\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 225, pp. 1--34.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/225\hfil Direct and inverse problems] {Direct and inverse problems for systems of singular differential boundary-value problems} \author[A. Favini, A. Lorenzi, H. Tanabe \hfil EJDE-2012/225\hfilneg] {Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe} % in alphabetical order \address{Angelo Favini \newline Dipartimento di Matematica, Universit\`{a} degli Studi di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy} \email{favini@dm.unibo.it} \address{Alfredo Lorenzi \newline Dipartimento di Matematica ``F. Enriques'', Universit\`a degli Studi di Milano, via Saldini 50, 20133 Milano, Italy} \email{alfredo.lorenzi@unimi.it} \address{Hiroki Tanabe \newline Takarazuka, Hirai Sanso 12-13, 665-0817, Japan} \email{h7tanabe@jttk.zaq.ne.jp} \thanks{Submitted November 10, 2012. Published December 11, 2012.} \subjclass[2000]{35R30, 34G10, 35K20, 35K50, 45N05, 45Q05} \keywords{Direct and inverse problems; first-order equations in Banach spaces; \hfill\break\indent linear parabolic integro-differential equations; existence and uniqueness} \begin{abstract} Real interpolation spaces are used for solving some direct and inverse linear evolution problems in Banach spaces, on the ground of space regularity assumptions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Several articles are devoted to studying identification problems of the type \begin{equation} \label{IP} \begin{gathered} y'(t) + Ay(t) = f(t)z + h(t), \quad 0 \leq t \leq \tau,\\ y(0) = y_0,\\ \Phi[y(t)] = g(t), \quad 0 \leq t \leq \tau, \end{gathered} \end{equation} Here $-A$ is a linear closed operator generating a $C_0$-semigroup in a Banach space $X$ or $C^\infty$-semigroup in $X$. Moreover, $z$ is a fixed element in $X$, $y_0 \in X$, $\Phi \in X^*$, $g \in C^1([0,\tau];\mathbb{C})$, $h \in C^1([0,\tau];X)$. Roughly speaking, we look for solutions $(y,f)$ in $[C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A)]\times C([0,\tau];\mathbb{C})$. More precisely, we recall that in \cite{AF1,AF2,AFL,FLT1,AS,FL1,FL2,FL3,LO}, all concerned with the parabolic case, the \emph{scalar} function $f$ is sought for in the more regular space $C^\theta([0,\tau];\mathbb{C})$, for some $\theta\in (0,1)$, so that known results of maximal H\"older regularity in time can be applied. For this purpose, $A$ is assumed (cfr. \cite{FY}) to satisfy the estimate \begin{equation} \label{e1.1} \| (\lambda+A)^{-1}\|_{\mathcal{L}(X)} \leq c(1+|\lambda|)^{-\beta} \end{equation} for all $\lambda$ in the sector \begin{equation} \label{e1.2} \Sigma_\alpha := \{ \lambda \in \mathbb{C}: \operatorname{Re}\lambda \geq -c (1+ |\operatorname{Im} \lambda|)^\alpha\}, \quad 0 < \beta \leq \alpha\le 1. \end{equation} Taira \cite{TA2} deals with the case $\alpha = 1$ and introduces the power $A^\gamma$ for $\gamma > 1-\beta$. He proves that $D(A^\gamma) \supseteq D(A)$ if $\beta > 1/2$ and $1-\beta < \gamma < \beta$. The results on the Cauchy problem \begin{gather} \label{e1.3} y'+Ay=f(t),\quad 0\le t\le\tau, \\ \label{e1.4} y(0)=y_0, \end{gather} $f\in C([0,\tau];X),y_0\in D(A)$, corresponding to the case $\alpha=\beta=1$ are by now classical after the works by Da Prato, Lunardi, Sinestrari, and their followers, concerning maximal in time and/or spatial regularity of the strict solutions. Further, when $0 < \beta \leq 1$, $\alpha = 1$, the existence (and also time regularity) of the solution was considered by Wild in \cite{WI}. In the case of a multi-valued operator $A$, for which \eqref{e1.3} takes the form of an inclusion, in \cite{FY} the authors introduced the spaces \[ X_A^{\theta,\infty} := \{ u \in X: \sup_{t>0} t^\theta \|A^\circ(t+A)^{-1}u \|_X < \infty \},\quad 0<\theta<1, \] where $A^\circ(t+A)^{-1}$ is the linear section of $A(t+A)^{-1}$ defined in Theorem 2.7 of the quoted monograph. If $A$ is not multivalued and $\alpha = \beta = 1$, then $X_A^{\theta,\infty} = (X,D(A))_{\theta,\infty}$, the latter being a real interpolation space between $X$ and $D(A)$. In general, the following inclusions hold true (cfr. \cite[p .26]{FY}) \begin{gather}\label{e1.6} X_A^{\theta,\infty} \subseteq (X,D(A))_{\theta,\infty}, \quad 0<\theta<1,\\ \label{e1.7} (X,D(A))_{\theta,\infty} \subseteq X_A^{\theta+\beta-1,\infty},\quad 1-\beta < \theta < 1. \end{gather} Whence it follows that $D(A) \subseteq X_A^{\theta,\infty}$ provided $0<\theta<\beta$. Therefore $X_A^{\theta,\infty}$ is not intermediate between $D(A)$ and $X$. However $D(A^2) \subseteq X_A^{\theta,\infty}$. Restricting ourselves to the \emph{univalent} case, if $u \in D(A^2)$, $t \geq 1$, $$ t^\theta A(A+t)^{-1} u = t^\theta(A+t)^{-1}A^{-1}A^2u = t^{\theta-1}(A^{-1}-(A+t)^{-1})A^2u,\quad 0<\theta<1. $$ Consequently, for $t \geq 1$, we obtain $$ t^\theta \| A(A+t)^{-1}u \|_X \leq t^{\theta-1} \|Au\|_X + c t^{\theta-1-\beta} \|A^2u\|_X. $$ Since $A$ is assumed to be invertible, the inclusion follows. Let us introduce some spaces ${\tilde X}_A^{\theta,\infty}$ which are intermediate between $X$ and $D(A)$ (and which reduce to $X_A^{\theta,\infty}$ if $\alpha = \beta = 1$). Such spaces seem more appropriate to solve \eqref{e1.3}, \eqref{e1.4} and to deduce the spatial regularity of the related solution. For the sake of brevity, we drop out ``$\infty$'' from $X_A^{\theta,\infty}$ and write $X_A^{\theta}$ and ${\tilde X}_A^{\theta}$, respectively. Section 2 is devoted to the intermediate spaces, while in Section 3 the spatial and temporal regularity of solutions to \eqref{e1.3}, \eqref{e1.4} is studied. Section 4 deals with the identification problem \eqref{e1.1}, \eqref{e1.2}, under suitable spatial regularity assumptions. Section 5 is devoted to the new identification problem \begin{gather*} y'(t)+Ay(t) = f_1(t)z_1 + f_2(t)z_2 +h(t),\\ \Phi_j[y(t)] = g_j(t), \quad t \in [0, \tau], \quad j = 0,1, \end{gather*} In Section 6 the results of Section 5 will be applied to solve an inverse problem for systems of evolution differential equations. Section 7 is devoted to general weakly coupled identification problems. Finally, in Sections 8 and 9 the previous abstract results will be applied to a few systems of PDE's, both regular and degenerate. \section{Interpolation spaces} Let $A$ be a closed linear operator acting in the complex space $X$ with \begin{equation}\label{e2.1} \|(\lambda+A)^{-1}\|_{\mathcal{L}(X)}\le C(1+|\lambda|)^{-\beta},\quad \lambda \in \Sigma_\alpha, \end{equation} for some \begin{equation} \label{e2.2} 0<\beta \le \alpha \le 1,\quad \alpha+\beta>1. \end{equation} Denote now (cf. \cite[p. 26]{FY}) \begin{equation} \label{e2.3} \begin{gathered} X^\theta_{A}=\{u\in X: [x]_{X^\theta_{A}} =\sup_{t>0} t^\theta\|A(t+A)^{-1}u\|_X<+\infty\}, \\ \|x\|_{X^\theta_{A}}=\|x\| + [x]_\theta. \end{gathered} \end{equation} It is known that \cite[Theorem 1.12, p. 26]{FY}, \begin{gather} \label{e2.4} X^\theta_{A}\subset (X,\mathcal{D}(A))_{\theta,\infty},\quad \theta\in (0,1), \\ \label{e2.5} (X,\mathcal{D}(A))_{\theta,\infty}\subset X^{\theta+\beta-1}_{A},\quad \theta\in (1-\beta,1). \end{gather} According to \cite[Proposition 3.4]{FY}, if $1-\beta<\theta<1$ we obtain \begin{equation} \label{e2.6} t^{(2-\beta-\theta)/\alpha}\|Ae^{-tA}x\|_X\le C\|x\|_{X^\theta_{A}}. \end{equation} Moreover, from \cite[Theorem 3.5]{FY} with $\theta\in (2-\alpha-\beta,1)$, we obtain \begin{equation} \label{e2.7} \|(e^{-tA}-I)x\|_X \le Ct^{(\alpha+\beta+\theta-2)/\alpha}\|x\|_{X^\theta_{A}}. \end{equation} This implies that, for any $x\in X^\theta_{A}$ and $\theta\in (2-\alpha-\beta,1)$, $e^{-tA}x\to x$ in $X$ as $t\to 0+$. Grounding on \eqref{e2.2}, let us now introduce the intermediate space \begin{equation} \label{e2.8} {\widetilde X}^\theta_{A}=\{u\in X: \sup_{t>0} t^{(2-\beta-\theta)/\alpha}\|Ae^{-tA}u\|_X<+\infty\},\quad 0<\theta<1, \end{equation} endowed with the norm \begin{equation} \label{e2.9} \|u\|_{{\widetilde X}^\theta_{A}}=\|u\|_X + \sup_{t>0} t^{(2-\beta-\theta)/\alpha}\|Ae^{-tA}u\|_X<+\infty. \end{equation} From the known semigroup estimate (cf. \cite[Proposition 3.2]{FLY}) \begin{equation} \label{e2.10} \|A^\theta e^{-tA}x\|_X\le Ct^{(\beta-\theta-1)/\alpha}\|x\|_X,\quad \theta \in [0,+\infty), \end{equation} in particular we deduce \begin{gather} \label{e2.11} \|Ae^{-tA}u\|_X\le Ct^{(\beta-2)/\alpha}\|u\|_X, \quad u\in X, \\ \begin{aligned} \|Ae^{-tA}u\|_X&=\|e^{-tA}Au\|_X\le Ct^{(\beta-1)/\alpha}\|Au\|_{X} \\ &\le Ct^{(\beta-1)/\alpha}\|u\|_{\mathcal{D}(A)}, \quad u\in \mathcal{D}(A), \end{aligned} \label{e2.12} \end{gather} where $\|u\|_{\mathcal{D}(A)}=\|u\|_X+\|Au\|_X$. By interpolation we obtain \begin{equation} \label{e2.13} \begin{aligned} \|Ae^{-tA}u\|_X &\le Ct^{(1-\theta)(\beta-2)/\alpha}t^{\theta(\beta-1)/\alpha} \|u\|_{(X,\mathcal{D}(A))_{\theta,\infty}}\\ & =t^{(\beta+\theta-2)/\alpha}\|u\|_{(X,\mathcal{D}(A))_{\theta,\infty}}. \end{aligned} \end{equation} This implies \begin{equation} \label{e2.14} \sup_{t>0} t^{(2-\beta-\theta)/\alpha}\|Ae^{-tA}u\|_X \le C\|u\|_{(X,\mathcal{D}(A))_{\theta,\infty}},\quad \theta\in (0,1). \end{equation} Therefore, we deduce the continuous embeddings \begin{equation} \label{e2.15} X^\theta_A\hookrightarrow (X,\mathcal{D}(A))_{\theta,\infty} \hookrightarrow {\widetilde X}^\theta_A,\quad \theta\in (0,1). \end{equation} \begin{lemma} \label{lem2.1} If $u\in {\widetilde X}^\theta_A$ and $\theta\in(2-\alpha-\beta,1)$, one has \[ \lim_{t\to +0}e^{-tA}u=u. \] \end{lemma} \begin{proof} If $u\in {\widetilde X}^\theta_{A}$ and $0c>0$, oriented from $\operatorname{Im}z=-\infty$ to $\operatorname{Im}z=+\infty$. Note then that \begin{equation} \label{e2.18} \begin{aligned} A^{-1}e^{-tA}u & = (2\pi i)^{-1}\int_\Gamma e^{t\lambda}A^{-1}(\lambda + A)^{-1}u\,d\lambda \\ &= (2\pi i)^{-1}\int_\Gamma e^{t\lambda}\lambda^{-1}[A^{-1}-(\lambda + A)^{-1}]u\,d\lambda \\ &= A^{-1}u - (2\pi i)^{-1}\int_\Gamma e^{t\lambda}\lambda^{-1}(\lambda + A)^{-1}u\,d\lambda. \end{aligned} \end{equation} As $t\to 0+$ the last integral converges to $\int_\Gamma \lambda^{-1}(\lambda + A)^{-1}u\,d\lambda=0$. Therefore, owing to the uniqueness of the limit, we obtain $A^{-1}u= A^{-1}\xi$, i.e. $\xi=u$. \end{proof} We have thus proved that, if $\theta\in(2-\alpha-\beta,1)$, the mapping $u\to e^{-tA}u$, $t\in [0,+\infty)$, maps ${\widetilde X}^\theta_A$ into $C([0,+\infty);X)$. Let $u\in \mathcal{D}(A)$ and $\lambda>0$. Then $t\to e^{-t\lambda}e^{-tA}$ belongs to $L^1(\mathbb{R}_+;X)$ since $1-\beta<\alpha$. Moreover, \begin{equation} \label{e2.19} \begin{aligned} \int_0^{+\infty} \lambda e^{-t\lambda}e^{-tA}u\,dt &= \int_0^{+\infty} D_t\big(-e^{-t\lambda}\big)e^{-tA}u\,dt \\ &=-[e^{-t\lambda}e^{-tA}u]_0^{+\infty} - \int_0^{+\infty} e^{-t\lambda}e^{-tA}Au\,dt\\ &= u - \int_0^{+\infty} e^{-t\lambda}e^{-tA}Au\,dt, \end{aligned} \end{equation} implying \[ \int_0^{+\infty} e^{-t\lambda}e^{-tA}(\lambda u+Au)\,dt=u,\quad \forall u\in {\mathcal{D}(A)}. \] But this implies the equality \begin{equation} \label{e2.20} (\lambda I+A)^{-1}u=\int_0^{+\infty} e^{-t\lambda}e^{-tA}u\,dt,\quad \forall u\in X. \end{equation} Indeed, if $u\in X$, there exists $v\in {\mathcal{D}(A)}$ such that $u=(\lambda I+A)v$. Then \[ (\lambda I+A)^{-1}u=v=\int_0^{+\infty} e^{-t\lambda}e^{-tA}(\lambda I+A)v\,dt =\int_0^{+\infty} e^{-t\lambda}e^{-tA}u\,dt. \] Consequently, for all $u\in {\widetilde X}^\theta_A$ and $t\in \mathbb{R}_+$, for $\theta>2-\alpha-\beta$ we have \begin{equation} \label{e2.21} \begin{aligned} \|A(t+A)^{-1}u\|_X &=\big\|\int_0^{+\infty} e^{-t\lambda}Ae^{-\lambda A}u\,d\lambda\big\|_X\\ & \le C\|u\|_{{\widetilde X}^\theta_{A}}\int_0^{+\infty} e^{-t\lambda}\lambda^{(\theta+\beta-2)/\alpha}\,d\lambda\\ & = C\|u\|_{{\widetilde X}^\theta_{A}}t^{-(\theta+\alpha+\beta-2)/\alpha} \int_0^{+\infty} e^{-\xi}\xi^{(\theta+\beta-2)/\alpha}\,d\xi. \end{aligned} \end{equation} Summing up, we have proved that the continuous embeddings \begin{equation} \label{e2.22} {\widetilde X}^\theta_A\hookrightarrow X^{(\theta+\alpha+\beta-2)/\alpha}_A\hookrightarrow (X,\mathcal{D}(A))_{(\theta+\alpha+\beta-2)/\alpha,\infty}, \end{equation} hold for any pair $(\alpha,\beta)\in (0,1]\times (0,1]$ satisfying $0<\beta\le \alpha\le 1$, $\alpha+\beta>1$ and $2-\alpha-\beta<\theta<1$. (Note that $(\theta+\alpha+\beta-2)<\alpha$ implies $\theta<2-\beta$.) \section{Spatial regularity of solutions to Cauchy problems} Consider the problem \begin{equation} \label{e3.1}%(DP) \begin{gathered} y'(t)+Ay(t)=f(t),\quad t\in [0,\tau], \\ y(0)=y_0. \end{gathered} \end{equation} We look for a strict solution to the Cauchy problem \eqref{e1.3}, \eqref{e1.4}, i.e. for a function $y\in C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A))$, related to spatial regular data. For this purpose we assume \begin{gather}\label{e3.2} f\in C([0,\tau];X)\cap B([0,\tau];{\widetilde X}^\theta_A),\quad y_0\in \mathcal{D}(A),\quad Ay_0\in {\widetilde X}^\theta_A,\\ \label{e3.3} 0<\beta\le \alpha \le 1,\quad \alpha+\beta>3/2,\quad 2(2-\alpha-\beta)<\theta<1. \end{gather} We recall that, for any Banach space $Y$, $B([0,\tau];Y)$ denotes the Banach space of all bounded $Y$-valued functions $f$, when endowed with the norm $\|f\|_{B([0,\tau];Y)}=\sup_{t\in [0,\tau]} \|f(t)\|_Y$. Necessarily the solution to \eqref{e1.3}, \eqref{e1.4} (cf. \cite{FY}) is given by \begin{equation} \label{e3.4} y(t)=e^{-tA}y_0+\int_0^t e^{-(t-s)A}f(s)\,ds,\quad t\in [0,\tau]. \end{equation} Set now \begin{equation} \label{e3.5} y_1(t)=e^{-tA}y_0,\quad