)} \\ &\quad + \|v_2\|_{L^2(\Omega)} \big(\|v_1\|_{L^2(\Omega)} + \|v_2\|_{L^2(\Omega)}\big) |\|v_1\|_{L^2(\Omega)} - \|v_2\|_{L^2(\Omega)}| \\ &\leq \Big(\|u_1\|_{L^2(\Omega)}^2 + \|u_1\|_{L^2(\Omega)}\|u_2\|_{L^2(\Omega)} + \|u_2\|_{L^2(\Omega)}^2 \Big) \|u_1-u_2\|_{L^2(\Omega)} \\ &\quad + \Big(\|v_1\|_{L^2(\Omega)}^2 + \|v_1\|_{L^2(\Omega)}\|v_2\|_{L^2(\Omega)} + \|v_2\|_{L^2(\Omega)}^2 \Big) \|v_1-v_2\|_{L^2(\Omega)}. \end{align*} It is easy to check that $h_1$ is not global Lipschitz. Let $\mathcal R >0$. For each $u_1,u_2,v_1, v_2$ such that $\mathcal R \geq \max \{\|u_1\|_{L^2(\Omega)},\|u_2\|_{L^2(\Omega)}, \|v_1\|_{L^2(\Omega)}, \|v_2\|_{L^2(\Omega)}\}$, we can choose $K(\mathcal R)=3\mathcal R^2$. However, this is not satisfied in many cases, e.g.\ $h_2(u,v) = a(u+v) - b(u^3+v^3)$, $(a, b > 0)$. Hence, we have to find another regularization method to study the problem with the locally Lipschitz source which is similar to the latter source. We assume that the functions $F,G:\overline{D_T} \times \mathbb R^2 \to \mathbb R$, are locally Lipschitz i.e., for each $\mathcal R>0$, there exists $ K_F(\mathcal R),K_G(\mathcal R)>0$ such that for all $(x,t) \in D_T$, we have \begin{equation} \label{local-lip-F} \begin{gathered} |F (x,t;u_1;v_1) - F(x,t;u_2;v_2)| \leq K_F(\mathcal R) ( |u_1-u_2|+|v_1-v_2|),\\ |G (x,t;u_1;v_1) - G(x,t;u_2;v_2)| \leq K_G(\mathcal R) ( |u_1-u_2|+|w_1-w_2|), \end{gathered} \end{equation} if $u_i,v_i \in \overline{\mathbb S}(\mathcal R)$, $i=1,2$, where $\overline{\mathbb S}(\mathcal R)$ is the closed ball in $L^2(\Omega)$ of center zero and radius $\mathcal R$, and \begin{gather*} \begin{aligned} K_F(\mathcal R):= \sup_{(x,t) \in D_T} \Big\{& \big|\frac{F (x,t;u_1;v_1) - F(x,t;u_2;v_2)}{ |u_1-u_2|+|v_1-v_2|} \big|\\ &: (u_1,v_1) \neq (u_2,v_2), \; u_i,w_i \in \overline{\mathbb S}(\mathcal R),\; i=1,2\Big\} < \infty, \end{aligned} \\ \begin{aligned} K_G(\mathcal R):= \sup_{(x,t) \in D_T} \Big\{ &\big|\frac{G (x,t;u_1;v_1) - G(x,t;u_2;v_2)}{ |u_1-u_2|+|v_1-v_2|} \big|\\ &: (u_1,v_1) \neq (u_2,v_2), \; u_i,v_i \in \overline{\mathbb S}(\mathcal R),\; i=1,2 \Big\} < \infty. \end{aligned} \end{gather*} Notice that $K_F(\mathcal R), K_G(\mathcal R)$ are increasing. The main idea is to approximate the locally Lipschitz functions $F, G$ by the sequences $\mathcal F_{\mathcal R^\delta}, \mathcal G_{\mathcal R^\delta}$ of globally Lipschitz functions: \begin{equation} \label{df-F} \mathcal F_{\mathcal R^\delta}(x,t;u;v) := F(x,t;\widetilde {u};\widetilde{ v}), \quad \mathcal G_{\mathcal R^\delta}(x,t;u;v) := G(x,t;\widetilde{ u};\widetilde{ v}), \end{equation} where \begin{equation} \label{definitionu} \widetilde{w} := \begin{cases} -\mathcal R^\delta, &\text{if } w \in (-\infty, -\mathcal R^\delta),\\ w , &\text{if } w \in [-\mathcal R^\delta, \mathcal R^\delta],\\ \mathcal R^\delta, &\text{if } w \in (\mathcal R^\delta, +\infty). \end{cases} \end{equation} Here, the term $\mathcal R^\delta$ is positive and depends on $\delta$ and satisfies $ \lim_{\delta \to 0} \mathcal R^\delta =+\infty $. Moreover, for $\delta$ sufficiently small, we have \[ \mathcal R^\delta \geq \sup_{(x,t) \in \overline{D_T} } \Big( |u(x,t)|+|v(x,t)| \Big). \] This implies immediately \begin{equation} \label{R-global} \mathcal F_{\mathcal R^\delta}(x,t;u;v) := F(x,t;u;v), \quad \mathcal G_{\mathcal R^\delta}(x,t;u;v) := G(x,t;u;v). \end{equation} \begin{remark} \rm The locally Lipschitz constants $K_F, K_G$ depend on $\delta$. It is also interesting that $\mathcal R^\delta$ is chosen suitable in order to obtain a convergence rate (our purpose is to improve the rate of convergence). \end{remark} Before presenting the main results, we need to some auxiliary results. We do not claim that these auxiliary results are new, but for completeness of the presentation we give their proofs here. \begin{lemma} \label{lem:F_est} Let $\mathcal F_{\mathcal R^\delta},\mathcal G_{\mathcal R^\delta} \in L^\infty(\overline{D_T} \times \mathbb{R}^2)$ given as in \eqref{df-F}. Then we have \begin{gather} |\mathcal F_{\mathcal R^\delta}(x,t;u_1;v_1) - \mathcal F_{\mathcal R^\delta}(x,t;u_2;v_2)| \leq K_{F}(\mathcal R^\delta) ( |u_1-u_2|+|v_1-v_2|), \label{Lip-localF}\\ |\mathcal G_{\mathcal R^\delta}(x,t;u_1;v_1) - \mathcal G_{\mathcal R^\delta}(x,t;u_2;v_2)| \leq K_{G}(\mathcal R^\delta) ( |u_1-u_2|+|v_1-v_2|), \label{Lip-localG} \end{gather} for any $(x,t)\in D_T$, $u_i, v_i \in \mathbb{R}$, $i =1,2$. \end{lemma} \begin{proof} First, we show that for any $w_1, w_2 \in \mathbb R$ and $\widetilde{w}_1, \widetilde{w}_2$ satisfying \eqref{definitionu} then \begin{equation} \label{ineu} |\widetilde{w}_1- \widetilde{w}_2| \le |w_1 - w_2|. \end{equation} The proof \eqref{ineu} is divided into three cases. \smallskip \noindent\textbf{ Case 1.} $w_1 <-\mathcal R^\delta$. \begin{itemize} \item[$\triangleright$] If $w_2<-\mathcal R^\delta$ then $ |\widetilde{w}_1- \widetilde{w}_2|=0$. \item[$\triangleright$] If $-\mathcal R^\delta \le w_2 \le \mathcal R^\delta$ then $|\widetilde{w}_1- \widetilde{w}_2| = w_2 + \mathcal R^\delta < w_2 - w_1 = | w_1 - w_2|$. \item[$\triangleright$] If $w_2>\mathcal R^\delta$ then $ |\widetilde{w}_1 - \widetilde{w}_2| =2\mathcal R^\delta \le w_2-w_1 =|w_1- w_2|$. \end{itemize} \smallskip \noindent\textbf{Case 2.} $ -\mathcal R^\delta \le w_1 \le \mathcal R^\delta$. \begin{itemize} \item[$\triangleright$] If $w_2<-\mathcal R^\delta$ then $ |\widetilde{w}_1 - \widetilde{w}_2|=|w_1 + \mathcal R^\delta| = w_1 + \mathcal R^\delta < w_1 - w_2 = | w_1- w_2|$. \item[$\triangleright$] If $-\mathcal R^\delta \le w_2 \le \mathcal R^\delta$ then $ |\widetilde{w}_1- \widetilde{w}_2| = |w_1- w_2|$. \item[$\triangleright$] If $w_2>\mathcal R^\delta$ then $ |\widetilde{w}_1 - \widetilde{w}_2| =\mathcal R^\delta- w_1 \le w_2-w_1 = | w_1- w_2|$. \end{itemize} \smallskip \noindent\textbf{Case 3.} $ w_1 > \mathcal R^\delta$. \begin{itemize} \item[$\triangleright$] If $w_2<-\mathcal R^\delta$ then $ |\widetilde{w}_1 - \widetilde{w}_2|=2\mathcal R^\delta < w_1-w_2=| w_1- w_2|$. \item[$\triangleright$] If $-\mathcal R^\delta \le w_2\le \mathcal R^\delta$ then $ |\widetilde{w}_1 - \widetilde{w}_2| = \mathcal R^\delta - w_2 \mathcal R^\delta$ then $ |\widetilde{w}_1- \widetilde{w}_2| =0 \le |w_1- w_2|$. \end{itemize} Summarizing the above discussions, we arrive at \eqref{ineu}. Now we return to the proof of Lemma \ref{lem:F_est}. Since $\widetilde u,~\widetilde