y_2(t)=\int_0^t e^{-(t-s)A}f(s)\,ds,\quad t\in [0,\tau]. \end{equation} It is immediate to check that the properties of the semigroup $\{e^{-tA}\}_{t>0}$ established previously guarantee that $y_1$ is differentiable in $(0,\tau]$. Moreover, for $0\le s < t\le \tau$, we have \begin{equation} \label{e3.6} \begin{aligned} \|y'_1(t)-y'_1(s)\|_X &=\|Ay_1(t)-Ay_1(s)\|_X=\big\|\int_s^t Ae^{-rA}Ay_0\,dr\big\|_X\\ & \le C\|Ay_0\|_{{\widetilde X}^\theta_A}\int_s^t r^{(\theta+\beta-2)/\alpha}\,dr\\ &\le C'\|Ay_0\|_{{\widetilde X}^\theta_A}(t-s)^{[\theta-(2-\alpha-\beta)]/\alpha}. \end{aligned} \end{equation} Therefore, we have proved that $y_1',Ay_1\in C^{[\theta-(2-\alpha-\beta)]/\alpha}([0,\tau];X)$. Consider now the relations \begin{equation}\label{e3.7} \begin{aligned} &\sup_{0\le s\le \tau} \sup_{t>0} s^{(2-\beta-\theta)/\alpha} \|Ae^{-sA}Ae^{-tA}y_0\|_X \\ & = \sup_{0\le s\le \tau} \sup_{t>0} s^{(2-\beta-\theta)/\alpha} \|Ae^{-(s+t)A}Ay_0\|_X\\ & = \sup_{0\le s\le \tau} \sup_{t>0} \Big(\frac{s}{s+t}\Big)^{(2-\beta-\theta)/\alpha} (s+t)^{(2-\beta-\theta)/\alpha}\|Ae^{-(s+t)A}Ay_0\|_X\\ &\le C\|Ay_0\|_{{\widetilde X}^\theta_A}. \end{aligned} \end{equation} Therefore, concerning the regularity of $y_1$, we obtain \begin{equation} \label{e3.8} y_1',Ay_1\in C^{[\theta-(2-\alpha-\beta)]/\alpha}([0,\tau];X) \cap B([0,\tau];{\widetilde X}^\theta_A). \end{equation} Let us now consider $y_2$ and let us notice that, for $0\le s < t\le \tau$, we have \begin{equation}\label{e3.9} \begin{aligned} &Ay_2(t)-Ay_2(s)\\ &=\int_0^s \big[Ae^{-(t-\sigma)A}-Ae^{-(s-\sigma)A}\big]f(\sigma)\,d\sigma + \int_s^t Ae^{-(t-\sigma)A}f(\sigma)\,d\sigma \\ & =: F_1(s,t)+F_2(s,t). \end{aligned} \end{equation} As far as $F_2$ is concerned we obtain \[ \|Ae^{-(t-\sigma)A}f(\sigma)\|_X\leq (t-\sigma)^{(\theta+\beta-2)/\alpha} \|f(\sigma)\|_{{\widetilde X}_\theta} \leq(t-\sigma)^{(\theta+\beta-2)/\alpha}\|f\|_{B([0,\tau];{\widetilde X}_\theta)}. \] Hence \begin{equation} \label{e3.10} \begin{aligned} \|F_2(s,t)\|_X &\leq \int_s^t (t-\sigma)^{(\theta+\beta-2)/\alpha}\|f\|_{B([0,\tau];{\widetilde X}_\theta)}\,d\sigma \\ &=\frac{(t-s)^{[\theta-(2-\alpha-\beta)]/\alpha}}{[\theta-(2-\alpha-\beta)]/\alpha} \|f\|_{B([0,\tau];{\widetilde X}_\theta)}. \end{aligned} \end{equation} Further, since \begin{align*} \|A^2e^{-rA}f(\sigma)\|_X &=\|Ae^{-(r/2)A}\big[Ae^{-(r/2)A}f(\sigma)\big]\|_X\\ &\leq Cr^{(\beta-2)/\alpha}r^{(\beta+\theta-2)/\alpha}\|f(\sigma)\|_{{\widetilde X}^\theta}\\ &\leq Cr^{-2+[\theta-2(2-\alpha-\beta)]/\alpha}\|f\|_{B([0,\tau];{\widetilde X}^\theta_A)}, \end{align*} we have \begin{equation}\label{e3.11} \begin{aligned} \|F_1(s,t)\|_X &=\big\|\int_0^s d\sigma \int_{s-\sigma}^{t-\sigma} A^2e^{-rA}f(\sigma)\,dr\big\|_X \\ &\leq C\int_0^s d\sigma \int_{s-\sigma}^{t-\sigma} r^{-2+[\theta-2(2-\alpha-\beta)]/\alpha}\,dr \|f\|_{B([0,\tau];{\widetilde X}^\theta_A)} \\ &\le C\|f\|_{B([0,\tau];{\widetilde X}^\theta_A)}(t-s)^{[\theta-2(2-\alpha-\beta)]/\alpha} \end{aligned} \end{equation} (recall that $\alpha+\beta>3/2)$. In other words, we have proved that \[ y_2\in C^{[\theta-2(2-\alpha-\beta)]/\alpha}([0,\tau];X). \] Concerning space regularity, first we consider the identity \begin{equation} \label{e3.12} Ae^{-\xi A}Ay_2(t) = \int_0^t A^2e^{-(t-s+\xi)A}f(s)\,ds. \end{equation} Recalling that $f\in B([0,\tau];{\widetilde X}^\theta_A)$ we have \begin{align*} \|A^2e^{-(t-s+\xi)A}f(s)\|_X &=\|Ae^{-[(t-s+\xi)/2]A}Ae^{-[(t-s+\xi)/2]A}f(s)\|_X\\ &\leq C(t-s+\xi)^{(\beta-2)/\alpha}(t-s+\xi)^{(\beta+\theta-2)/\alpha} \|f(s)\|_{{\widetilde X}^\theta_A}\\ &\leq C\|f\|_{B([0,\tau];{\widetilde X}^\theta_A)} (t-s+\xi)^{[\theta-2(2-\beta)]/\alpha}. \end{align*} Hence noting that \[ [\theta-2(2-\beta)]/\alpha=(\theta+2\beta-4)/\alpha< (2\beta-3)/\alpha\le -1, \] we have \begin{align*} \|Ae^{-\xi A}Ay_2(t)\|_X &\leq C\|f\|_{B([0,\tau];{\widetilde X}^\theta_A)} \int_{-\infty}^t (t-s+\xi)^{[\theta-2(2-\beta)]/\alpha}\,ds\\ &= C\|f\|_{B([0,\tau];{\widetilde X}^\theta_A)} \frac{\alpha\xi^{(\alpha+2\beta+\theta-4)/\alpha}}{4-\alpha-2\beta-\theta}. \end{align*} Therefore, \begin{equation} \label{e3.13} \sup_{0\le t\le\tau} \sup_{\xi>0} \xi^{(4-\alpha-2\beta-\theta)/\alpha} \|Ae^{-\xi A}Ay_2(t)\|_X<+\infty. \end{equation} Since $(4-\alpha-2\beta-\theta)/\alpha=[2-\beta-(\alpha+\beta+\theta-2)]/\alpha$, \eqref{e3.9}, \eqref{e3.10}, \eqref{e3.11}, \eqref{e3.13} imply \begin{gather*} \label{e3.14} Ay_2\in C^{[\theta-2(2-\alpha-\beta)]/\alpha}([0,\tau];X) \cap B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_A),\\ \label{e3.15} \|Ay_2\|_{B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_A)} \le C\|f\|_{B([0,\tau];{\widetilde X}^\theta_A)}. \end{gather*} It follows from \eqref{e3.2} and \eqref{e3.14} that $y_2'=f-Ay_2\in C([0,\tau];X)$. Summing up, we have proved the following theorem. \begin{theorem} \label{thm3.1} Let the pairs $(f,y_0)$ and $(\alpha,\beta)$ satisfy \eqref{e3.2} and \eqref{e3.3}, respectively. Then Problem \eqref{e3.1} admits a unique strict solution $y$ with the following regularity properties: \begin{gather} \label{e3.16} y'\in C([0,\tau];X)\cap B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_A),\\ \label{e3.17} Ay\in C^{[\theta-2(2-\alpha-\beta)]/\alpha}([0,\tau];X)\cap B([0,\tau]; {\widetilde X}^{\theta-(2-\alpha-\beta)}_A). \end{gather} \end{theorem} Taking into account the inclusions proved in Section 2, we can also establish the following result concerning spaces $X_A^\theta$. \begin{theorem} \label{thm3.2} Let $2\alpha+\beta>2$, $3-2\alpha-\beta<\theta<1$, $y_0\in D(A)$, $Ay_0\in(X,D(A))_{\theta,\infty}$, $f\in C([0,\tau];X)\cap B([0,\tau];(X,D(A))_{\theta,\infty})$. Then Problem \eqref{e3.1} admits a unique strict solution $y$ such that \begin{gather*} y'\in C([0,\tau];X)\cap B([0,\tau];X_A^{[\theta-(3-2\alpha-\beta)]/\alpha}),\\ Ay\in C^{[\theta-(3-2\alpha-\beta)]/\alpha}([0,\tau];X)\cap B([0,\tau]; X_A^{[\theta-(3-2\alpha-\beta)]/\alpha}). \end{gather*} \end{theorem} \begin{proof} We use the notation in the proof of Theorem \ref{thm3.1}. One has $Ay_0\in{\widetilde X}_A^{\theta}$ by virtue of our assumption and \eqref{e2.14}. Hence, owing to the proof of Theorem \ref{thm3.1} $y_1(t)=e^{-tA}y_0$ satisfies \begin{equation} \label{e3.18} \begin{aligned} y_1', Ay_1&\in C^{[\theta-(2-\alpha-\beta)]/\alpha}([0,\tau];X)\cap B([0,\tau]; {\widetilde X}_A^{\theta})\\ &\subset C^{[\theta-(2-\alpha-\beta)]/\alpha}([0,\tau];X)\cap B([0,\tau]; X_A^{[\theta-(2-\alpha-\beta)]/\alpha}). \end{aligned} \end{equation} From \eqref{e2.15} and \eqref{e3.10} one deduces the estimate \begin{equation} \label{e3.19} \begin{aligned} \|F_2(s,t)\|_X &\leq C\|f\|_{B([0,\tau]; {\tilde X}_A^{\theta})}(t-s)^{[\theta-(2-\alpha-\beta)]/\alpha} \\ &\leq C\|f\|_{B([0,\tau];(X,D(A))_{\theta,\infty})}(t-s)^{[\theta-(2-\alpha-\beta)]/\alpha}. \end{aligned} \end{equation} Likewise, from the inequalities \begin{equation}\label{e4.20} \begin{aligned} \|A^2e^{-rA}f(\sigma)\|_X &\leq Cr^{(\beta-3+\theta)/\alpha}\|f(\sigma)\|_{(X,D(A))_{\theta,\infty}}\\ & \leq Cr^{(\beta-3+\theta)/\alpha}\|f\|_{B([0,\tau];(X,D(A))_{\theta,\infty})}, \end{aligned} \end{equation} one obtains \begin{equation} \begin{aligned}\label{e3.21} \|F_1(s,t)\|_X &= \big\|\int_0^s d\sigma \int_{s-\sigma}^{t-\sigma}A^2e^{-rA}f(\sigma)dr\big\|_X\\ &\leq C\int_0^s d\sigma \int_{s-\sigma}^{t-\sigma}r^{(\beta-3+\theta)/\alpha}\|f\|_{B([0,\tau];(X,D(A))_{\theta,\infty})}dr\\ &\leq C\|f\|_{B([0,\tau];(X,D(A))_{\theta,\infty})}(t-s)^{(2\alpha+\beta-3+\theta)/\alpha}. \end{aligned} \end{equation} It follows from \eqref{e3.19} and \eqref{e3.21} that \begin{equation} \label{e3.22} Ay_2\in C^{[\theta-(3-2\alpha-\beta)]/\alpha}([0,\tau];X). \end{equation} Using \begin{align*} \|A^2e^{-(t-s+\xi)A}f(s)\|_X &\leq C(t-s+\xi)^{(\beta-3+\theta)/\alpha}\|f(s)\|_{(X,D(A))_{\theta,\infty}}\\ &\leq C(t-s+\xi)^{(\beta-3+\theta)/\alpha}\|f\|_{B([0,\tau];(X,D(A))_{\theta,\infty})} \end{align*} one obtains \begin{equation} \label{e3.23} \begin{aligned} \|Ae^{-\xi A}Ay_2(t)\|_X &=\big\|\int_0^tA^2e^{-(t-s+\xi)A}f(s)ds\big\|_X\\ &\leq C\|f\|_{B([0,\tau];(X,D(A))_{\theta,\infty})} \int_0^t(t-s+\xi)^{(\beta-3+\theta)/\alpha}\,ds \\ &\leq C\|f\|_{B([0,\tau];(X,D(A))_{\theta,\infty})}\xi^{(\alpha+\beta-3+\theta)/\alpha}. \end{aligned} \end{equation} Hence one obtains \begin{equation} Ay_2\in B([0,\tau];{\widetilde X}_A^{\alpha+\theta-1}) \subset B([0,\tau];X_A^{[\theta-(3-2\alpha-\beta)]/\alpha}). \label{e3.24} \end{equation} From \eqref{e3.18}, \eqref{e3.22} and \eqref{e3.24}, it follows that \begin{equation} Ay\in C^{[\theta-(3-2\alpha-\beta)]/\alpha}([0,\tau];X) \cap B([0,\tau];X_A^{[\theta-(3-2\alpha-\beta)]/\alpha}). \label{e3.25} \end{equation} The hypothesis on $f$ and \eqref{e3.24} imply \begin{equation} y_2'=f-Ay_2\in B([0,\tau];X_A^{[\theta-(3-2\alpha-\beta)]/\alpha}). \label{e3.26} \end{equation} By \eqref{e3.18} and \eqref{e3.26}, one concludes that \[ y'\in B([0,\tau];X_A^{[\theta-(3-2\alpha-\beta)]/\alpha}). \] \end{proof} In view of embedding \eqref{e1.6}, Theorem \ref{thm3.2} leads to the following corollary, where $Y_A^\gamma$ stands for anyone of the spaces $X_A^\gamma$ or $(X,D(A))_{\gamma,\infty}$. \begin{corollary} \label{coro3.3} Let $2\alpha+\beta>2$, $3-2\alpha-\beta<\theta<1$, $y_0\in D(A)$, $Ay_0\in Y_A^\theta$, $f\in C([0,\tau];X)\cap B([0,\tau];Y_A^\theta)$. Then Problem \eqref{e3.1} admits a unique strict solution $y$ such that \begin{gather*} y'\in C([0,\tau];X)\cap B([0,\tau];Y_A^{[\theta-(3-2\alpha-\beta)]/\alpha}), \\ Ay\in C^{[\theta-(3-2\alpha-\beta)]/\alpha}([0,\tau];X)\cap B([0,\tau]; Y_A^{[\theta-(3-2\alpha-\beta)]/\alpha}). \end{gather*} \end{corollary} \section{A first identification problem} Consider the identification problem \eqref{IP} in Section 1. We want to determine a pair $(y,f)\in [C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A))]\times C([0,\tau];\mathbb{C})$ satisfying \eqref{IP} under the following assumptions: \begin{gather} \label{e4.1} \begin{gathered} y_0\in \mathcal{D}(A),\quad Ay_0,z\in {\widetilde X}^\theta_A,\\ g\in C^1([0,\tau];\mathbb{C}),\quad h\in C([0,\tau];X)\cap B([0,\tau];{\widetilde X}^\theta_A); \end{gathered} \\ \label{e4.2} 0<\beta\le \alpha\le 1,\quad \alpha+\beta>3/2,\quad 1>\theta>2(2-\alpha-\beta);\\ \label{e4.3} \Phi\in X^*,\quad \Phi[y_0]=g(0),\quad \Phi[z]\ne 0. \end{gather} If $(y,f)$ is the solution sought for, we immediately deduce that $(y,f)$ solves the equation \begin{equation} \label{e4.4} g'(t)+\Phi[Ay(t)]=f(t)\Phi[z]+\Phi[h(t)],\quad t\in [0,\tau]. \end{equation} Therefore, taking advantage of Theorem \ref{thm3.1}, we obtain the integral equation for $f$ \begin{equation} \label{e4.5} \begin{aligned} f(t)&=\frac{g'(t)-\Phi[h(t)]+\Phi[Ay(t)]}{\Phi[z]}\\ &=\frac{g'(t)-\Phi[h(t)]+\Phi[Ae^{-tA}y_0]}{\Phi[z]} +\frac{1}{\Phi[z]}\int_0^t \Phi[Ae^{-(t-s)A}z]f(s)\,ds \\ &\quad +\frac{1}{\Phi[z]}\int_0^t \Phi[Ae^{-(t-s)A}h(s)]\,ds =: b(t)+Sf(t),\quad t\in [0,\tau]. \end{aligned} \end{equation} Note that $t\to Ae^{-tA}y_0$ is continuous in $[0,\tau]$ by Lemma \ref{lem2.1}. Since $z\in {\widetilde X}^\theta_A$, we obtain \begin{equation} \label{e4.6} \|Ae^{-(t-s)A}z\|_X\le C(t-s)^{-(2-\beta-\theta)/\alpha}\le C(t-s)^{-(1-\theta_0)}. \end{equation} where $\theta_0=[\theta-(2-\alpha-\beta)]/\alpha$. Whence we deduce the inequality \[ |Sf(t)|\le \frac{C}{|\Phi[z]|}\|\Phi\|_{X^*}\|z\|_{{\widetilde X}^\theta_A} \int_0^t (t-s)^{-1+\theta_0}|f(s)|\,ds,\quad t\in (0,\tau]. \] Repeating the arguments and techniques in \cite{AF2} we can deduce the following estimates involving the iterates $S^n$ of operator $S$: \begin{equation} \label{e4.7} |S^nf(t)|\le \big[C(\Phi[z])^{-1}\|\Phi\|_{X^*}\|z\|_{{\widetilde X}^\theta_A}\big]^n \frac{\Gamma(\theta_0)^nt^{n\theta_0}}{\Gamma(n\theta_0)n\theta_0}\|f\|_{C([0,\tau];\mathbb{C})}, \quad t\in (0,\tau]. \end{equation} Since $[\Gamma(n\theta_0)]^{1/n}\to +\infty$ as $n\to +\infty$, we conclude that the operator $S$ has spectral radius equal to $0$. Therefore equation \eqref{e4.5} admits a unique solution $f\in C([0,\tau];\mathbb{C})$. In view of Theorem \ref{thm3.1} we conclude that the solution $y$ corresponding to such an $f$ has the regularity \begin{gather} \label{e4.8} \begin{gathered} y\in C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A)),\\ y'\in C([0,\tau];X)\cap B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_A), \end{gathered}\\ \label{e4.9} Ay\in C^{[\theta-2(2-\alpha-\beta)]/\alpha}([0,\tau];X) \cap B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_A). \end{gather} Summing up, we have proved the following theorem. \begin{theorem} \label{thm4.1} Under assumptions \eqref{e4.1} and \eqref{e4.2}, the identification problem \eqref{IP} in Section 1 admits a unique strict solution $(y,f)$ satisfying \eqref{e4.8}, \eqref{e4.9}. \end{theorem} We change now a bit our assumptions on the data: \eqref{e4.1} is replaced with the following, where we change the condition on the pair $(y_0,z)$: \begin{equation} \label{e4.10} \begin{gathered} y_0\in \mathcal{D}(A),\quad Ay_0,z\in X^\theta_A,\quad g\in C^1([0,\tau];\mathbb{C}),\\ h\in C([0,\tau];X)\cap B([0,\tau];X^\theta_A), \end{gathered} \end{equation} If $(y,f)$ is the solution sought for, we deduce, as above, that $f$ solves the integral equation \eqref{e4.5}. Reasoning as above and taking advantage of Corollary \ref{coro3.3}, we obtain the following result. \begin{theorem} \label{thm4.2} Let $Y_A^\gamma$ be anyone of the spaces $(X,D(A))_{\gamma,\infty}$ or $X_A^\gamma$. Let $2\alpha+\beta>2$, $\theta>3-2\alpha-\beta$ and let \begin{equation} \begin{gathered} y_0\in D(A),\quad Ay_0,z \in Y_A^{\theta,\infty},\quad g\in C^1([0,\tau];\mathbb{C}),\\ h\in C([0,\tau];Y_A^{\theta,\infty}),\quad \Phi[z]\ne 0. \end{gathered} \end{equation} Then the identification problem \begin{equation} \label{Ip} \begin{gathered} y'(t)+Ay(t)=f(t)z+h(t),\quad t\in[0,\tau], \\ y(0)=y_0, \\ \Phi[y(t)]=g(t), \quad t\in[0,\tau]. \end{gathered} \end{equation} admits a unique strict solution $(y,f)\in [C^1([0,\tau];X)\cap C([0,\tau];D(A))]\times C([0,\tau];{\mathbb C})$ such that \begin{gather*} y'\in C([0,\tau];X)\cap B([0,\tau];Y_A^{(2\alpha+\beta-3+\theta)/\alpha,\infty}), \\ Ay\in C^{(2\alpha+\beta-3+\theta)/\alpha}([0,\tau];X)\cap B([0,\tau];Y_A^{(2\alpha+\beta-3+\theta)/\alpha,\infty}). \end{gather*} \end{theorem} \begin{proof} When $Y_A^\gamma=(X,D(A))_{\gamma,\infty}$, it suffices to observe that from \eqref{e2.14} we deduce estimate \eqref{e4.6} and that the same argument in the proof of Theorem \ref{thm4.1} runs well, since $3-2\alpha-\beta=(2-\alpha-\beta)+1-\alpha\ge 2-\alpha-\beta$. When $Y_A^\gamma=X_A^\gamma$, the assertion follows from \cite[Corollary 3.3 and Proposition 3.4]{FY}. \end{proof} \section{A latter identification problem} In this section we consider the problem consisting in recovering two unknown scalar functions $f_1,f_2\in C([0,\tau];\mathbb{C})$ and a vector function $y\in C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A))$ such that \begin{equation} \label{Q} % e5.1 \begin{gathered} y'(t)+Ay(t)=f_1(t)z_1+f_2(t)z_2 + h(t),\quad t\in [0,\tau],\\ y(0)=y_0,\\ \Phi_j[y(t)]=g_j(t),\quad t\in [0,\tau],\ j=1,2, \end{gathered} \end{equation} where $\Phi_j\in X^*$, $g_j\in C^1([0,\tau];\mathbb{C})$, $z_j\in X$, $j=1,2$, $h\in C([0,\tau];X)\cap B([0,\tau];X^\theta_A)$, and $y_0\in \mathcal{D}(A)$ are given. Let \[ \mathcal{A}=\begin{bmatrix} \Phi_1[z_1] & \Phi_1[z_2]\\ \Phi_2[z_1] & \Phi_2[z_2] \end{bmatrix},\quad \det\mathcal{A}\ne 0. \] Then we obtain the following fixed-point integral system for $(f_1,f_2)$, \begin{equation} \label{e5.2} \begin{split} \begin{bmatrix} f_1(t) \\ f_2(t) \end{bmatrix} &= \mathcal{A}^{-1} \begin{bmatrix} g'_1(t)\\ g'_2(t) \end{bmatrix} + \mathcal{A}^{-1} \begin{bmatrix} \Phi_1[e^{-tA}Ay_0]\\ \Phi_2[e^{-tA}Ay_0] \end{bmatrix}\\ &\quad - \mathcal{A}^{-1} \begin{bmatrix} \Phi_1[h(t)]\\ \Phi_2[h(t)] \end{bmatrix} + \mathcal{A}^{-1}\int_0^t \begin{bmatrix} \Phi_1[Ae^{-(t-s)A}h(s)]\\ \Phi_2[Ae^{-(t-s)A}h(s)] \end{bmatrix} ds \\ &\quad + \mathcal{A}^{-1}\int_0^t \begin{bmatrix} \Phi_1[Ae^{-(t-s)A}z_1]&\Phi_1[Ae^{-(t-s)A}z_2]\\ \Phi_2[Ae^{-(t-s)A}z_1]&\Phi_2[Ae^{-(t-s)A}z_2] \end{bmatrix} \begin{bmatrix} f_1(s)\\f_2(s) \end{bmatrix} ds \\ &\quad =:\begin{bmatrix} b_1(t) \\ b_2(t) \end{bmatrix} + S\begin{bmatrix} f_1\\ f_2 \end{bmatrix}(t), \quad t\in [0,\tau]. \end{split} \end{equation} We introduce in $C([0,\tau];\mathbb{C}^2)$ the sup-norm \[ \|(f_1,f_2)\|_{C([0,\tau];\mathbb{C}^2)} =\max_{t\in [0,\tau]} |f_1(t)| + \max_{t\in [0,\tau]} |f_2(t)|. \] For any pair $(z_1,z_2)\in ({\widetilde X}_A^\theta)^2$ and $(f_1,f_2)\in C([0,\tau];\mathbb{C}^2)$ we obtain the bounds \begin{align*} &\big\| S\begin{bmatrix} f_1\\ f_2 \end{bmatrix}(t) \big\| \le \|\mathcal{A}^{-1}\|_{\mathcal{L}(\mathbb{C}^2)} \int_0^t \sum_{j,k=1}^2 |\Phi_j[ A e^{-(t-s)A}z_k]| |f_k(s)|\,ds \\ & \le \|\mathcal{A}^{-1}\|_{\mathcal{L}(\mathbb{C}^2)}\sum_{j=1}^2 \|\Phi_j\|_{X^*}\int_0^t \sum_{k=1}^2 \| A e^{-(t-s)A}z_k\|_X |f_k(s)|\,ds \\ & \le C\|\mathcal{A}^{-1}\|_{\mathcal{L}(\mathbb{C}^2)}\sum_{j=1}^2 \|\Phi_j\|_{X^*} \int_0^t \sum_{k=1}^2 \|z_k\|_{{\widetilde X}_A^\theta} (t-s)^{(\beta+\theta-2)/\alpha}|f_k(s)|\,ds \\ & \le C\|\mathcal{A}^{-1}\|_{\mathcal{L}(\mathbb{C}^2)} \sum_{j=1}^2 \|\Phi_j\|_{X^*}\max_{1\le k\le 2}\|z_k\|_{{\widetilde X}_A^\theta} \int_0^t (t-s)^{(\beta+\theta-2)/\alpha}\sum_{k=1}^2 |f_k(s)|\,ds. \end{align*} Proceeding by induction, we can prove the bounds for the iterates $S^n$ of operator $S$ (cf. Section 4): \[ \| S^n\begin{bmatrix} f_1\\ f_2 \end{bmatrix} (t) \|_2 \le C_1^n\frac{\Gamma(\theta_0)^nt^{n\theta_0}}{\Gamma(n\theta_0)n\theta_0} \| \begin{bmatrix} f_1\\ f_2 \end{bmatrix} \|_{C([0,\tau];\mathbb{C}^2)}, \] where we have set \[ C_1=C\|\mathcal{A}^{-1}\|_{\mathcal{L}(\mathbb{C}^2)} \sum_{j=1}^2 \|\Phi_j\|_{X^*}\max_{1\le k\le 2} \|z_k\|_{{\widetilde X}_A^\theta}. \] Since $[\Gamma(n\theta_0)]^{1/n}\to +\infty$ as $n\to +\infty$, we can conclude that operator $S$ has spectral radius equal to $0$, so that problem \eqref{Q} admits a unique solution $(f_1,f_2)\in C([0,\tau];\mathbb{C}^2)$. Using Theorem \ref{thm3.1} we easily deduce the following result. \begin{theorem} \label{thm5.1} Let $\alpha+\beta>3/2$ and $\theta\in (2(2-\alpha-\beta),1)$. Let $y_0\in \mathcal{D}(A)$, $Ay_0\in {\widetilde X}^\theta_A$, $z_j\in {\widetilde X}^\theta_A$, $\Phi_j\in X^*$, $g_j\in C^1([0,\tau];\mathbb{C})$, $j=1,2$ and $h\in C([0,\tau];X)\cap B([0,\tau];\widetilde X^\theta_A)$ such that \begin{equation} \label{e5.3} \Phi_1[z_1]\Phi_2[z_2] - \Phi_2[z_1]\Phi_1[z_2] \ne 0,\quad \Phi_j[y_0]=g_j(0),\; j=1,2. \end{equation} Then problem \eqref{Q} admits a unique strict solution $(y,f_1,f_2)\in [C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A))] \times C([0,\tau];\mathbb{C})\times C([0,\tau];\mathbb{C})$ such that \[ y'\in B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_A),\ Ay\in C^{[\theta-2(2-\alpha-\beta)]/\alpha}([0,\tau];X)\cap B([0,\tau]; {\widetilde X}^{\theta-(2-\alpha-\beta)}_A). \] \end{theorem} We conclude this section by two easy extensions of Problem \eqref{Q} to the case of $n$ unknown functions $f$. \begin{corollary} \label{coro5.2} Let $\alpha+\beta>3/2$ and $\theta\in (2(2-\alpha-\beta),1)$. Let $y_0\in \mathcal{D}(A)$, $Ay_0\in {\widetilde X}^\theta_A$, $z_j\in {\widetilde X}^\theta_A$, $g_j\in C^1([0,\tau];{\mathbb R})$, $h\in C([0,\tau];X)\cap B([0,\tau];{\widetilde X}^\theta_A)$, $\Phi_j\in X^*$, $\Phi_j[y_0]=g_j(0)$, $j=1,\dots,n$ be such that \[ \det \begin{bmatrix} \Phi_1[z_1]& \dots & \Phi_1[z_n]\\ \dots& \dots & \dots\\ \Phi_n[z_1]& \dots & \Phi_n[z_n] \end{bmatrix} \ne 0. \] Then the identification problem \begin{equation}\label{e5.4} \begin{gathered} y'(t)+Ay(t)= \sum_{j=1}^n f_j(t)z_j+ h(t),\quad t\in [0,\tau], \\ y(0)=y_0, \\ \Phi_j[y(t)]=g_j(t),\quad t\in [0,\tau],\ j=1,\dots,n, \end{gathered} \end{equation} admits a unique strict solution $(y,f_1,\dots,f_n)\in [C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A))] \times C([0,\tau];{\mathbb R})^n$ such that \begin{gather*} y'\in B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_A), \\ Ay\in C^{[\theta-2(2-\alpha-\beta)]/\alpha}([0,\tau];X)\cap B([0,\tau]; {\widetilde X}^{\theta-(2-\alpha-\beta)}_A). \end{gather*} \end{corollary} \begin{corollary} \label{coro5.3} Let $Y_A^\gamma$ be anyone of the spaces $(X,D(A))_{\gamma,\infty}$ or $X_A^\gamma$. Let $2\alpha+\beta>2$ and $\theta\in (3-2\alpha-\beta,1)$. Let $y_0\in \mathcal{D}(A)$, $Ay_0\in Y_A^{\theta,\infty}$, $z_j\in Y_A^{\theta,\infty}$, $g_j\in C^1([0,\tau];{\mathbb R})$, $h\in C([0,\tau];X)\cap B([0,\tau];Y_A^{\theta,\infty})$, $\Phi_j\in X^*$, $\Phi_j[y_0]=g_j(0)$, $j=1,\dots,n$ be such that \[ \det \begin{bmatrix} \Phi_1[z_1]& \dots & \Phi_1[z_n] \\ \dots& \dots & \dots \\ \Phi_n[z_1]& \dots & \Phi_n[z_n] \end{bmatrix} \ne 0. \] Then the identification problem \eqref{e5.4} admits a unique strict solution $(y,f_1,\dots,f_n)$ in $[C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A))] \times C([0,\tau];{\mathbb R})^n$ such that \begin{gather*} y'\in B([0,\tau];Y_A^{[\theta-(3-2\alpha-\beta)]/\alpha,\infty}), \\ Ay\in C^{[\theta-(3-2\alpha-\beta)]/\alpha}([0,\tau];X)\cap B([0,\tau];Y_A^{[\theta-(3-2\alpha-\beta)]/\alpha,\infty}). \end{gather*} \end{corollary} \section{Inverse problems for systems of differential boundary value problems} Let $A$, $B$, $C$, $D$ be linear closed operators acting in the Banach space $X$ satisfying the following relations: \begin{gather} \label{e6.1} \mathcal{D}(A)\subset \mathcal{D}(C),\quad \mathcal{D}(D)\subset \mathcal{D}(B),\\ \label{e6.2} \|(\lambda+A)^{-1}\|_{\mathcal{L}(X)}\le c|\lambda|^{-\beta_1},\quad \|(\lambda+D)^{-1}\|_{\mathcal{L}(X)}\le c|\lambda|^{-\beta_2},\quad \lambda\in \Sigma_\alpha,\\ \label{e6.3} \|C(\lambda+A)^{-1}\|_{\mathcal{L}(X)}\le c|\lambda|^{-\gamma_1},\quad \|B(\lambda+D)^{-1}\|_{\mathcal{L}(X)}\le c|\lambda|^{-\gamma_2},\quad \lambda\in \Sigma_\alpha, \end{gather} with \begin{equation} \label{e6.4} \gamma_1+\gamma_2\in \mathbb{R}_+. \end{equation} In the Banach space $X\times X$ we consider the problem consisting in determining a quadruplet $(y_1,y_2,f_1,f_2)$, with \begin{gather} \label{e6.5} (y_1,y_2)\in [C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(A))] \times [C^1([0,\tau];X)\cap C([0,\tau];\mathcal{D}(D))], \\ \label{e6.6} (f_1,f_2)\in C([0,\tau];\mathbb{C})\times C([0,\tau];\mathbb{C}), \end{gather} solving the identification problem \begin{gather} \label{e6.7} \begin{aligned} &\frac{d}{dt} \begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix} + \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} y_1(t)\\ y_2(t) \end{bmatrix}\\ & = f_1(t) \begin{bmatrix} z_{1,1}\\ z_{2,1} \end{bmatrix} + f_2(t) \begin{bmatrix} z_{1,2}\\ z_{2,2} \end{bmatrix} + \begin{bmatrix} h_{1}(t) \\ h_2(t) \end{bmatrix},\quad t\in [0,\tau], \end{aligned} \\ \label{e6.8} y_j(0)=y_{0,j},\quad j=1,2,\\ \label{e6.9} \Psi_j[y_j(t)]=g_j(t),\quad t\in [0,\tau],\; j=1,2, \end{gather} with \begin{equation} \label{e6.10} \Psi_j\in X^*,\quad \Psi_j[y_{0,j}]=g_j(0),\quad j=1,2. \end{equation} Let us now introduce the linear operator $\mathcal{A}$ defined by \[ \mathcal{D}(\mathcal{A})=\mathcal{D}(A)\times \mathcal{D}(D),\quad \mathcal{A} \begin{bmatrix} y_1\\ y_2 \end{bmatrix} = \begin{bmatrix} Ay_1 + By_2 \\ Cy_1 + Dy_2 \end{bmatrix}. \] It is shown in \cite{FLT2} that, for large positive $R$, \begin{equation} \label{e6.11} \|(\lambda+\mathcal{A})^{-1}\|_{\mathcal{L}(X)}\le c|\lambda|^{-\beta},\quad \lambda\in \Sigma_\alpha\cap B(0,R)^c, \end{equation} where \begin{equation} \label{e6.12} \beta=\min\{\beta_1,\beta_2,\beta_1+\gamma_2,\beta_2+\gamma_1\}. \end{equation} Using the change of variables $(y_1(t),y_2(t))\mapsto(e^{-kt}y_1(t),e^{-kt}y_2(t))$ with a sufficiently large positive constant $k$, we can assume that bound \eqref{e6.11} holds for all $\lambda\in \Sigma_\alpha$. Set now \[ E=X\times X,\quad \xi = \begin{bmatrix} y_1\\ y_2 \end{bmatrix},\quad z_1 = \begin{bmatrix} z_{1,1}\\ z_{2,1} \end{bmatrix},\quad z_2 = \begin{bmatrix} z_{1,2}\\ z_{2,2} \end{bmatrix}. \] Then the direct problem \eqref{e6.7}, \eqref{e6.8} takes the simpler form \begin{gather} \label{e6.13} \xi'(t)+\mathcal{A}\xi(t)=f_1(t)z_1 + f_2(t)z_2,\quad t\in [0,\tau],\\ \label{e6.14} \xi(0)= \begin{bmatrix} y_{0,1}\\ y_{0,2} \end{bmatrix} =: \xi_0. \end{gather} Define the norm in $X\times X$ by $\|(y_1,y_2)\|_{X\times X}=(\|y_1\|_X^2+\|y_2\|_X^2)^{1/2}$ and introduce the functionals $\Phi_1,\Phi_2\in E^*\sim X^*\times X^*$ (cf. \cite[p. 164]{KA}) defined by \[ \Phi_j[\xi]=\Phi_j \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} =\Psi_j[y_j],\quad j=1,2. \] Applying $\Phi_j$, $j=1,2$, to both sides in \eqref{e6.7}, we easily obtain the following system, for all $t\in [0,\tau]$, \begin{gather*} g'_1(t)+\Psi_1[Ay_1(t)+By_2(t)]=f_1(t)\Psi_1[z_{1,1}] + f_2(t)\Psi_1[z_{1,2}] + \Psi_1[h(t)], \\ g'_2(t)+\Psi_2[Cy_1(t)+Dy_2(t)]=f_1(t)\Psi_2[z_{2,1}] + f_2(t)\Psi_2[z_{2,2}] + \Psi_2[h(t)], \end{gather*} Now assume that \begin{equation} \label{e6.15} \Phi_1[z_1]\Phi_2[z_2]-\Phi_1[z_2]\Phi_2[z_1] =\Psi_1[z_{1,1}]\Psi_2[z_{2,2}]-\Psi_1[z_{1,2}]\Psi_2[z_{2,1}]\ne 0. \end{equation} Then it is easy to realize that Theorem \ref{thm5.1} and Corollary \ref{coro5.2} yield the following results. \begin{theorem} \label{thm6.1} Let operators $A$, $B$, $C$, $D$ satisfy conditions \eqref{e6.1}--\eqref{e6.4} and let $\beta$ be defined by \eqref{e6.12}. Let $\alpha+\beta>3/2$, $\theta\in (2(2-\alpha-\beta),1)$, $g_1,g_2\in C^1([0,\tau];\mathbb{C})$ and $h\in C([0,\tau];X^2)\cap B([0,\tau];{\widetilde X}^\theta_\mathcal{A})$. Moreover, let $[y_{0,1},y_{0,2}]\in \mathcal{D}(\mathcal{A})$, $\mathcal{A}[y_{0,1},y_{0,2}]^T\in {\widetilde X}^\theta_\mathcal{A}$, $[z_{1,1},z_{2,1}]^T,[z_{1,2},z_{2,2}]^T\in {\widetilde X}^\theta_\mathcal{A}$ and let \eqref{e6.10} and \eqref{e6.15} be satisfied. Then problem \eqref{e6.7}-\eqref{e6.9} admits a unique strict solution $(y_1,y_2,f_1,f_2)$ in the space defined by \eqref{e6.5}, \eqref{e6.6} such that \begin{gather*} [y'_1,y'_2]^T\in B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_\mathcal{A}), \\ \begin{aligned} &[Ay_1+By_2,Cy_1+Dy_2]^T\\ &\in C^{[\theta-2(2-\alpha-\beta)]/\alpha}([0,\tau];X\times X) \cap B([0,\tau];{\widetilde X}^{\theta-(2-\alpha-\beta)}_\mathcal{A}). \end{aligned} \end{gather*} \end{theorem} We can conclude this section by stating the following corollaries that take into account Corollary \ref{coro5.3}. \begin{corollary} \label{coro6.2} Let operators $A$, $B$, $C$, $D$ satisfy conditions \eqref{e6.1}--\eqref{e6.4} and let $\beta$ be defined by \eqref{e6.12}. Let $2\alpha+\beta>2$, $\theta\in (3-2\alpha-\beta,1)$, $g_1,g_2\in C^1([0,\tau];{\mathbb R})$ and $h\in C([0,\tau];X^2)\cap B([0,\tau];X_\mathcal{A}^\theta)$. Moreover, let $[y_{0,1},y_{0,2}]\in \mathcal{D}(\mathcal{A})$, $\mathcal{A}[y_{0,1},y_{0,2}]^T\in