v \le \mathcal R^\delta$ and using \eqref{local-lip-F}, we have \begin{align*} |\mathcal F_{\mathcal R^\delta}(x,t;u_1;v_1) - \mathcal F_{\mathcal R^\delta}(x,t;u_2;v_2)| &= | F(x,t;\widetilde{u}_1;\widetilde{v}_1)- F(x,t;\widetilde{u}_2;\widetilde{v}_2)| \\ &\le K_{ F}(\mathcal R^\delta) ( |\widetilde{u}_1-\widetilde{u}_2| +|\widetilde{v}_1-\widetilde{v}_2|) \\ &\le K_{ F}(\mathcal R^\delta) ( |u_1-u_2|+|v_1-v_2|), \end{align*} where we have used \eqref{ineu} in the last estimate. We use a similar argument to ensure the local Lipschitzian condition f the function $\mathcal G_{\mathcal R^\delta}$. \end{proof} We first consider a perturbed model yielding a well-posed system whose solution will approximate $u,v$. In particular, we define the approximate system \begin{equation} \label{reg-sys-lip} \begin{gathered} \begin{aligned} I_\delta^{\Lambda}(x,t)&= \mathcal A^{\Lambda}(T-t) u_T^\delta(x) + \mathcal B^{\Lambda}(T-t) \widetilde u_T^\delta(x) \\ &\quad + \int_t^T \mathcal B^{\Lambda}(\tau-t) \mathcal F_{\mathcal R^\delta} ( I_\delta^\Lambda(x,\tau), J_\delta^\Lambda(x,\tau)) \mathrm d\tau, \end{aligned} \\ \begin{aligned} J_\delta^\Lambda(x,t)&= \mathcal A^{\Lambda}(T-t) v_T^\delta(x) + \mathcal B^{\Lambda}(T-t) \widetilde v_T^\delta(x) \\ &\quad + \int_t^T \mathcal B^{\Lambda}(\tau-t) \mathcal G_{\mathcal R^\delta} ( I_\delta^\Lambda(x,\tau), J_\delta^\Lambda(x,\tau)) \mathrm d\tau. \end{aligned} \end{gathered} \end{equation} Our principal result, based on the analysis above, is then the following theorem. \begin{theorem}\label{The-main2} Let $m\geq 1$, $n \geq 2aT $ and $\Lambda:=\Lambda(\delta)$ be as in Theorem \ref{theo-main1}. Assume that the system \eqref{u_ex1} has a unique solution $\mathcal S := ( u, v) \in [C([0,T];L^2(\Omega)) \cap L^\infty(0,T;\mathbb G_{m,n}^\gamma)]^2$ which satisfies \begin{equation} \label{priori2} \|\mathcal S(t)\|_{[C([0,T];L^2(\Omega))]^2} + \|\mathcal S(t)\|_{[L^\infty(0,T;\mathbb G_{m,n}^\gamma)]^2}\le Q. \end{equation} Assuming that we can choose a sequence $\mathcal{R}^\delta > 0$ such that $\lim_{\delta \to 0^+} \mathcal{R}^\delta = \infty$ and \begin{equation} \label{condK} \mathbb K(\mathcal{R}^\delta) \leq \frac{\eta}{TT_a} \log[\log(\Lambda^{-1})],\quad \text{for some } \eta \in (0,\frac{1}{2}), \end{equation} where, $\mathbb K(\mathcal{R}^\delta) = \max\{K_{F}(\mathcal{R}^\delta), K_{G}(\mathcal{R}^\delta)\}$, and $T_a$ is defined in Lemma \ref{lem1}. Suppose that System \eqref{reg-sys-lip} has a unique solution $(I_\delta^\Lambda,J_\delta^\Lambda) \in [C([0,T];L^2(\Omega))]^2$. Then, the error estimate between the solution $(I_\delta^\Lambda,J_\delta^\Lambda) \in [C([0,T];L^2(\Omega))]^2$ of problem \eqref{reg-sys-lip} and the sought solution $(u,v)$ to \eqref{system} is given by \begin{equation} \label{seconderror} \|Z_\delta^\Lambda(t) - \mathcal S(t)\|_{[L^2(\Omega)]^2} \leq \big(2a Q + T_a\delta \Lambda^{-1} \big) \log^{2\eta}(\Lambda^{-1}) \Big[ \frac{T}{ \log (\frac{T}{\Lambda })} \Big]^{1-\frac{t}{T}} \Lambda^{t/T} , \end{equation} where $Z_\delta^\Lambda:=(I_\delta^\Lambda,J_\delta^\Lambda) \in [C([0,T];L^2(\Omega))]^2$. \end{theorem} \begin{remark} \rm In \eqref{seconderror}, if $t>0$, the error estimate is of order $\Lambda^{t/T} [ \frac{T}{ \log (\frac{T}{\Lambda })} ]^{2\eta+ \frac{t}{T}-1} $ which tends to zero as $\delta \to 0^+$. And if $t= 0$, the error \eqref{seconderror} becomes \begin{equation} \label{t=0} \|Z_\delta^\Lambda(0) - \mathcal S(0)\|_{[L^2(\Omega)]^2} \leq C \Big[ \frac{T}{ \log (\frac{T}{\Lambda })} \Big]^{2\eta -1}, \quad 0<\eta<\frac{1}{2}. \end{equation} We also note that the term $[ \frac{T}{ \log (\frac{T}{\Lambda })} ]^{2\eta -1}$, $\eta \in (0,\frac{1}{2})$ tends to zero when $\delta \to 0^+$. From above observations, we conclude that the right-hand side in estimation \eqref{seconderror} tends to zero for all $t \in [0,T]$. \end{remark} \begin{proof} First, we note that the proof of the existence and uniqueness of the solution to problem \eqref{reg-sys-lip} is the same as in Theorem \ref{theorem1}. Next, we denote $ Z^\Lambda:=(I^\Lambda,J^\Lambda)$ the solution of system \eqref{reg-sys-lip} with exact data $(u_T,\widetilde{u}_T )$ and $(v_T,\widetilde{v}_T)$. We know that \begin{equation} \| Z_\delta^\Lambda(t) - \mathcal S(t) \|_{[L^2(\Omega)]^2} \leq \| Z_\delta^\Lambda(t) - Z^\Lambda(t) \|_{[L^2(\Omega)]^2} + \| Z^\Lambda(t) - \mathcal S(t) \|_{[L^2(\Omega)]^2}. \end{equation} By an argument analogous to that used for the proof of Theorem \ref{theo-main1}, we emphasize that the proof in Step 1 and Step 2 of Theorem \ref{theo-main1} remains valid. Also, replace the globally Lipschitz conditions \eqref{Lip-F} and \eqref{Lip-G} by the locally Lipschitz conditions \eqref{Lip-localF} and \eqref{Lip-localG}, respectively. Since $\lim_{\delta \to 0^+} \mathcal R^\delta = \infty$, for a sufficiently small $\delta >0$, there is an $\mathcal R^\delta>0$ such that $\mathcal R^\delta \geq \|\mathcal S\|_{[L^\infty(0,T;L^2(\Omega))]^2}$. For this value of $\mathcal R^\delta$ (from \eqref{definitionu}) we have \begin{equation} \mathcal F_{\mathcal R^\delta}(x,t; u; v)=F(x,t; u; v), \quad \mathcal G_{\mathcal R^\delta}(x,t; u; v)=G(x,t; u; v). \end{equation} Using the global Lipschitz property of $\mathcal F_{\mathcal R^\delta}, \mathcal G_{\mathcal R^\delta}$ (see Lemma \ref{lem:F_est}), yields \begin{equation} \|Z_\delta^\Lambda(t) - Z^\Lambda(t) \|_{[L^2(\Omega)]^2} \leq \exp\Big(2T_a \mathbb K(\mathcal R^\delta)(T-t)\Big) \Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{1-\frac{t}{T}} T_a\delta. \label{ZDL-ZL} \end{equation} Also, one has \begin{equation} \|Z^\Lambda(t) - \mathcal S(t) \|_{[L^2(\Omega)]^2} \leq 2a \Lambda \exp\Big(2T_a\mathbb K(\mathcal R^\delta)(T-t)\Big) \Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{1-\frac{t}{T}} Q. \label{ZL-Z} \end{equation} Combining \eqref{ZDL-ZL} and \eqref{ZL-Z}, we obtain \[ % \label{est2} \|Z_\delta^\Lambda(t) - \mathcal S(t) \|_{[L^2(\Omega)]^2} \leq \big(2a Q + T_a\delta \Lambda^{-1}\big) \exp\Big(2T_a\mathbb K(\mathcal R^\delta)(T-t)\Big) \Big[\frac{T}{\Lambda \log(\frac{T}{\Lambda })}\Big]^{1-\frac{t}{T}}. \] Using \eqref{condK} in this inequality, estimate \eqref{seconderror} follows. \end{proof} \subsection*{Conclusion} In this article, we showed that the inverse backward system \eqref{system} is ill-posed in the sense of Hadamard. 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