X^\theta_\mathcal{A}$, $[z_{1,1},z_{2,1}]^T,[z_{1,2},z_{2,2}]^T\in X^\theta_\mathcal{A}$, and let \eqref{e6.10} and \eqref{e6.15} be satisfied. Then problem \eqref{e6.7}--\eqref{e6.9} admits a unique strict solution $(y_1,y_2,f_1,f_2)$ in the space defined by \eqref{e6.5}, \eqref{e6.6} such that \begin{gather*} [y'_1,y'_2]^T\in B([0,\tau];X^{[\theta-(3-2\alpha-\beta)]/\alpha}_\mathcal{A}),\\ \begin{aligned} &[Ay_1+By_2,Cy_1+Dy_2]^T\\ &\in C^{[\theta-(3-2\alpha-\beta)]/\alpha}([0,\tau];X\times X) \cap B([0,\tau];X^{[\theta-(3-2\alpha-\beta)]/\alpha}_\mathcal{A}). \end{aligned} \end{gather*} \end{corollary} \begin{corollary} \label{coro6.3} Let operators $A$, $B$, $C$, $D$ satisfy conditions \eqref{e6.1}--\eqref{e6.4} and let $\beta$ be defined by \eqref{e6.12}. Let $2\alpha+\beta>2$, $\theta\in (3-2\alpha-\beta,1)$ and $g_1,g_2\in C^1([0,\tau];{\mathbb R})$. Moreover, let $[y_{0,1},y_{0,2}]\in \mathcal{D}(\mathcal{A})$, $\mathcal{A}[y_{0,1},y_{0,2}]^T\in (X\times X,\mathcal{D}(\mathcal{A}))_{\theta,\infty}$, $[z_{1,1},z_{2,1}]^T$, $[z_{1,2},z_{2,2}]^T\in (X\times X,\mathcal{D}(\mathcal{A}))_{\theta,\infty}$, and let \eqref{e6.10} and \eqref{e6.15} be satisfied. Then problem \eqref{e6.7}-\eqref{e6.9} admits a unique strict solution $(y_1,y_2,f_1,f_2)$ in the space defined by \eqref{e6.5}, \eqref{e6.6} such that \begin{gather*} [y'_1,y'_2]^T\in B([0,\tau];(X\times X,\mathcal{D} (\mathcal{A}))_{[\theta-(3-2\alpha-\beta)]/\alpha,\infty}), \\ \begin{aligned} &[Ay_1+By_2,Cy_1+Dy_2]^T\\ &\in C^{[\theta-(3-2\alpha-\beta)]/\alpha}([0,\tau];X\times X) \cap B([0,\tau];(X\times X,\mathcal{D}(\mathcal{A}))_{[\theta-(3-2\alpha-\beta)]/ \alpha,\infty}). \end{aligned} \end{gather*} \end{corollary} \begin{remark} \label{rmk6.1} \rm The conclusions of Theorem \ref{thm6.1} and Corollaries 6.2 and 6.3 may be true even in cases when the domain of the operator matrix $\mathcal{A}$ is not $\mathcal{D}(A)\times \mathcal{D}(D)$ (cf. Problem 8.1 in Section 8). \end{remark} \begin{remark} \label{rmk6.2} \rm In the optimal situation, when $\alpha=\beta=1$ and, e.g., the operators $A$ and $D$ generate two analytic semigroups on $X$ and $B$ and $C$ are bounded operators, the previous conditions reduce to the following for some $\theta\in (0,1)$: \begin{gather*} (Ay_{0,1}+By_{0,2},Cy_{0,1}+Dy_{0,2})\in (X,\mathcal{D}(A))_{\theta,\infty})\times (X,\mathcal{D}(D))_{\theta,\infty}) \\ z_{1,1},z_{1,2}\in (X,\mathcal{D}(A))_{\theta,\infty}),\quad z_{2,1},z_{2,2}\in (X,\mathcal{D}(D)_{\theta,\infty}), \end{gather*} Then $(y_1,y_2,f_1,f_2)$ also satisfies \begin{gather*} y'_1,Ay_1+By_2\in B([0,\tau];(X,\mathcal{D}(A))_{\theta,\infty}),\\ y'_2,Cy_1+Dy_2\in B([0,\tau];(X,\mathcal{D}(A))_{\theta,\infty}), \\ Ay_1+By_2\in C^\theta([0,\tau];X),\quad Cy_1+Dy_2\in C^\theta([0,\tau];X). \end{gather*} \end{remark} \section{Weakly coupled identification problems} In this section we deal with the following weakly coupled identification problem \begin{gather} \label{e7.1} \begin{aligned} & \frac{d}{dt}\begin{bmatrix} y_1(t)\\ \dots\\ y_n(t) \end{bmatrix} + \begin{bmatrix} A_{1,1}+ B_{1,1} & B_{1,2} & \dots & B_{1,n} \\ B_{2,1} & A_{2,2}+ B_{2,2} & \dots & B_{2,n} \\ \dots &\dots &\dots &\dots \\ B_{n,1} & B_{n,2} & \dots & A_{n,n}+ B_{n,n} \end{bmatrix} \begin{bmatrix} y_1(t) \\ \dots\\ y_n(t) \end{bmatrix} \\ &= \begin{bmatrix} h_1(t)\\ \dots\\ h_n(t) \end{bmatrix} + \sum_{j=1}^n f_j(t) \begin{bmatrix} z_{1,j}\\ \dots\\ z_{n,j} \end{bmatrix},\quad t\in [0,\tau], \end{aligned} \\ \label{e7.2} y_j(0)=y_{0,j},\quad j=1,\dots,n,\\ \label{e7.3} \Psi_j[y_j(t)]=g_j(t),\quad t\in [0,\tau],\; j=1,\dots,n, \end{gather} with \begin{equation} \label{e7.4} \Psi_j\in X^*,\quad \Psi_j[y_{0,j}]=g_j(0),\quad j=1,\dots,n, \end{equation} where $A_{i,i}$, $B_{i,j}$ are closed linear operators acting in the Banach space $X$. Now we introduce the operator matrices $A$ and $B$ defined by \[ A=\begin{bmatrix} A_{1,1} & O & \dots & O \\ O & A_{2,2} & \dots & O \\ \dots &\dots &\dots &\dots \\ O & O & \dots & A_{n,n} \end{bmatrix}, \quad B= \begin{bmatrix} B_{1,1} & B_{1,2} & \dots & B_{1,n} \\ B_{2,1} & B_{2,2} & \dots & B_{2,n} \\ \dots &\dots &\dots &\dots \\ B_{n,1} & B_{n,2} & \dots & B_{n,n} \end{bmatrix} \] Assume now that $\rho(A_{j,j})\supset \Sigma_\alpha$, $j=1,\dots,n$, and \begin{equation} \label{e7.5} \|(\lambda I+A_{j,j})^{-1}\|_{\mathcal{L}(X)}\le C(1+|\lambda|)^{-\beta},\quad \lambda\in \Sigma_\alpha, \end{equation} Then $\lambda I-A$ is invertible for all $\lambda\in \Sigma_\alpha$ and \[ (\lambda I+A)^{-1}=\begin{bmatrix} (\lambda I+A_{1,1})^{-1} & O& \dots & O\\ O& (\lambda I+A_{2,2})^{-1} & \dots & O\\ \dots &\dots &\dots &\dots \\ O& O& \dots & (\lambda I+A_{n,n})^{-1} \end{bmatrix}. \] Further, let the linear closed operators $B_{i,j}:\mathcal{D}(B_{i,j})\subset X\to X$, $\mathcal{D}(B_{i,j})\supset \mathcal{D}(A_{j,j})$, $i,j=1,\dots,n$, satisfy, for some $\sigma>0$, the estimates \begin{equation} \label{e7.6} \|B_{i,j}(\lambda I+A_{j,j})^{-1}\|_{\mathcal{L}(X)}\le C(1+|\lambda|)^{-\sigma}, \quad \lambda\in \Sigma_\alpha. \end{equation} Observe now that \[ \mathcal{D}(A+B)=\mathcal{D}(A)=\prod_{j=1}^n \mathcal{D}(A_{j,j}),\quad \lambda I+A+B=\big[I+B(\lambda I+A)^{-1}](\lambda I+A). \] Since \begin{align*} &\begin{bmatrix} B_{1,1} & \dots & B_{1,n} \\ \dots &\dots &\dots \\ B_{n,1} & \dots & B_{n,n} \end{bmatrix} \begin{bmatrix} (\lambda I+A_{1,1})^{-1} & \dots & O \\ \dots &\dots &\dots \\ O & \dots & (\lambda I+A_{n,n})^{-1} \end{bmatrix} \\ &= \begin{bmatrix} B_{1,1}(\lambda I+A_{1,1})^{-1} & \dots & B_{1,n}(\lambda I+A_{n,n})^{-1}\\ \dots &\dots &\dots \\ B_{n,1}(\lambda I+A_{1,1})^{-1} & \dots & B_{n,n}(\lambda I+A_{n,n})^{-1} \end{bmatrix}, \end{align*} assumption \eqref{e7.6}) implies that $I+B(\lambda I+A)^{-1}$ has a bounded inverse for each $\lambda \in \Sigma_\alpha$, with a large enough modulus, satisfying \[ \|\big[I+B(\lambda I+A)^{-1}\big]^{-1}\|_{\mathcal{L}(X^n)}\le 2,\quad \lambda\in \Sigma_\alpha\cap B(0,C)^c, \] where $X^n$ is the product space $X\times \dots \times X$ ($n$ times). On the other hand, the change of the vector unknown defined by $y(t)=e^{kt}w(t)$, where $y(t)=[y_1(t),\dots,y_n(t)]^T$ and $w(t)=[w_1(t),\dots,w_n(t)]^T$, transforms our equation into \[ w'(t)=-(kI+A+B)w(t) + e^{-kt}\sum_{j=1}^n f_j(t)z_j + e^{-kt}h(t),\quad t\in [0,\tau], \] where $h(t)=[h_1(t),\dots,h_n(t)]^T$. Observe now that \[ (\lambda I+kI+A+B)^{-1}=((\lambda+k)I+A+B)^{-1} =((\lambda+k)I+A)^{-1}\big[I+B((\lambda+k)I+A)^{-1}]. \] Therefore, we conclude that $\lambda I+kI+A+B$ is invertible for large enough $k$ and $(\lambda I+kI+A+B)^{-1}\in \mathcal{L}(X^n)$ for $\lambda \in \Sigma_\alpha$. Then set \begin{gather*} \mathcal{A}=A+B,\quad \mathcal{D}(\mathcal{A})=\prod_{j=1}^n \mathcal{D}(A_{j,j}),\quad y=(y_1,\dots,y_n)^T, \\ z_j=(z_{1,j},\dots,z_{n,j})^T,\quad \Phi_j\in (X^n)^*=(X^*)^n,\quad \Phi_j[y]=\Psi_j[y_j],\quad j=1,\dots,n. \end{gather*} Consider the equality \[ \begin{bmatrix} \Phi_1[z_{1}] & \dots & \Phi_1[z_{n}] \\ \dots &\dots &\dots \\ \Phi_n[z_{1}] & \dots & \Phi_n[z_{n}] \end{bmatrix} =\begin{bmatrix} \Psi_1[z_{1,1}] & \dots & \Psi_1[z_{1,n}]\\ \dots &\dots &\dots \\ \Psi_n[z_{n,1}] & \dots & \Psi_n[z_{n,n}] \end{bmatrix}. \] Thus we need to assume that \begin{equation} \label{e7.7} \det \begin{bmatrix} \Psi_1[z_{1,1}] & \dots & \Psi_1[z_{1,n}]\\ \dots &\dots &\dots \\ \Psi_n[z_{n,1}] & \dots & \Psi_n[z_{n,n} \end{bmatrix} \ne 0. \end{equation} Then we characterize the space $X^\theta_\mathcal{A}$ in the following Lemma. \begin{lemma} \label{lem7.1} The following relations hold for all $\theta\in (0,\beta)$: \begin{equation} \label{e7.8} X^{\theta+1-\beta}_\mathcal{A}\hookrightarrow X^\theta_A,\quad X^{\theta+1-\beta}_A\hookrightarrow X^\theta_\mathcal{A},\quad X^\theta_A=\prod_{j=1}^n X^{\theta}_{A_{j,j}}. \end{equation} \end{lemma} We postpone the proof of this lemma to the end of this section and state our conclusive theorem. \begin{theorem} \label{thm7.2} Let $\alpha,\beta\in (0,1]$, $\alpha+2\beta+\alpha\beta>3$ and $3-\alpha-\beta-\alpha\beta<\theta<\beta$. Let $z_j=[z_{1,j},\dots,z_{n,j}]\in X^{\theta+1-\beta}_A(\subset X^\theta_\mathcal{A})$, $j=1,\dots,n$, $y_0 = (y_{0,1},\dots,y_{0,n})\in \mathcal{D}(A)$, $(A+B)y_0\in X^{\theta+1-\beta}_A$, $h\in C([0,\tau];X^n)\cap B([0,\tau];X_A^{\theta+1-\beta})$ satisfy condition \eqref{e7.7}. Then the identification problem \eqref{e9.40}-\eqref{e9.42} admits a unique strict solution $(y,f_1,\dots,f_n)\in C([0,\tau];X)\cap C([0,\tau];\mathbb{C})^n$ such that \begin{gather*} y'\in B([0,\tau];X_A^{[\theta-(3-2\alpha-\beta)-\alpha(1-\beta)]/\alpha}),\\ (A+B)y\in C^{[\theta-(3-2\alpha-\beta)]/\alpha}([0,\tau];X^n)\cap B([0,\tau]; X_A^{[\theta-(3-2\alpha-\beta)-\alpha(1-\beta)]/\alpha}). \end{gather*} \end{theorem} The proof of the above theorem follows easily from our assumptions and Corollary \ref{coro5.3}. We conclude this section with the proof of Lemma \ref{lem7.1}. \begin{proof}[Proof of Lemma \ref{lem7.1}] To show the first embedding in \eqref{e7.8} we recall that $\tau+A+B$ admits a continuous inverse for $\tau\ge t_0$, $t_0$ being positive and large enough. Then we make use of the following identity, with $t>0$: \begin{equation} \label{e7.9} \begin{aligned} &(t_0+A+B)(t+t_0+A+B)^{-1}-(t_0+A+B)(t+A)^{-1}\\ &=- (t_0+A+B)(t+t_0+A+B)^{-1}(t_0+B)(t+A)^{-1} \end{aligned} \end{equation} Whence we deduce \begin{align*} &\sup_{t>0} (1+t)^\theta\|(t_0+A+B)(t+t_0+A+B)^{-1}u\|\\ &\le \|(t_0+A+B)A^{-1}\|_{\mathcal{L}(X)} \sup_{t>0} (1+t)^\theta\|A(t+A)^{-1}u\| \\ &\quad + \sup_{t>0} C(1+t)^{1-\beta}\|(t_0+B)A^{-1}\|_{\mathcal{L}(X)}(1+t)^\theta \|A(t+A)^{-1}u\| \\ &\le C'\sup_{t>0} (1+t)^{\theta+1-\beta}\|A(t+A)^{-1}u\|. \end{align*} These inequalities imply the embedding \[ X^{\theta+1-\beta}_A\hookrightarrow X^\theta_{t_0+A+B}. \] Interchanging the roles of $t_0+A+B$ and $A$, we obtain the embedding \[ X^{\theta+1-\beta}_{t_0+A+B}\hookrightarrow X^\theta_A,\quad \text{if } \theta\in (0,\beta). \] We have thus shown the first two relations in \eqref{e7.7}. Now we show that $X^\theta_A=X^\theta_{t_0+A}$ for all $\theta\in (0,1)$ and $t_0\in \mathbb{R}_+$. For this purpose first we consider the following identities: \begin{align*} &(t_0+A)(t+t_0+A)^{-1}-A(t+A)^{-1} \\ & = \big[(t_0+A)(t+t_0+A)^{-1}(t+A)A^{-1}-I\big]A(t+A)^{-1}\\ & = \big[(t_0+A)(t+t_0+A)^{-1}(t+t_0+A-t_0)A^{-1}-I\big]A(t+A)^{-1}\\ & = \big[t_0A^{-1}+I-I-t_0(t_0A^{-1}+I)(t+t_0+A)^{-1}\big]\big[A(t+A)^{-1}\big] \end{align*} Observe that \[ \|(t+t_0+A)^{-1}\|_{\mathcal{L}(X)}\le \frac{M}{(1+t+t_0)^\beta} \le M,\quad t\in [0,+\infty). \] Therefore we easily get the estimate \[ \sup_{t>0} (1+t)^\theta\|(t_0+A)(t+t_0+A)^{-1}u\| \le C(t_0,A)\sup_{t>0} (1+t)^\theta\|A(t+A)^{-1}u\|, \] where \[ C(t_0,A)\le 1+t_0\|A^{-1}\|_{\mathcal{L}(X)} +t_0M\big[t_0\|A^{-1}\|_{\mathcal{L}(X)}+1\big]. \] This inequality implies, for all $t_0>0$, the embedding \[ X^\theta_{A}\hookrightarrow X^{\theta}_{t_0+A}. \] Interchanging the roles of $A$ and $t_0+A$, we obtain the identities \begin{align*} &A(t+A)^{-1}-(t_0+A)(t+t_0+A)^{-1}\\ & = \big[A(t+A)^{-1}(t+A+t_0)(t_0+A)^{-1}-I\big](t_0+A)(t+t_0+A)^{-1}\\ & = \big[A(t_0+A)^{-1}+t_0A(t_0+A)^{-1}(t+A)^{-1}-I\big](t_0+A)(t+t_0+A)^{-1} \end{align*} Recalling that \begin{gather*} \|(t+A)^{-1}\|_{\mathcal{L}(X)}\le c(1+t)^{-\beta}\le c,\\ \|A(t_0+A)^{-1}\|_{\mathcal{L}(X)}\le \|I-t_0(t_0+A)^{-1}\|_{\mathcal{L}(X)} \le 1+ct_0^{1-\beta}, \end{gather*} we deduce the estimate \[ \|A(t+A)^{-1}u\| \le \big[(1+ct_0^{1-\beta})(1+ct_0)+1\big]\|(t_0+A) \big(t+t_0+A\big)^{-1}u\| \] Whence we deduce the following inequality holding for all $\theta\in (0,1)$: \begin{align*} &\sup_{t>0} (1+t)^\theta\|A(t+A)^{-1}u\|\\ & \le \big[(1+ct_0^{1-\beta})(1+ct_0)+1\big]\sup_{t>0} (1+t)^\theta\|(t_0+A)(t+t_0+A)^{-1}u\|. \end{align*} Whence we deduce the following inequality holding for all $\theta\in (0,1)$: \[ \sup_{t>0} (1+t)^\theta\|A(t+A)^{-1}u\| \le C'\sup_{t>0} (1+t)^\theta\|(t_0+A)(t+t_0+A)^{-1}u\|. \] We have thus proved the reverse embedding, holding for all $t_0>0$ and all $\theta\in (0,1)$: \[ X^{\theta}_{t_0+A}\hookrightarrow X^\theta_{A}. \] Finally, we have shown the first two relations in \eqref{e7.8}. The third equality follows from the fact that $A$ is a diagonal operator-matrix operator, so that, for all $t>0$, we have \[ (tI+A)^{-1}=\begin{bmatrix} (tI+A_{1})^{-1} & O & \dots & O \\ O & (tI+A_2)^{-1} & \dots & O \\ \dots &\dots &\dots &\dots \\ O & O & \dots & (tI+A_{n})^{-1} \end{bmatrix}. \] If we define the norm in $X^n$ by \[ \|(x_1,\dots,x_n)\|_{X^n} = \sum_{j=1}^n \|x_j\|_X, \] then \[ \sup_{t>0} (1+t)^\theta\|A(t+A)^{-1}x\|_{X^n} =\sup_{t>0} (1+t)^\theta \sum_{j=1}^n \|A_j(t+A_j)^{-1}x_j\|_X \le \sum_{j=1}^n \|x_j\|_{X^\theta_{A_j}}. \] Therefore, \[ \prod_{j=1}^n X^\theta_{A_j}\hookrightarrow (X^n)^\theta_{A}. \] Conversely, if $\sup_{t>0} (1+t)^\theta\|A(t+A)^{-1}x\|_{X^n}<+\infty$, then \[ \sup_{t>0} (1+t)^\theta \sum_{j=1}^n \|A_j(t+A_j)^{-1}x_j\|_X \le \sup_{t>0} (1+t)^\theta\|A(t+A)^{-1}x\|_{X^n}<+\infty, \] for all $j=1,\dots,n$, so that the embedding \[ (X^n)^\theta_{A}\hookrightarrow \prod_{j=1}^n X^\theta_{A_j}. \] follows immediately. Now we show that $X^\theta_A=X^\theta_{t_0+A}$. For this purpose first we consider the following identity obtained from \eqref{e7.9} setting $B=O$: \[ (t_0+A)(t+t_0+A)^{-1}-(t_0+A)(t+A)^{-1} = - t_0(t_0+A)(t+t_0+A)^{-1}(t+A)^{-1}. \] Whence we deduce \begin{align*} &\sup_{t>0} (1+t)^\theta\|(t_0+A)(t+t_0+A)^{-1}u\|\\ &\le \|(t_0+A)A^{-1}\|_{\mathcal{L}(X)} \sup_{t>0} (1+t)^\theta\|A(t+A)^{-1}u\|\\ &\quad + \sup_{t>0} M(1+t)^\theta(1+t+t_0)^{-\beta}t_0 \|(t_0+A)A^{-1}\|_{\mathcal{L}(X)}\|A(t+A)^{-1}u\|\\ &\le C\sup_{t>0} (1+t)^{\theta}\|A(t+A)^{-1}u\|. \end{align*} So, we have proved the embedding $X^\theta_A\hookrightarrow X^{\theta}_{t_0+A}$. Interchanging the roles of $t_0+A$ and $A$, we obtain the set equality $X^\theta_A = X^{\theta}_{t_0+A}$ with equivalence of the corresponding norms. Finally, the third embedding in \eqref{e7.8} is obvious. \end{proof} \begin{remark} \label{rmk7.1} \rm Corollary \ref{coro5.3} applies if the regularity assumptions on the data concern the spaces $(X,\mathcal{D}(A))_{\theta,\infty} =(X,\mathcal{D}(\lambda_0+A+B))_{\theta,\infty}$. Notice that, if operator $B$, with $\mathcal{D}(A)\subset \mathcal{D}(B)$ satisfies the following estimate, similar to the ones satisfied by $A$ (cf.\eqref{e1.1}, \eqref{e1.2}): \begin{equation} \label{e7.10} \| (\lambda+\lambda_0+A+B)^{-1}\|_{\mathcal{L}(X)} \leq c'(1+|\lambda|)^{-\beta} \end{equation} for all $\lambda$ in the sector \begin{equation} \label{e6.30} \Sigma_\alpha := \{ \lambda \in \mathbb{C}: \operatorname{Re}\lambda \geq -c'(1+ |\operatorname{Im} \lambda|)^\alpha\}, \quad 0 < \beta \leq \alpha\le 1, \end{equation} then $(X,\mathcal{D}(A))_{\theta,\infty}=(X,\mathcal{D}(\lambda_0+A+B))_{\theta,\infty}$ with the equivalence of their norms. \end{remark} \section{Identification problems for singular non-classical first-order in time systems of PDE's corresponding to $\beta=1$} In this section some applications related to the regular and singular parabolic equations will be given. \subsection*{Problem 8.1} We will consider a problem related to a reaction diffusion model describing a man-environment epidemic system investigated in \cite{CK}. Such a model consists in a parabolic equation coupled with an ordinary differential equation via a boundary feedback operator (cf. also \cite{E}). To obtain stability results in \cite{CK} the authors linearize the model and arrive at the following evolution system, where $u(t,x)$ and $v(t,x)$ stand, respectively, for the concentration of the infection agent and the density of the infective population at time $t$ and point $x$: \begin{equation} \label{e8.1} \begin{gathered} D_{t}u(t,x)=\Delta u(t,x)-a(x)u(t,x) +f_1(t)z_{1,1}(x)+f_2(t)z_{1,2}(x),\\ (t,x)\in(0,\tau)\times \Omega,\\ D_{t}v(t,x)=c(x)u(t,x)-d(x)v(t,x) +f_1(t)z_{2,1}(x)+f_2(t)z_{2,2}(x),\\ (t,x)\in(0,\tau)\times \Omega, \\ u(0,x)=u_{0}(x),\quad v(0,x)=v_{0}(x),\quad x\in\Omega, \\ D_{\nu}u(t,x)+\beta(x)u(t,x)=\int_{\Omega}k(x,y)v(t,y)\,dy,\quad (t,x)\in (0,\tau)\times \partial\Omega, \\ \int_{\overline \Omega} u(t,x)\,d\mu_1(x)=g_1(t),\quad t\in [0,\tau], \\ \int_{\overline \Omega} v(t,x)\,d\mu_2(x)=g_2(t),\quad t\in [0,\tau], \end{gathered} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with a smooth boundary $\partial\Omega$, $\Delta$ is the Laplacian, $a$, $c$, $d\in C(\overline{\Omega})$, $\beta\in C(\partial\Omega)$, $k\in W^{1,\infty}(\partial\Omega;L^\infty(\Omega))$ are non-negative functions and $D_{\nu}$ denotes the outward normal derivative on $\partial\Omega$. Finally, $\mu_1$ and $\mu_2$ are two positive Borel measure on $\overline \Omega$. We define $E=C(\overline{\Omega}), X=E\times E$ and denote by $M_{h}$ the multiplication operator induced by the function $h$. Moreover, we introduce the operator-matrix \begin{gather} \mathcal{A}=\begin{bmatrix} \Delta-M_{a} & O\\ M_{c} & -M_{d} \end{bmatrix}, \label{e8.2} \\ \begin{aligned} \mathcal{D}(\mathcal{A})=\Big\{&(u,v)\in X: u\in H^{2}(\Omega),\quad \Delta u\in E, \\ &D_{\nu}u(\cdot)+\beta(\cdot)u(\cdot) =\int_{\Omega}k(\cdot,y)v(y)\,dy\text{ on } \partial\Omega\Big\}. \end{aligned} \end{gather} It can be proved (cf. \cite[p. 26]{E}) that $A$ generates an analytic semigroup on $X$ with $\alpha=\beta=1$ and $\theta\in (0,1)$. Therefore we can apply Theorem \ref{thm6.1} and its Corollaries 6.2, 6.3. Let us assume that $(u_0,v_0) \in D({\mathcal{A}})$, $((\Delta-a(\cdot))u_0, c(\cdot)u_0 - d(\cdot)v_0) \in (C(\overline\Omega)\times C(\overline\Omega),D({\mathcal{A}}))_{\theta,\infty}$, \[ \int_\Omega u_0(x)\mu_1(dx) = g_1(0),\quad \int_\Omega v_0(x)\,d\mu_2(x) = g_2(0), \] $g_1,g_2\in C^1([0,\tau]; \mathbb{C})$, $z_{ik} \in C(\overline\Omega)$, $i,k = 1,2$, $(z_{11},z_{21}), (z_{12},z_{22}) \in (C(\overline\Omega) \times C(\overline\Omega), \mathcal{D} ({\mathcal{A}}))_{\theta,\infty}$, $$ \int_\Omega z_{1,1}\,d\mu_1(x)\int_\Omega z_{2,2}\,d\mu_2(x) - \int_\Omega z_{1,2}\,d\mu_1(x) \int_\Omega z_{2,1}\,d\mu_2(x) \neq 0. $$ Then the identification problem \eqref{e8.1}, admits a unique global strict solution $$ ((u,v),f_1,f_2) \in C([0,\tau];D({\mathcal{A}})) \times C([0,\tau];\mathbb{C}) \times C([0,\tau];\mathbb{C}) $$ such that $$ (D_t u,D_t v),\ {\mathcal{A}}(u,v)^T\in B([0,\tau];(C(\overline\Omega)\times C(\overline\Omega),\mathcal{D} ({\mathcal{A}}))_{\theta,\infty}). $$ We can characterize the interpolation space $(C(\overline\Omega)\times C(\overline\Omega),D({\mathcal{A}}))_{\theta,\infty}$ taking advantage of (cf. \cite[Theorem 1.14.3, p. 93]{TR}) and the representation of $\mathcal{A} - \lambda I$ as a product of suitable operator matrices (cf. \cite{E}, p. 126). \subsection*{Problem 8.2} Let us consider the weakly coupled identification vector problem occurring in the theory of semiconductors. Here we will deal with the problem consisting of recovering the three scalar functions $f_{j}: [0,\tau]\to\mathbb{R}$, $1\le j\le 3$, in the singular problem \begin{equation}\label{e8.4} \begin{gathered} D_{t}u_{\mathrm{l}}=a\Delta u_{1}-d\Delta u_{3}+f_{1}(t)\zeta_{1},\quad \text{in }(0,\tau)\times \Omega, \\ D_{t}u_2=b\Delta u_2+e\Delta u_{3}+f_2(t)\zeta_2,\quad \text{in }(0,\tau)\times \Omega, \\ 0=u_{1}-u_2-c\Delta u_{3}+f_{3}(t)\zeta_{3},\quad \text{in }(0,\tau)\times \Omega, \\ u_{1}(0,\cdot)=u_{0,1},\quad u_2(0,\cdot)=u_{0,2},\quad \text{in } \Omega, \\ u_{1}=u_2=u_{3}=0, \quad \text{in } (0,\tau)\times \partial\Omega, \end{gathered} \end{equation} under the following three additional conditions \begin{equation} \langle u_{i}(t,\cdot),\varphi_i\rangle:=\int_\Omega u_{i}(t,x)\varphi_i(x)\,dx =g_{i}(t),\quad t\in(0,\tau),\; i=1,2,3, \label{e8.5} \end{equation} where $\zeta_{i}\in L^{p}(\Omega)$, $i=1,2,3$, $p\in (1,+\infty]$, $a, b\in \mathbb{R}_{+}$, $c,e\in \mathbb{R}\backslash \{0\}$, $d\in \mathbb{R}$ and $\varphi_i\in L^{p'}(\Omega)$, $1/p+1/p'=1$, $i=1,2,3$. We notice that Theorem \ref{thm7.2} cannot be directly applied to this identification problem, since such a problem is singular due to the lack of the term $D_tu_3$. However, since $\Delta: W_{0}^{1,p}(\Omega)\cap W^{2,p}(\Omega)\to L^{p}(\Omega)$ is a linear isomorphism, we can solve the elliptic equation for $u_3$: \begin{equation} u_{3}=c^{-1}\Delta^{-1}[u_{1}-u_2+f_{3}(t)\zeta_{3}],\quad \text{in } (0,\tau)\times \Omega. \label{e8.6} \end{equation} Assume now \begin{equation} \chi_{3}^{-1}:=(\Delta^{-1}\zeta_{3},\varphi_3)_{L^{2}(\Omega)}\neq 0. \label{e8.7} \end{equation} Consequently, from \eqref{e8.6} and the additional equation $(u_{3}(t,\cdot),\varphi_3)_{L^{2}(\Omega)}=g_{3}(t)$ we deduce the following formula for $f_{3}$: \begin{equation} f_{3}(t)=c\chi_{3}g_{3}(t)-\chi_{3}\langle \Delta^{-1}(u_{1} -u_2)(t,\cdot),\varphi_3\rangle. \label{e8.8} \end{equation} Therefore, our inverse problem is equivalent to the following problem: \begin{equation} \label{e8.9} \begin{gathered} \begin{aligned} D_{t}u_1&=a\Delta u_{1}+\Big[-dc^{-1}u_{1}+dc^{-1}\chi_3\langle \Delta^{-1}u_{1},\varphi_3\rangle \zeta_3 +dc^{-1}u_2\\ &\quad -dc^{-1}\chi_3\langle \Delta^{-1}u_2,\varphi_3\rangle\zeta_3\Big] -d\chi_3g_3(t)\zeta_{3}+f_{1}(t)\zeta_{1},\quad \text{in } (0,\tau)\times \Omega, \end{aligned}\\ \begin{aligned} D_{t}u_2&=b\Delta u_2+\Big[ec^{-1}u_{1}-ec^{-1}\chi_3\langle \Delta^{-1}u_{1},\varphi_3\rangle \zeta_3 -ec^{-1}u_2\\ &\quad +ec^{-1}\chi_3\langle \Delta^{-1}u_2,\varphi_3\rangle \zeta_3\Big] +e\chi_3g_3(t)\zeta_{3} + f_2(t)\zeta_2,\quad \text{in }(0,\tau)\times \Omega, \end{aligned}\\ u_{1}(0,\cdot)=u_{0,1},\quad u_2(0,\cdot)=u_{0,2},\quad \text{in } \Omega, \\ u_{1}=u_2=0, \quad \text{in } (0,\tau)\times \partial\Omega, \\ \langle u_i(t,\cdot),\varphi_i\rangle =g_i(t),\quad t\in (0,\tau),\; i=1,2. \end{gathered} \end{equation} Define $\{e^{t\Delta}\}_{t>0}$ as the analytic semigroup generated by $\Delta$ with the domain $\Delta: W_{0}^{1,p}(\Omega)\cap W^{2,p}(\Omega)\to L^{p}(\Omega)$ and observe that the semigroups $\{T_{1}(t)\}_{t>0}$ and $\{T_2(t)\}_{t>0}$ generated by $a\Delta$ and $b\Delta$ are defined, respectively, by \begin{equation} T_{1}(t)=e^{at\Delta},\quad T_2(t)=e^{bt\Delta}. \label{e8.10} \end{equation} In this case we have $$ X^\theta_A=(L^p(\Omega);W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))_{\theta,\infty}\times (L^p(\Omega);W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))_{\theta,\infty}. $$ Such spaces were characterized by Grisvard (a proof can be found, for the reader's convenience, in \cite[p. 321]{TR}). Then we define \begin{gather} \label{e8.11} B_{1,1}u_1=-dc^{-1}u_1+dc^{-1}\chi_3\langle\Delta^{-1}u_1, \varphi_3\rangle\zeta_3,\\ \label{e8.12} B_{1,2}u_2=dc^{-1}u_2-dc^{-1}\chi_3\langle\Delta^{-1}u_2, \varphi_2\rangle\zeta_3,\\ \label{e8.13} B_{2,1}u_1=ec^{-1}u_1-ec^{-1}\chi_3\langle\Delta^{-1}u_1, \varphi_3\rangle\zeta_3,\\ \label{e8.14} B_{2,2}u_2=-ec^{-1}u_2+ec^{-1}\chi_3\langle\Delta^{-1}u_2,\varphi_3\rangle\zeta_3, \\ \label{e8.15} h_{1}(t)=-d\chi_{3}g_{3}(t)\zeta_{3},\quad h_2(t)=e\chi_{3}g_{3}(t)\zeta_{3}, \\ \label{e8.16} z_{1,1}=\zeta_{1},\quad z_{2,2}=\zeta_2,\quad z_{1,2}=z_{2,1}=0. \end{gather} Assume further that \begin{gather} \label{e8.17} \begin{aligned} &\Big[a\Delta u_{0,1}-dc^{-1}u_{0,1}+dc^{-1}\chi_3\langle\Delta^{-1}u_{0,1}, \varphi\rangle\zeta_3 +dc^{-1}u_{0,2} -dc^{-1}\chi_3\langle\Delta^{-1}u_{0,2},\varphi_3\rangle\zeta_3\Big]\\ &\in (L^p(\Omega);W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))_{\theta,\infty}, \end{aligned} \\ \label{e8.18} \begin{aligned} &\Big[ec^{-1}u_{0,1}-ec^{-1}\chi_3\langle\Delta^{-1}u_{0,1}, \varphi_3\rangle\zeta_3 + b\Delta u_{0,2}-ec^{-1}u_{0,2} +ec^{-1}\chi_3\langle\Delta^{-1}u_{0,2},\varphi_3\rangle\zeta_3\Big]\\ &\in (L^p(\Omega);W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))_{\theta,\infty}, \end{aligned} \\ \label{e8.19} u_{0,1},u_{0,2}\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega),\quad \zeta_3\in (L^p(\Omega);W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))_{\theta,\infty}, \\ \label{e8.20} \langle\zeta_1,\varphi_1\rangle \langle\zeta_2,\varphi_2\rangle \langle\Delta^{-1}\zeta_3,\varphi_3\rangle\ne 0. \end{gather} Then we can apply Theorem \ref{thm7.2} with $(\alpha,\beta)=(1,1)$, to problem \eqref{e8.9} to ensure that there exists a quadruplet $(u_{1},u_2,f_{1},f_2)\in {\mathcal X}=\big\{C^1([0,\tau];L^p(\Omega)^2) \cap C([0,\tau];\big[W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)\big]^2)\big\}\times C([0,\tau];\mathbb{C}^{2})$ solving \eqref{e8.9}. Finally, we observe that the pair $(u_{3},f_{3})$ is defined by formulae \eqref{e8.6} and \eqref{e8.7}. Therefore it belongs to $C([0,\tau];W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))\times C([0,\tau];\mathbb{C})$. The same technique applies when our additional information is \[ (u_{i}(t,\cdot),\varphi_i)_{L^{2}(\Omega)}=g_{i}(t),\; i=1,2,\quad (u_{1}(t,\cdot),\varphi_3)_{L^{2}(\Omega)}=g_{3}(t),\quad t\in(0,\tau). \] In this case the solvability condition changes to \begin{equation}\label{e8.21} \int_\Omega \varphi_2(x)\zeta_2(x)\,dx \begin{bmatrix} \int_\Omega \varphi_1(x)\zeta_1(x)\,dx & \int_\Omega \varphi_1(x)\zeta_3(x)\,dx \\ \int_\Omega \varphi_3(x)\zeta_1(x)\,dx & \int_\Omega \varphi_3(x)\zeta_3(x)\,dx \end{bmatrix} \ne 0. \end{equation} In fact, from \eqref{e8.6}, we easily derive the new identification problem \begin{equation} \label{e8.22} \begin{gathered} D_{t}u_1=\big(a\Delta -dc^{-1}\big)u_{1}+bc^{-1}u_2 +f_{1}(t)\zeta_{1}-dc^{-1}f_{3}(t)\zeta_{3},\quad \text{in }(0,\tau)\times \Omega, \\ D_{t}u_2=ec^{-1}u_{1}+\big(b\Delta -ec^{-1}\big)u_2 +f_2(t)\zeta_2+ec^{-1}f_{3}(t)\zeta_{3}, \quad \text{in }(0,\tau)\times \Omega, \\ u_{1}(0,\cdot)=u_{0,1},\quad u_2(0,\cdot)=u_{0,2},\quad \text{in } \Omega, \\ u_{1}=u_2, \quad \text{in } (0,\tau)\times \partial\Omega, \\ (u_i(t,\cdot),\varphi_i)_{L^{2}(\Omega)}=g_i(t),\quad t\in (0,\tau),\ i=1,2, \\ (u_1(t,\cdot),\varphi_3)_{L^{2}(\Omega)}=g_3(t),\quad t\in (0,\tau). \end{gathered} \end{equation} Now Corollary \ref{coro5.3} applies if the following solvability condition is satisfied \[ \begin{vmatrix} \int_\Omega \varphi_1(x)\zeta_1(x)\,dx& 0 & -dc^{-1}\int_\Omega \varphi_1(x)\zeta_3(x)\,dx \\ 0 & \int_\Omega \varphi_2(x)\zeta_2(x)\,dx & ec^{-1}\int_\Omega \varphi_2(x)\zeta_3(x)\,dx \\ \int_\Omega \varphi_3(x)\zeta_1(x)\,dx& 0 & -dc^{-1}\int_\Omega \varphi_3(x)\zeta_3(x)\,dx \end{vmatrix} \ne 0. \] But this condition is nothing but \eqref{e8.21}. However, notice that the consistency conditions \[ (u_{0,i}(t,\cdot),\varphi_i)_{L^{2}(\Omega)}=g_i(0),\; i=1,2,\quad (u_{0,1}(t,\cdot),\varphi_3)_{L^{2}(\Omega)}=g_3(0), \] must hold. Assume now that the boundary condition involving $u_3$ is changed to the Neumann one, i.e. $D_\nu u_3=0$ on $(0,\tau)\times \partial\Omega$, where $\nu$ and $D_{\nu}$ denote, respectively, the outward unit vector normal to $\partial\Omega$ and the normal derivative on $\partial\Omega$. Then the elliptic problem \begin{equation} \label{e8.23} \begin{gathered} 0=u_{1}-u_2-c\Delta u_{3}+f_{3}(t)\zeta_{3},\quad \text{in }(0,\tau)\times \Omega, \\ D_\nu u_{3}=0, \quad \text{in } (0,\tau)\times \partial\Omega, \end{gathered} \end{equation} admits a unique solution in $W^{2,p}(\Omega)$, if and only if the following condition is satisfied \[ f_3(t)\langle \zeta_3,1\rangle = -\langle (u_1-u_2)(t,\cdot),1\rangle,\quad t\in [0,\tau], \] where $\langle h,1\rangle=\int_\Omega h(x)\,dx$, $h\in L^1(\Omega)$. Assuming that \[ \chi_3^{-1}:= \langle \zeta_3,1\rangle \ne 0, \] we obtain \begin{equation} \label{e8.24} f_3(t) = -\chi_3\langle (u_1-u_2)(t,\cdot),1\rangle,\quad t\in [0,\tau]. \end{equation} Note that in this case we can get rid off of the third additional condition in \eqref{e8.5}. Consequently, an equivalent problem for $(u_1,u_2,f_1,f_2)$ turns out to be the following: \begin{equation} \label{e8.25} \begin{gathered} D_{t}u_1=a\Delta u_{1}-dc^{-1}\Big[u_{1}-u_2 -\chi_3\zeta_3\langle (u_{1}-u_2),1\rangle\big] +f_1(t)\zeta_1,\quad \text{in }(0,\tau)\times \Omega, \\ D_{t}u_2=b\Delta u_2+ec^{-1}\big[u_{1}-u_2 -\chi_3\zeta_3\langle (u_{1}-u_2),1\rangle\big] +f_2(t)\zeta_2,\quad \text{in }(0,\tau)\times \Omega, \\ u_{1}(0,\cdot)=u_{0,1},\quad u_2(0,\cdot)=u_{0,2},\quad \text{in } \Omega, \\ u_{1}=u_2=0, \quad \text{in } (0,\tau)\times \partial\Omega, \\ \langle u_i(t,\cdot),\varphi_i\rangle =g_i(t),\quad t\in (0,\tau),\; i=1,2. \end{gathered} \end{equation} Then we define \begin{gather*} B_{1}(u_{1},u_2)=-dc^{-1}\big[u_{1}-u_2-\chi_3\zeta_3 \langle (u_{1}-u_2),1\rangle\big], \\ B_2(u_{1},u_2)=ec^{-1}\big[u_{1}-u_2-\chi_3\zeta_3 \langle (u_{1}-u_2),1\rangle\big], \\ h_{1}(t)=h_2(t)=0, \\ z_{1,1}=\zeta_{1},\quad z_{2,2}=\zeta_2,\quad z_{1,2}=z_{2,1}=0. \end{gather*} Assume further \begin{equation}\label{e8.26} \begin{gathered} \begin{aligned} &a\Delta u_{0,1}-dc^{-1}\big[ u_{0,1}-u_{0,2} -\chi_3\zeta_3\langle (u_{0,1}-u_{0,2}),1\rangle\big] \\ &\in (L^p(\Omega);W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))_{\theta,\infty}, \end{aligned} \\ \begin{aligned} &b\Delta u_{0,2}-dc^{-1}\big[u_{0,1}-u_{0,2}-\chi_3\zeta_3 \langle (u_{0,1}-u_{0,2}),1\rangle\big]\\ &\in (L^p(\Omega);W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))_{\theta,\infty}, \end{aligned} \\ u_{0,1},u_{0,2}\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega),\quad \zeta_3\in (L^p(\Omega);W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))_{\theta,\infty}, \\ \langle\zeta_1,\varphi_1\rangle \langle\zeta_2,\varphi_2\rangle \langle \zeta_3,1\rangle \ne 0. \end{gathered} \end{equation} Then we can apply Theorem \ref{thm7.2} with $(\alpha,\beta)=(1,1)$, to problem \eqref{e8.25} to ensure that there exists a quadruplet \begin{align*} &(u_{1},u_2,f_{1},f_2)\in {\mathcal X}\\ &=\big\{C^1([0,\tau];L^p(\Omega)^2) \cap C^1([0,\tau];\big[W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)\big]^2)\big\}\times C([0,\tau];\mathbb{C}^{2}) \end{align*} solving \eqref{e8.9}. Finally, we observe that the pair $(u_{3},f_{3})$ is defined by formulae \eqref{e8.6} and \eqref{e8.24}. Therefore it belongs to $C([0,\tau];W^{2,p}(\Omega))\times C([0,\tau];\mathbb{R})$. Now we change the boundary conditions in the previous direct problem to the following ones of mixed Dirichlet-Neumann type \begin{gather}\label{e8.30} u_{1}=u_2=u_{3}=0,\quad \text{in }(0,\tau)\times \Gamma_D,\\ D_{\nu}u_{1}=D_{\nu}u_2=D_{\nu}u_{3}=0,\quad \text{in } (0,\tau)\times\Gamma_{N}.\label{e8.31} \end{gather} Here $\Gamma_D$ is an non-empty open subset and $\Gamma_{N}=\partial\Omega \setminus \Gamma_D$. Moreover, $\Omega$ must satisfy the exterior sphere condition \begin{equation} m_n(B(x_0,R)\cap \Omega^c)\ge cR^n,\quad \forall x_0\in \partial \Omega, \end{equation} $m_n$ denoting the $n$-dimensional Lebesgue measure. In particular the latter property holds if $\partial \Omega$ is Lipschitz (cf. \cite{BF,YA}). Explicitly, we consider the identification problem \eqref{e8.11} consisting in recovering the three scalar functions $f_i\in C([0,\tau];\mathbb{C})$ such that \begin{equation} \label{e8.33} \begin{gathered} D_{t}u_{\mathrm{l}}=a\Delta u_{1}-d\Delta u_{3}+f_{1}(t)\zeta_{1},\quad \text{in }(0,\tau)\times \Omega,\\ D_{t}u_2=b\Delta u_2+e\Delta u_{3}+f_2(t)\zeta_2,\quad \text{in }(0,\tau)\times \Omega,\\ 0=u_{1}-u_2-c\Delta u_{3}+f_{3}(t)\zeta_{3},\quad \text{in } (0,\tau)\times \Omega, \\ u_{1}(0,\cdot)=u_{0,1},\quad u_2(0,\cdot)=u_{0,2},\quad \text{in } \Omega, \\ u_{1}=u_2=u_{3}=0, \quad \text{in } (0,\tau)\times \Gamma_D, \\ D_\nu u_{1}=D_\nu u_2=D_\nu u_{3}=0, \quad \text{in } (0,\tau)\times \Gamma_N, \\ H_D^1(\Omega) \langle u_i(t,\cdot),\varphi_i \rangle_{H_D^1(\Omega)^{*}} =g_i(t),\quad t\in (0,\tau),\; i=1,2,3, \end{gathered} \end{equation} for given $\varphi_i\in H^1_D(\Omega)$ and $g_i\in C^1([0,\tau];\mathbb{C})$, $i=1,2,3$. Identifying $L^{2}(\Omega)$ with its antidual space, we introduce the Hilbert space \begin{equation} H_D^1(\Omega)=\{ u\in H^1(\Omega): u=0 {\rm\ on\ } \Gamma_D\} \label{e8.34} \end{equation} and we denote its antidual space by $H_D(\Omega)^{*}$. Then we define the linear operator $\Lambda\in \mathcal{L}(H_D^1(\Omega);H_D^1(\Omega)^{*})$ by the bilinear form \begin{equation} (\Lambda u,v)_{L^{2}(\Omega)}=\int_{\Omega}\nabla u\cdot {\overline {\nabla v}}\,dx,\quad u\in H_D^1(\Omega),\ v\in H_D^1(\Omega)^{*}. \label{e8.35} \end{equation} We note that $\Lambda$ is the realization of $-\Delta$ in $H_D^1(\Omega)^{*}$ under the homogeneous Dirichlet condition on $\Gamma_D$ and the homogeneous Neumann condition on $\Gamma_{N}$ and that $-\Lambda$ generates an analytic semigroup on $H_D^1(\Omega)^{*}$ (cf. \cite{BF}, \cite[p.114]{FLY} and \cite{YA}). Moreover, $-\Lambda$ is an isomorphism from $H_D^1(\Omega)$ to $H_D^1(\Omega)^{*}$. Let us observe that, for any $\theta\in [1/2,1]$, we have \begin{align*} D(\Lambda^{\theta}) &=[L^2(\Omega),H_D^1(\Omega)]_{2\theta-1} =\big[[H_D^1(\Omega)^{*},H_D^1(\Omega)]_{1/2}, H_D^1(\Omega)\big]_{2\theta-1} \\ &=[H_D^1(\Omega)^{*},H_D^1(\Omega)]_{1/2-(2\theta-1)/2}\\ &=[H_D^1(\Omega)^{*},H_D^1(\Omega)]_{\theta}\hookrightarrow (H_D^1(\Omega)^{*},H_D^1(\Omega))_{\theta,\infty}. \end{align*} Notice that $\Lambda u\in L^{2}(\Omega)$ needs not to imply $u\in H^{2}(\Omega)$ due to the boundary conditions of mixed type, while $D(\Lambda^{1/2})= L^{2}(\Omega)$. Choose now $X=H_D^1(\Omega)^{*}$. Then from the equation \[ c\Lambda u_3=-u_1+u_2-f_3(t)\zeta_3 \] we have \[ u_3=-c^{-1}\Lambda^{-1} u_1+c^{-1}\Lambda^{-1} u_2-c^{-1} \Lambda^{-1} f_3(t)\zeta_3 \] and \begin{align*} g_3(t)&=_{H_D^1(\Omega)}\langle u_3(t,\cdot),\varphi_3 \rangle_{H_D^1(\Omega)^{*}}\\ &= -c^{-1}_{H_D^1(\Omega)}\langle \Lambda^{-1}u_1(t,\cdot), \varphi_3 \rangle_{H_D^1(\Omega)^{*}} +c^{-1}_{H_D^1(\Omega)}\langle \Lambda^{-1}u_2(t,\cdot), \varphi_3 \rangle_{H_D^1(\Omega)^{*}}\\ &\quad -c^{-1}f_3(t) _{H_D^1(\Omega)}\langle \Lambda^{-1}\zeta_3,\varphi_3 \rangle_{H_D^1(\Omega)^{*}}. \end{align*} If \[ \eta^{-1}=: _{H_D^1(\Omega)}\langle \Lambda^{-1}\zeta_3,\varphi_3 \rangle_{H_D^1(\Omega)^{*}} \ne 0, \] the latter equation uniquely determines $f_3$ as \[ f_3(t)=-c\eta g_3(t) -\eta _{H_D^1(\Omega)}\langle \Lambda^{-1}u_1(t,\cdot),\varphi_3 \rangle_{H_D^1(\Omega)^{*}} +\eta _{H_D^1(\Omega)}\langle \Lambda^{-1}u_2(t,\cdot),\varphi_3 \rangle_{H_D^1(\Omega)^{*}}. \] Whence we easily deduce the formula \begin{align*} \Lambda u_3 &=-c^{-1}u_1+c^{-1}u_2 +c^{-1}\eta _{H_D^1(\Omega)} \langle \Lambda^{-1}u_1(t,\cdot),\varphi_3 \rangle_{H_D^1(\Omega)^{*}}\zeta_3 \\ &\quad -c^{-1}\eta _{H_D^1(\Omega)}\langle \Lambda^{-1}u_2(t,\cdot), \varphi_3 \rangle_{H_D^1(\Omega)^{*}}\zeta_3 + \eta g_3(t)\zeta_3. \end{align*} So, problem \eqref{e8.33} reduces to the following \begin{equation} \label{e8.36} \begin{gathered} \begin{aligned} D_{t}u_{\mathrm{l}} &=-a\Lambda u_{1}-c^{-1}du_1+c^{-1}du_2 +c^{-1}d\eta _{H_D^1(\Omega)}\langle \Lambda^{-1}u_1(t,\cdot), \varphi_3 \rangle_{H_D^1(\Omega)^{*}}\zeta_3 \\ &\quad -c^{-1}d\eta _{H_D^1(\Omega)}\langle \Lambda^{-1}u_2(t,\cdot),\varphi_3 \rangle_{H_D^1(\Omega)^{*}}\zeta_3 + d\eta g_3(t)\zeta_3 +f_{1}(t)\zeta_{1},\\ &\quad \text{in }(0,\tau)\times \Omega, \end{aligned}\\ \begin{aligned} D_{t}u_2&=-b\Lambda u_2+c^{-1}eu_1-c^{-1}eu_2 -c^{-1}e\eta _{H_D^1(\Omega)}\langle \Lambda^{-1}u_1(t,\cdot),\varphi_3 \rangle_{H_D^1(\Omega)^{*}}\zeta_3 \\ &\quad +c^{-1}e\eta _{H_D^1(\Omega)}\langle \Lambda^{-1}u_2(t,\cdot),\varphi_3 \rangle_{H_D^1(\Omega)^{*}}\zeta_3 - e\eta g_3(t)\zeta_3 +f_2(t)\zeta_2,\\ &\quad \text{in }(0,\tau)\times \Omega, \end{aligned}\\ u_{1}(0,\cdot)=u_{0,1},\quad u_2(0,\cdot)=u_{0,2},\quad \text{in } \Omega, \\ _{H_D^1(\Omega)}\langle u_i(t,\cdot),\varphi_i \rangle_{H_D^1 (\Omega)^{*}}=g_i(t),\quad t\in (0,\tau),\; i=1,2. \end{gathered} \end{equation} Now assume that the data $(\varphi_1,\varphi_2,\varphi_3,\zeta_1,\zeta_2,\zeta_3, u_{0,1},u_{0,2},g_1,g_2,g_3)$ satisfy the following properties: \begin{gather*} \varphi_i\in H_D^1(\Omega),\quad \zeta_i \in (H_D^1(\Omega)^{*},H_D^1(\Omega))_{\theta,\infty},\quad i=1,2,3; \\ u_{0,1},u_{0,2}\in H_D^1(\Omega),\quad \Delta u_{0,1},\Delta u_{0,2}\in (H_D^1(\Omega)^{*},H_D^1(\Omega))_{\theta,\infty}, \\ g_1,g_2,g_3\in C([0,\tau];\mathbb{R}), \\ _{H_D^1 (\Omega)}\langle u_{i,0},\varphi_i \rangle_{H_D^1(\Omega)^{*}}=g_i(0),\quad i=1,2,3, \\ _{H_D^1(\Omega)}\langle \Lambda^{-1}\zeta_3,\varphi_3 \rangle_{H_D^1(\Omega)^{*}}\, _{H_D^1(\Omega)}\langle \zeta_1,\varphi_1 \rangle_{H_D^1(\Omega)^{*}}\, _{H_D^1(\Omega)}\langle \zeta_2, \varphi_2 \rangle_{H_D^1(\Omega)^{*}}\ne 0. \end{gather*} Then, according to Theorem \ref{thm7.2}, with $\alpha=\beta=1$, we can conclude that the identification problem \eqref{e8.33} admits a unique solution \begin{gather*} (u_1,u_2,u_3,f_1,f_2,f_3)\in C([0,\tau];[H_D^1(\Omega)^{*}]^3) \times C([0,\tau];\mathbb{C}), \\ D_tu_1,D_tu_2\in B([0,\tau];(H_D^1(\Omega)^{*},H_D^1(\Omega))_{\theta,\infty}), \end{gather*} \begin{remark} \label{rmk8.1} \rm Note that sufficient conditions could be deduced simply by replacing the interpolation space $(H_D^1(\Omega)^{*},H_D^1(\Omega))_{\theta,\infty}$ with $\mathcal{D}(\Lambda^\theta)$, since $\mathcal{D}(\Lambda^\theta)$ for $\theta\in [1/2,1]$ coincides with the complex interpolation space $[H_D^1(\Omega)^{*},H_D^1(\Omega)]_{\theta}$, which is included in $(H_D^1(\Omega)^{*},H_D^1(\Omega))_{\theta,\infty}$, as we have already pointed out. \end{remark} \subsection*{Problem 8.3} Let us consider the identification problem consisting of recovering the $m$ scalar functions $f_{j}: [0,\tau]\to\mathbb{R}$ in the singular problem \begin{equation} \label{e8.39} \begin{gathered} D_{t}u=a_{1,1}\Delta u + a_{1,2}\Delta v + b_{1,1}(x)u + b_{1,2}(x)v + h_1(t,x) + \sum_{j=1}^m f_{j}(t)z_{1,j},\\ \text{in }(0,\tau)\times \Omega, \\ D_{t}v=a_{2,1}\Delta u + a_{2,2}\Delta v + b_{2,1}(x)u + b_{2,2}(x)v + h_2(t,x) + \sum_{j=1}^m f_{j}(t)z_{2,j},\\ \text{in }(0,\tau)\times \Omega, \\ u(0,\cdot)=u_{0,1},\quad v(0,\cdot)=u_{0,2},\quad \text{in } \Omega, \\ u=v=0, \quad \text{in } (0,\tau)\times \partial\Omega, \end{gathered} \end{equation} under the following $m$ additional conditions \begin{gather} \Psi_j[u(t,\cdot)]=g_{j}(t),\quad t\in(0,\tau),\; j=1,\dots,r,\label{e8.40}\\ \label{e8.41} \Psi_j[v(t,\cdot)]=g_{j}(t),\quad t\in(0,\tau),\; j=r+1,\dots,m, \end{gather} where $\Omega$ is a (possibly unbounded) domain in ${\mathbb R}^n$ with a smooth boundary, $a_{i,j}\in {\mathbb R}$, $b_{i,j}\in C({\overline \Omega};{\mathbb R})$, $i,j=1,2$. Therefore, choosing $X_0=L^p(\Omega)$, $p\in (1,+\infty)$, and $\mathcal{D}(\Delta)=W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$, the well known resolvent estimates for our operator $\Delta$ hold in $L^p(\Omega)$, so that $\Delta$ generates an analytic semigroup of linear bounded operators. To develop our strategy, we generalize to the case $p\in (1,+\infty)$ the results proved for $p=2$ in \cite{DM}. For this purpose we introduce in the space $X=L^p(\Omega)\times L^p(\Omega)$ the linear unbounded operator $\mathcal{A}$ defined by \begin{equation} \mathcal{A}=\begin{bmatrix} a_{1,1}\Delta & a_{1,2}\Delta\\ a_{2,1}\Delta & a_{2,2}\Delta \end{bmatrix}, \quad \mathcal{D}(\mathcal{A})=\mathcal{D}(\Delta)\times \mathcal{D}(\Delta). \end{equation} Some simple algebraic computations yield the following formula for the resolvent \begin{equation} \label{e8.43} \begin{aligned} &(\mathcal{A}-\lambda I)^{-1}\\ &=\begin{bmatrix} a_{2,2}\Delta -\lambda I & - a_{1,2}\Delta\\ - a_{2,1}\Delta & a_{1,1}\Delta -\lambda I \end{bmatrix} \big[(a_{1,1}\Delta -\lambda I)(a_{2,2}\Delta -\lambda I) - a_{1,2}a_{2,1}\Delta^2\big]^{-1}. \end{aligned} \end{equation} Observe that the determinant operator $D=(a_{1,1}\Delta -\lambda I)(a_{2,2}\Delta -\lambda I) - a_{1,2}a_{2,1}\Delta^2$ coincides with \begin{equation} \label{e8.45} D=\lambda ^2I - \lambda (a_{1,1}+a_{2,2})\Delta+(a_{1,1}a_{2,2}- a_{1,2}a_{2,1})\Delta^2. \end{equation} Suppose now that \begin{gather} \label{e8.46} a_{1,1}\ge 0,\quad a_{2,2}\ge 0,\quad a_{1,1}+a_{2,2}>0,\quad a_{1,1}a_{2,2}- a_{1,2}a_{2,1}>0,\\ \label{e8.47} (a_{1,1}-a_{2,2})^2+4a_{1,2}a_{2,1}\ge 0. \end{gather} We note that the last inequality in \eqref{e8.46} can be weakened to $\ge$ if $\Omega$ is bounded, while condition \eqref{e8.47}, not required in \cite[Lemma 1, p.185]{DM}, is necessary to ensure that the equation $\lambda ^2 - \lambda (a_{1,1}+a_{2,2})+(a_{1,1}a_{2,2}- a_{1,2}a_{2,1})=0$ admits two real solutions $0<\lambda_1\le\lambda_2$, since (cf. \eqref{e8.47}) \begin{equation} (a_{1,1}+a_{2,2})^2-4(a_{1,1}a_{2,2}- a_{1,2}a_{2,1}) =(a_{1,1}-a_{2,2})^2+4a_{1,2}a_{2,1}\ge 0. \end{equation} Now, in contrast with \cite{DM} we use the factorization $D=(\lambda - \lambda_1\Delta)(\lambda - \lambda_2\Delta)$. From the identity \begin{equation} \label{e8.48} D=\lambda_1\lambda_2(\lambda_1^{-1}\lambda - \Delta)(\lambda_2^{-1}\lambda - \Delta) \end{equation} we deduce the resolvent estimate \begin{equation} \label{e8.49} \|(\mathcal{A}-\lambda I)^{-1}\|_{\mathcal{L}(X)}\le C(1+|\lambda|)^{-1},\quad \operatorname{Re}\lambda\ge 0. \end{equation} Therefore, $\mathcal{A}$ generates an analytic semigroup on $X$. On the other hand the matrix operator \begin{equation} \label{e8.50} \mathcal{B}=\begin{bmatrix} b_{1,1}I & b_{1,2}I\\ b_{2,1}I & b_{2,2}I \end{bmatrix} \end{equation} belongs to $\mathcal{L}(X)$, so that $\mathcal{A}+\mathcal{B}$, with $\mathcal{D}(\mathcal{A}+\mathcal{B})=\mathcal{D}(\mathcal{A})$, generates an analytic semigroup on $X$, too. Now we make the following assumptions: \begin{gather*} \det \begin{bmatrix} \Psi_{1}[z_{1,1}] & \dots & \Psi_{1}[z_{1,m}] \\ \dots & \dots & \dots \\ \Psi_{r}[z_{1,1}] & \dots & \Psi_{r}[z_{1,m}] \\ \Psi_{r+1}[z_{2,1}] & \dots & \Psi_{r+1}[z_{2,m}] \\ \dots & \dots & \dots \\ \Psi_{m}[z_{2,1}] & \dots & \Psi_{1}[z_{2,m}] \end{bmatrix}\ne 0, \\ u_0,v_0\in \mathcal{D}(\Delta),\quad (\mathcal{A}+\mathcal{B})(u_0,v_0)^T\in (X,\mathcal{D}(\Delta))_{\theta,\infty} \times (X,\mathcal{D}(\Delta))_{\theta,\infty}:=Z_{\theta}\times Z_{\theta}, \\ (z_{1,j},z_{2,j})^T\in Z_{\theta},\quad g_j\in C^1([0,\tau]; {\mathbb R}),\ \varphi_j\in L^q(\Omega),\ j=1,\dots,m,\; 1/p+1/q=1, \\ (h_1,h_2)^t\in C([0,\tau];Z_{\theta}\times Z_{\theta}). \end{gather*} The characterization of the space $Z_\theta$ can be found in \cite[Theorem 4.4.1, p.321]{TR}. Now we define the linear bounded functionals $\Psi_j$, $j=1,\dots,m$, by \begin{equation} \label{e85.1} \Psi_j[u]=\int_{\Omega} \varphi_j(x)u(x)\,dx. \end{equation} Then, by Corollary \ref{coro5.2} we conclude that the identification problem \eqref{e8.39}--\eqref{e8.41} admits a strict solution $(u,v,f_1,\dots,f_m)$ with the following additional regularity: \begin{gather*} D_tu,D_tv\in B([0,\tau];Z_{\theta}),\\ a_{1,1}\Delta u + a_{1,2}\Delta v + b_{1,1}(\cdot)u + b_{1,2}(\cdot)v\in C([0,\tau];L^p(\Omega))\cap B([0,\tau];Z_{\theta}), \\ a_{2,1}\Delta u + a_{2,2}\Delta v + b_{2,1}(\cdot)u + b_{2,2}(\cdot)v\in C([0,\tau];L^p(\Omega))\cap B([0,\tau];Z_{\theta}). \end{gather*} Observe that this strategy works also if $L^p(\Omega)$ is replaced with $C(\overline{\Omega})$ and related functionals $\Psi_i\in C(\overline{\Omega})^*$. \section{Identification problems for PDE's corresponding to $\beta\in (0,1)$} \subsection*{Problem 9.1} Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with a $C^\infty$-boundary $\partial\Omega$. We want to recover the scalar functions $f_j:[0,\tau]\to \mathbb{C}$, $j=1,\dots,m$, in the initial boundary value problem \begin{gather} \label{e9.1} \frac{\partial u}{\partial t}(t,x) + A(x,D_x)u(t,x) =\sum_{j=1}^m f_j(t)z_{j}(x),\quad (t,x) \in (0,\tau) \times \Omega, \\ \label{e9.2} u(0,x) = u_0(x),\quad x \in \Omega,\\ \label{e9.3} a(x)D_\nu u(t,x) + \alpha(x)\cdot \nabla u(t,x)+b(t,x)u(t,x)= 0, \quad (t,x) \in (0,\tau) \times \partial\Omega, \end{gather} under the $m$ additional pieces of information \begin{equation}\label{e9.4} \int_\Omega \eta_i(x)u(t,x)dx = g_i(t),\quad 0 \leq t \leq \tau,\; i=1,\dots,m, \end{equation} along with the consistency conditions \begin{equation} \label{e9.5} \int_\Omega \eta_i(x)u_0(x)dx = g_i(0), \quad i=1,\dots,m. \end{equation} Here \begin{equation} \label{e9.6} -A(x,D_x) = \sum_{i,j=1}^n a_{i,j}(x)D_{x_i}D_{x_j} + \sum_{i=1}^n a_{i}(x)D_{x_i}+a_{0}(x) \end{equation} is a second-order elliptic differential operator with real-valued $C^\infty$-coefficients on ${\overline \Omega}$ such that \begin{equation}\label{e9.7} a_{j,i}(x)=a_{i,j}(x),\quad \sum_{i,j=1}^n a_{i,j}(x)\xi_i\xi_j\ge c_0|\xi|^2, \quad (x,\xi)\in {\overline \Omega}\times \mathbb{R}^n, \end{equation} $c_0$ being a positive constant. Concerning the linear boundary differential operator defined, for all $(t,x) \in (0,\tau) \times \partial\Omega$, by \begin{equation} \label{e9.8} {\widehat A}(x,D_x)u(t,x)=a(x)D_\nu u(t,x) + \alpha(x)\cdot \nabla u(t,x)+b(x)u(t,x), \end{equation} we assume that $a$, $b$ and $\alpha$ are real-valued $C^\infty$-functions and a vector field on $\partial \Omega$, respectively, such that $Tu=\alpha\cdot \nabla u$ is a real $C^\infty$-tangential operator on $\partial \Omega$, $D_\nu$ standing for the conormal derivative associated with the matrix $\big(a_{i,j}(x)\big)$; i.e., \begin{equation} \label{e9.9} D_\nu = \Big(\sum_{i,j=1}^n a_{i,j}(x)n_i(x)n_j(x)\Big)^{-1} \sum_{i,j=1}^n a_{i,j}(x)D_{x_i}, \end{equation} $n(x)=\big(n_1(x),\dots,n_n(x)\big)$ denoting the outward unit normal vector to $\partial \Omega$ at $x$. Assume further (cf. \cite[p. 515]{TA2} that the vector field $\alpha$ does not vanish on $\Gamma_0=\{x\in \partial \Omega: a(x)=0\}$ and the function $t\to a(x(t,x_0))$ has zeros of even order not exceeding some value $2k_1$ along the integral curve $x'(t,x_0)=\alpha(x(t,x_0))$ satisfying the initial condition $x(0,x_0)=x_0$, with $x_0\in \Gamma_0$. In other words, the so-called $(H)_\delta$-condition holds with $\delta=\delta_1=(1+2k_1)^{-1}$. It is shown on p. 516 in \cite{TA2} that the operator $L$ defined by \begin{gather} \label{e9.10} \mathcal{D}(L)=\big\{u\in L^2(\Omega): A(\cdot,D_x)u\in L^2(\Omega),\; {\widehat A}(\cdot,D_x)u=0\ {\rm on\ \partial \Omega} \big\}, \\ Lu=A(\cdot,D_x)u,\quad u\in \mathcal{D}(L). \label{e9.11} \end{gather} satisfies in $L^2(\Omega)$ the resolvent estimate \begin{equation} \label{e9.12} \| (\lambda+L)^{-1}\|_{\mathcal{L}(L^2(\Omega))} \leq C(1+|\lambda|)^{-(1+\delta)/2} \end{equation} for all $\lambda$ with a large enough modulus belonging to the sector \begin{equation} \label{e9.13} \Sigma_\varphi := \{ \lambda \in \mathbb{C}\setminus \{0\}: |\operatorname{arg} \lambda| \le \varphi\}, \quad \varphi \in (\pi/2,\pi). \end{equation} If we consider the subelliptic case $k=(1+\delta)/2>1/2$, we can immediately apply Corollary \ref{coro5.3}. \subsection*{Problem 9.2} Here we deal with a problem - similar to Problem 9.1 - in the reference space of H\"older-continuous functions $X=\big(C^\alpha({\overline \Omega}),\|\cdot\|_\alpha\big)$, when $\alpha\in (0,1)$ and the boundary $\partial\Omega$ of $\Omega$ is of class $C^{4m}$ for some positive integer $m$. In this case the linear differential operator $A$ is defined by \begin{gather}\label{e9.14} \mathcal{D}(A)=\big\{u\in C^{2m+\alpha}({\overline \Omega}): D^\gamma u=0 \text{ on }\partial \Omega,\; |\gamma|\le m-1 \big\},\\ \label{9.14} Au(x)=\sum_{|\gamma|\le 2m} a_{\gamma}(x)D^\gamma u(x),\quad x\in \Omega,\; u\in \mathcal{D}(A), \end{gather} where $\beta$ is a usual multi-index with $|\beta|=\sum_{j=1}^n \beta_j$ and $D^\beta =\prod_{j=1}^n (-iD_{x_j})^{\beta_j}$. Assume that the coefficients $a_\gamma:{\overline \Omega}\to \mathbb{C}$ of $A$ satisfy the following conditions \begin{itemize} \item[(i)] $a_\gamma\in C^\alpha({\overline \Omega};\mathbb{C})$ for all $|\gamma|\le 2m$; \item[(ii)] $a_\gamma(x)\in \mathbb{R}$ for all $x\in {\overline \Omega}$ and $|\gamma|=2m$; \item[(iii)] there exists a positive constant $M\ge 1$ such that \begin{equation} \label{e9.16} M^{-1}|\xi|^{2m}\le \sum_{|\gamma|=2m} a_{\gamma}(x)\xi^\gamma\le M|\xi|^{2m}, \quad (x,\xi)\in {\overline \Omega}\times \mathbb{R}^n. \end{equation} \end{itemize} Then there exist $\lambda, \varepsilon\in \mathbb{R}_+$ such that the spectrum of the operator $A+\lambda$ satisfies \begin{equation} \label{e9.17} \sigma(A+\lambda)\subset S_{(\pi/2)-\varepsilon} =\big\{z\in \mathbb{C}\setminus \{0\}: |\operatorname{arg} z|\le \frac{\pi}{2}-\varepsilon \big\} \cup \{0\}. \end{equation} Moreover, for any $\mu\in (\pi/2,\pi)$ there exists a positive constant $C(\mu)$ such that \begin{equation} \label{e9.18} \| (\lambda-A)^{-1}\|_{\mathcal{L}(C^{\alpha}({\overline \Omega}))} \leq C(\mu)|\lambda|^{(\alpha/2m)-1},\quad \lambda\in S_\mu. \end{equation} For details cf. Satz 1 and Satz 2 in \cite{WA}, where we choose $l=\alpha$, $\beta=1-(\alpha/2m)$. As an example, we can consider the problem consisting in recovering the vector-function $(u,f_1,\dots,f_p)$, where $f_j:[0,\tau]\to \mathbb{C}$, $j=1,\dots,p$, satisfying \begin{gather}\label{e9.19} \frac{\partial u}{\partial t}(t,x) + (A+\lambda)u(t,x) =\sum_{j=1}^p f_j(t)z_{j}(x)+h(t,x),\quad (t,x) \in [0,\tau] \times \Omega, \\ \label{e9.20} u(0,x) = u_0(x),\quad x \in \Omega,\\ \label{e9.21} D^\gamma u(t,x) = 0, \quad (t,x) \in [0,\tau] \times \partial\Omega,\; |\gamma|\le m-1, \end{gather} under the $p$ additional conditions \begin{equation} \label{e9.22} u(t,{\overline x}_j) = g_j(t),\quad t \in [0,\tau],\; j=1,\dots,p, \end{equation} where ${\overline x}_j$, $j=1,\dots,p$, are $p$ fixed points in $\Omega$. We remark that $A$ is not sectorial and $\mathcal{D}(A)\subset \big\{u\in C^{\alpha}(\Omega): u=0\ {\rm on\ \partial \Omega}\}$. In view of Corollary \ref{coro5.3} we can establish our identification result. \begin{theorem} \label{thm9.1} Let $\theta\in (\alpha/(2m),1)$ and $\beta=1-\alpha/(2m)>0$. Let $Y_A^{\gamma,\infty}$ be either of the spaces $(X,\mathcal{D}(A))_{\gamma,\infty}$ or $X^\gamma_A$. Let $u_0\in \mathcal{D}(A)$, $Au_0\in Y_A^{\gamma,\infty}$, $g_j\in C^1([0,\tau];\mathbb{C})$, $j=1,\dots,p$, $h\in C([0,\tau];X)\cap B([0,\tau];Y_A^{\gamma,\infty})$, $u_0({\overline x}_j) = g_j(0)$, $j=1,\dots,p$, with \[ \det \begin{bmatrix} z_{1}({\overline x}_1) & \dots & z_{p}({\overline x}_1)\\ \dots & \dots & \dots\\ z_{1}({\overline x}_p) & \dots & z_{p}({\overline x}_p) \end{bmatrix} \ne 0. \] Then problem \eqref{e9.19}-\eqref{e9.22} admits a unique strict solution \[ (u,f_1,\dots,f_p)\in \big[C^1([0,\tau];C^\alpha({\overline \Omega})\cap C([0,\tau]; \mathcal{D}(A))\big]\times C([0,\tau];\mathbb{C})^p \] such that $u(t,\cdot)\in C^{2m+\alpha}({\overline \Omega})$ for all $t\in [0,\tau]$, $D_tu\in B([0,\tau];Y_A^{\theta-(\alpha/2m),\infty})$, $Au\in \big[C^{\theta-(\alpha/2m)}([0,\tau];C^\alpha({\overline \Omega})) \cap B([0,\tau];Y_A^{\theta-(\alpha/2m),\infty})\big]$. \end{theorem} \subsection*{Problem 9.3} Under the same assumptions on $m$ and $\Omega$ as in Problem 9.2 we introduce the linear operators $A$ and $D$ by the formulae. \begin{gather} \label{e9.23} \mathcal{D}(A)=\big\{u\in C^{2m+\alpha}({\overline \Omega}): D^\beta u=0\text{ on } \partial \Omega,\; |\beta|\le m-1 \big\}, \\ \label{e9.24} Au(x)=\sum_{|\beta|\le 2m} a_{\beta}(x)D^\beta u(x),\quad x\in \Omega,\ u\in \mathcal{D}(A), \\ \label{e9.25} \mathcal{D}(D)=\big\{v\in C^{2p+\alpha}({\overline \Omega}): D^\gamma v=0\ {\rm on\ \partial \Omega}, \; |\gamma|\le p-1 \big\}, \\ \label{e9.26} Dv(x)=\sum_{|\gamma|\le 2p}\,d_{\gamma}(x)D^\gamma v(x),\quad x\in \Omega,\; v\in \mathcal{D}(D), \end{gather} where $a_\beta,d_\gamma\in C^{\alpha}({\overline \Omega})$ and $D^\beta$, $D^\gamma$ are defined as in Problem 9.2. Let us introduce the operators $B$ and $C$ defined by \begin{gather}\label{9.26} Bv(x)=\sum_{|\gamma|\le 2p-1} b_{\gamma}(x)D^\gamma v(x),\quad x\in \Omega,\; v\in C^{2p-1+\alpha}({\overline \Omega}), \\ \label{9.27} Cu(x)=\sum_{|\beta|\le 2m-1} c_{\beta}(x)D^\beta u(x),\quad x\in \Omega,\; u\in C^{2m-1+\alpha}({\overline \Omega}). \end{gather} In view of \cite[Satz 1]{WA} the following estimate holds in the set $|\operatorname{arg} \lambda|\le (\pi/2)+\varepsilon$, $\operatorname{Re}\lambda\ge \lambda_0$: \begin{equation} \label{e9.29} \begin{aligned} &|\lambda|\|u\|_{C({\overline \Omega})} + |\lambda|^{(2m-\alpha)/(2m)}\|u\|_{C^{\alpha}({\overline \Omega})} + |\lambda|^{(1-\alpha)/(2m)}\|u\|_{C^{2m-1+\alpha}({\overline \Omega})} + \|u\|_{C^{2m+\alpha}({\overline \Omega})} \\ &\le C_1\|(A+\lambda)u\|_{C^{\alpha}({\overline \Omega})}. \end{aligned} \end{equation} Whence we deduce the estimates \begin{gather} \label{e9.30} \|C(A+\lambda)^{-1}f\|_{C^{\alpha}({\overline \Omega})}\le C_2|\lambda|^{(-1+\alpha)/(2m)} \|f\|_{C^{\alpha}({\overline \Omega})}, \\ \label{e9.31} \|B(D+\lambda)^{-1}f\|_{C^{\alpha}({\overline \Omega})}\le C_3|\lambda|^{(-1+\alpha)/(2p)}\|f\|_{C^{\alpha}({\overline \Omega})}. \end{gather} Consequently, conditions \eqref{e6.1}-\eqref{e6.3} hold with \begin{equation} \label{e9.32} \beta_1=1-\frac{\alpha}{2m},\quad \beta_2=1-\frac{\alpha}{2p},\quad \gamma_1=\frac{1-\alpha}{2m},\quad \gamma_2=\frac{1-\alpha}{2m}. \end{equation} Therefore, we are allowed to apply Theorem \ref{thm6.1} and Corollaries 6.2, 6.3 to the problem consisting in finding a quadruplet $(u,v,f_1,f_2)$ solving \begin{gather} \label{e9.33} \begin{aligned} &\frac{\partial u}{\partial t}(t,x) + A(x,D_x)u(t,x)+ B(x,D_x)v(t,x)\\ &=f_1(t)z_{1,1}(x) +f_2(t)z_{1,2}(x)+h_1(t,x),\quad (t,x) \in (0,\tau) \times \Omega, \end{aligned}\\ \label{e9.34} \begin{aligned} &\frac{\partial v}{\partial t}(t,x) + C(x,D_x)u(t,x)+ D(x,D_x)v(t,x)\\ &=f_1(t)z_{2,1}(x) +f_2(t)z_{2,2}(x) +h_2(t,x),\quad (t,x) \in (0,\tau) \times \Omega, \end{aligned}\\ \label{e9.35} u(0,x) = u_0(x), \quad v(0,x) = v_0(x), \quad x \in {\overline \Omega},\\ \label{e9.36} u(t,{\overline x})=g_1(t),\quad v(t,{\widetilde x})=g_2(t),\quad t \in [0,\tau], \end{gather} where \begin{equation} \label{e9.37} {\overline x}, {\widetilde x}\in {\overline \Omega},\quad u(0,{\overline x})=g_1(0),\quad v(0,{\widetilde x})=g_2(0). \end{equation} Indeed, operator $\mathcal{A}$ defined in $X\times X=C^\alpha({\overline \Omega})\times C^\alpha({\overline \Omega})$ by \[ \mathcal{D}(\mathcal{A})=\mathcal{D}(A)\times \mathcal{D}(D),\quad \mathcal{A} \begin{bmatrix} u\\ v \end{bmatrix} = \begin{bmatrix} Au + Bv\\ Cu + Dv \end{bmatrix} \] satisfies, for all $\lambda$ with large modulus belonging to the sector $|\operatorname{arg} \lambda|\le (\pi/2)+\varepsilon$, the resolvent estimate \begin{equation} \label{e9.38} \|(\lambda+\mathcal{A})^{-1}\|_{\mathcal{L}(C^\alpha({\overline \Omega}) \times C^\alpha({\overline \Omega}))}\le c|\lambda|^{-\beta}, \end{equation} where \begin{equation}\label{e9.39} \beta=\min\Big\{1-\frac{\alpha}{2m},1-\frac{\alpha}{2p}\Big\}. \end{equation} We confine ourselves to translating Corollary \ref{coro6.3} to this new situation. \begin{theorem} \label{thm9.2} Let $\beta$ be defined by \eqref{e9.39} and $\theta\in (1-\beta,1)$, $g_1,g_2\in C^1([0,\tau];\mathbb{C})$, $u_0\in \mathcal{D}(A)$, $v_0\in \mathcal{D}(D)$, $Au_0+Bv_0\in (C^\alpha({\overline \Omega});\mathcal{D}(A))_{\theta,\infty}$, \begin{gather*} Cu_0+Dv_0\in (C^\alpha({\overline \Omega});\mathcal{D}(D))_{\theta,\infty},\\ z_{1,1},z_{1,2}\in (C^\alpha({\overline \Omega});\mathcal{D}(A))_{\theta,\infty},\quad z_{2,1},z_{2,2}\in (C^\alpha({\overline \Omega});\mathcal{D}(D))_{\theta,\infty},\\ h_1\in C([0,\tau];C^\alpha({\overline \Omega}))\cap B([0,\tau]; (C^\alpha({\overline \Omega});\mathcal{D}(A))_{\theta,\infty}),\\ h_2\in C([0,\tau];C^\alpha({\overline \Omega}))\cap B([0,\tau]; (C^\alpha({\overline \Omega});\mathcal{D}(D))_{\theta,\infty}), \end{gather*} satisfying the consistency conditions \eqref{e9.37} as well the solvability condition \[ z_{1,1}({\overline x})z_{2,2}({\widetilde x})-z_{1,2}({\overline x})z_{2,1}({\widetilde x})\ne 0. \] Then problem \eqref{e9.33}-\eqref{e9.36} admits a unique strict solution $(u,v,f_1,f_2)$ in the space $ \big[C^1([0,\tau];C^\alpha({\overline \Omega})\cap C([0,\tau];\mathcal{D}(A))\big] \times \big[C^1([0,\tau];C^\alpha({\overline \Omega}))\cap C([0,\tau]; \mathcal{D}(D))\big]$\\ $\times C([0,\tau];\mathbb{C})\times C([0,\tau];\mathbb{C})$ such that \begin{gather*} D_tu\in B([0,\tau];(C^\alpha({\overline \Omega};\mathcal{D}(A))_{\theta+\beta-1,\infty}),\quad D_tv\in B([0,\tau];(C^\alpha({\overline \Omega};\mathcal{D}(D))_{\theta+\beta-1,\infty}), \\ Au+Bv\in C^{\theta+\beta-1}([0,\tau];C^\alpha({\overline \Omega})),\quad Cu+Dv\in C^{\theta+\beta-1}([0,\tau];C^\alpha({\overline \Omega})). \end{gather*} \end{theorem} We could also handle the system \begin{gather} \label{e9.40} \begin{aligned} &\frac{\partial}{\partial t} \begin{bmatrix} y_1(t,x)\\ \dots\\ y_n(t.x) \end{bmatrix} + \begin{bmatrix} A_{1}+ B_{1,1} & B_{1,2} & \dots & B_{1,n}\\ B_{2,1} & A_2+ B_{2,2} & \dots & B_{2,n}\\ \dots &\dots &\dots &\dots \\ B_{n,1} & B_{n,2} & \dots & A_{n}+ B_{n,n} \end{bmatrix} \begin{bmatrix} y_1(t.x)\\ \dots\\ y_n(t.x) \end{bmatrix} \\ &= \begin{bmatrix} h_1(t.x)\\ \dots\\ h_n(t.x) \end{bmatrix} + \sum_{j=1}^n f_j(t.x) \begin{bmatrix} z_{1,j}\\ \dots\\ z_{n,j} \end{bmatrix},\quad t\in [0,\tau], \end{aligned}\\ \label{e9.41} y_j(0)=y_{0,j},\quad j=1,\dots,n,\\ \label{e9.42} \Psi_j[y_j(t)]=g_j(t),\quad t\in [0,\tau],\; j=1,\dots,n, \end{gather} in the space $\big[C^\alpha({\overline \Omega})\big]^n$. Here the $A_j$'s and the $B_{i,j}$'s are linear differential operators like in Problem 9.2 and Problem 9.1, respectively such that $\operatorname{ord} B_{i,j}<\operatorname{ord} A_j$, for all $i,j=1,\dots,p$. Since a bound of type \eqref{e7.5} follows from \cite[Satz 1]{WA} the previous argument applies immediately, e.g., when functionals $\Psi_j$ are defined by $\Psi_j[y_j(\cdot)]=y_j({\overline x}^{(j)})$, $j=1,\dots,p$, ${\overline x}^{(1)},\dots {\overline x}^{(p)}$ being $p$ fixed points in ${\overline \Omega}$. The details are left to the reader. \subsection*{Problem 9.4} Let us consider the degenerate parabolic system \begin{gather} \label{e9.43} \frac{\partial u}{\partial t}(t,x) = \Delta (a(x)u(t,x)) + b(x)v(t,x) + f_1(t)z_{1,1}(x)+f_2(t)z_{1,2}(x),\\ \label{e9.44} \frac{\partial v}{\partial t}(t,x) = c(x)u(t,x) + \Delta(d(x)v(t,x)) + f_1(t)z_{2,1}(x) + f_2(t)z_{2,2}(x), \\ (t,x) \in (0,\tau) \times \Omega, \nonumber\\ \label{e9.45} u(0,x) = u_0(x), \quad v(0,x) = v_0(x), \quad x \in \Omega, \\ \label{e9.46} a(x)u(t,x) = 0 = d(x)v(t,x), \quad (t,x) \in (0,\tau) \times \partial\Omega, \\ \label{e9.47} \int_\Omega \eta_1(x)u(t,x)dx = g_1(t),\quad \int_\Omega \eta_2(x)v(t,x)dx = g_2(t), \ 0 \leq t \leq \tau, \end{gather} along with the consistency conditions \begin{equation} \label{e9.48} \int_\Omega \eta_1(x)u_0(x)dx = g_1(0), \quad \int_\Omega \eta_2(x)v_0(x)dx = g_2(0), \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n \geq 1$, with a $C^2$-boundary $\partial\Omega$, while $a$, $b$, $c$, $d$ are functions in $C(\overline\Omega;\mathbb{R})$ such that $a(x)>0$ and $d(x) > 0$ a.e. in $\Omega$. Moreover, $z_{i,j} \in L^2(\Omega)$, $i,j=1,2$, $u_0, v_0 \in H_0^1(\Omega) \cap H^2(\Omega)$, $g_i\in C^1([0,\tau];\mathbb{C})$, $i = 1,2$. Our task consists in recovering $(u,v,f_1,f_2)$. We recall \cite[p. 83]{FY} that, if $$ a^{-1} \in L^r(\Omega) \quad\text{with } \begin{cases} r \geq 2 & \text{when } n = 1,\\ r > 2, &\text{when } n = 2,\\ r \geq n, &\text{when } n \geq 3, \end{cases} $$ then, for any function $e$ enjoying the same properties as $a$, operator $K(e)$ defined by \[ \mathcal{D}(K(e)) := \{ u \in L^2(\Omega): eu \in H_0^1(\Omega) \cap H^2(\Omega) \}, \quad K(e)u := -\Delta (eu),\quad u \in \mathcal{D}(K) \] satisfies the estimate $$ \| (\lambda I + K(e))^{-1} f \|_{L^2(\Omega)} \leq c|\lambda|^{-(2r-n)/2r}\|f\|_{L^2(\Omega)}, $$ for all $\lambda$ in a sector containing the half-plane $\operatorname{Re} z \geq 0$. Therefore, $\alpha = 1$, $\beta = (2r-n)/2r$. Let us assume $1/a\in L^{r_1}(\Omega)$, $1/d\in L^{r_2}(\Omega)$. Consequently, estimates \eqref{e6.2}) hold with $\alpha = 1$, $\beta_1 = (2r_1-n)/2r_1$, $\beta_2 = (2r_2 - n)/2r_2$ for operators $K(a)$ and $K(d)$. Since the multiplication operators generated by $b$ and $c$ are bounded in $L^2(\Omega)$, $\beta$ in \eqref{e6.12} is given by $$ \beta = \min\{\beta_1,\beta_2\} = \min \Big\{ 1 - \frac{n}{2r_1}, 1 - \frac{n}{2r_2} \Big\} \ge \frac{1}{2}, $$ since $r_j\ge n$, $j=1,2$. Let us assume \begin{gather*} \int_\Omega \eta_1(x) z_{1,1}(x)dx\int_\Omega \eta_2(x) z_{2,2}(x)dx - \int_\Omega \eta_1(x) z_{1,2}(x)dx \int_\Omega \eta_2(x) z_{2,1}(x)dx \neq 0, \\ g_1,g_2 \in C^1([0,\tau];\mathbb{C}), \quad 1-\beta < \theta < 1. \end{gather*} Let \begin{equation} \label{e9.49} \mathcal{A}=\begin{bmatrix} K(a) & -M_{b}\\ -M_{c} & K(d) \end{bmatrix}, \end{equation} As $(L^2(\Omega) \times L^2(\Omega),\mathcal{D}(\mathcal{A}))_{\theta,\infty} = (L^2(\Omega),\mathcal{D}(K(a)))_{\theta,\infty} \times (L^2(\Omega),\mathcal{D}(K(d)))_{\theta,\infty}$ if \begin{align*} &(\Delta(a(\cdot)u_0)+ b(\cdot)v_0, c(\cdot)u_0 + \Delta(d(\cdot)v_0))\\ &\in (L^2(\Omega),\mathcal{D}(K(a)))_{\theta,\infty} \times (L^2(\Omega), \mathcal{D}(K(d)))_{\theta,\infty}, \end{align*} $z_{11},z_{12} \in (L^2(\Omega),D(K(a)))_{\theta,\infty}$, $z_{21}, z_{22} \in (L^2(\Omega),\mathcal{D}(K(d)))_{\theta,\infty}$, from Corollary \ref{coro6.3} we can conclude that problem \eqref{e9.43}--\eqref{e9.47}, endowed with the consistency condition \eqref{e9.48} admits a unique global strict solution $((u,v),f_1,f_2) \in C([0,\tau];\mathcal{D}(K(a)) \times \mathcal{D}(K(d))) \times C([0,\tau];\mathbb{C})\times C([0,\tau];\mathbb{C})$ such that $(D_tu,D_tv)^T\in B([0,\tau]; (L^2(\Omega)\times L^2(\Omega), \mathcal{D}(K(a))\times \mathcal{D}(K(d)))_{\theta-(1-\beta),\infty})$, $\mathcal{A}(u,v)^T\in C^{\theta-(1-\beta)}([0,\tau];L^2(\Omega) \times L^2(\Omega))\cap B([0,\tau];(L^2(\Omega)\times L^2(\Omega), \mathcal{D}(K(a))\times \mathcal{D}(K(d)))_{\theta-(1-\beta),\infty})$. More generally, we could deal with an analogous doubly degenerate problem related to the system \begin{gather*} \frac{\partial}{\partial t}(m(x)u(t,x)) = \Delta (a(x)u(t,x)) + b(x)v(t,x) + f_1(t)z_{1,1}(x)+f_2(t)z_{1,2}(x), \\ \frac{\partial}{\partial t}(n(x)v(t,x)) = c(x)u(t,x) + \Delta(d(x)v(t,x)) + f_1(t)z_{2,1}(x) + f_2(t)z_{2,2}(x), \\ (t,x) \in (0,\tau) \times \Omega \end{gather*} $m$ and $n$ being positive and continuous functions on $\Omega$, using the change of unknowns defined by $m(x)u = u_1$, $n(x)v = v_1$. Notice that then we must make continuity assumptions on the behaviour on the boundary of functions $b/n$, $c/m$, $a/m$, $d/n$. \subsection*{Acknowledgments} This research was partially financed by the funds P.U.R. of the Universit\`a degli Studi di Milano and the project PRIN 2008 ``Analisi Matematica nei Problemi Inversi per le Applicazioni" of the Italian Ministero dell'Istruzione, dell'Universit\`a e della Ricerca (M.I.U.R.). The first two authors are members of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM) \begin{thebibliography}{99} \bibitem{AF1} AL Horani, M. H. and Favini, A.; \emph{Degenerate First Order Problems in Banach Spaces}, in Differential Equations Inverse and Direct Problems, edited by A.Favini and A.Lorenzi. Taylor and Francis Group, pp.1-15, Boca Raton, USA, 2006. \bibitem{AF2} AL Horani, M. H. and Favini A.; \emph{An identification problem for first order degenerate differential equations}, Journal Optim. Th. Appl. 130 (2006), 41--60. \bibitem{AFL} AL Horani, M. H. Favini, A. and Lorenzi A.; \emph{Second-order degenerate identification differential problems}, Journal Optim. Th. Appl. 141 (2009), 13--36. \bibitem{AS} Asanov, A. and Atamanov E.R.; \emph{Nonclassical and inverse problems for pseudoparabolic equations}, VSP, Utrecht, Holland, 1997. \bibitem{BF} Bensoussan, A. and Frehse, J.; \emph{Regularity results for nonlinear elliptic systems and applications}, Applied Mathematical Sciences, 151. Springer-Verlag, Berlin, 2002. \bibitem{CK} Capasso, V. and Kunisch K.; \emph{A reaction diffusion system arising in modelling man-environment diseases}, Quart. Appl. Math. 46 (1988), 431-450. \bibitem{CR} Cross, R.; \emph{Multivalued linear operators}, M. Dekker, New York, USA, 1998. \bibitem{DM} Dal Passo, R. and de Mottoni, P.; \emph{On a paper: ``On the general linear coupled system for diffusion in media with two diffusivities''} [by A. I. Lee and J. M. Hill, J. Math. Anal. Appl. 89 (1982), 530--557; MR0677744 (84j:76047)], J. Math. Anal. Appl. 106 (1985), 184--187. \bibitem{E} K. Engel; \emph{Operator matrices and systems of evolution equations}, preprint, 125-127. \bibitem{FL1} Favini, A. and Lorenzi A.; \emph{Singular integro-differential equations of parabolic type and inverse problems}, Mathem. Models and Methods in Applied Sciences 13 (2003), 1745--1766. \bibitem{FL2} Favini, A. and Lorenzi, A.; \emph{Identification problems in singular integro-differential equations of parabolic type I}, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Math. Anal. 12 (2005), 303--321. \bibitem{FL3} Favini, A. and Lorenzi, A.; \emph{Identification problems in singular integro-differential equations of parabolic type II}, Nonlinear Analysis: Theory, Methods and Applications 56 (2004), 879--904. \bibitem{FLT1} Favini, A., Lorenzi, A., Tanabe, H., and Yagi, A.; \emph{An $L^{p}$-Approach to Singular Linear Parabolic Equations in Bounded Domains}, Osaka Journal Mathematics, 42, pp. 385-406, 2005. \bibitem{FLT2} Favini, A., Lorenzi, A. and Tanabe, H.; \emph{First-order regular and degenerate identification differential problems}, Preprint. \bibitem{FLY} Favini, A., Lorenzi, A. and Yagi, A.; \emph{Exponential attactors for semiconductor equations}, in "Differential Equations: Inverse and Direct Problems", Angelo Favini and Alfredo Lorenzi eds., Taylor and Francis Group, Chapman \& Hall CRC, Boca Raton, USA, 2006, 111--130. \bibitem{FY} Favini, A., and Yagi, A.; \emph{Degenerate Differential Equations in Banach Spaces}, 1$^{st}$ ed., Dekker, New York-Basel-Hong Kong, 1999. \bibitem{KA} Kato, T.; \emph{Perturbation Theory of linear bounded operators}, Springer, Berlin-Heidelberg-New York, 1980. \bibitem{LO} Lorenzi, A.; \emph{Introduction to identification problems via functional analysis}, VSP, Utrecht, Holland, 2001. \bibitem{LU} Lunardi, A.; \emph{Analytic semigroups and optimal regularity in parabolic problems}, Birkh\"auser, Basel, Switzerland, 1995. \bibitem{TA2} Taira, K.; \emph{The Theory of Semigroups with weak Singularity and its Applications to Partial Differential Equations}, Tsukuba. J. Math. 13 (1989), pp. 513-562. \bibitem{TR} Triebel, H.; \emph{Interpolation theory, function spaces, differential operators}, North-Holland, Amsterdam, Holland, 1978. \bibitem{WA} v. Wahl, W.; \emph{Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen,} Nachr. Akad. Wiss. Göttingen Math.-Phys. (1972), 231-258. \bibitem{WI} Wild, C.; \emph{Semigroupes de croissance $\alpha < 1$ holomorphes}, C.R. Acad. Sc. Paris 285 (1977), 437--440. \bibitem{YA} Yagi, A.; \emph{Abstract parabolic evolution equations and their applications}, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. \bibitem{YO} Yosida, K.; \emph{Functional Analysis}, 2$^{nd}$ ed., Springer Verlag, Berlin, 1969. \end{thebibliography} \end{document}