\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 92, pp. 1--25.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/92\hfil Nonautonomous ill-posed evolution problems] {Nonautonomous ill-posed evolution problems \\ with strongly elliptic differential operators} \author[M. A. Fury\hfil EJDE-2013/92\hfilneg] {Matthew A. Fury} % in alphabetical order \address{Matthew Fury \newline Division of Science \& Engineering \\ Penn State Abington \\ 1600 Woodland Road \\ Abington, PA 19001, USA\newline Tel: 215-881-7553 \\ Fax: 215-881-7333} \email{maf44@psu.edu} \thanks{Submitted November 3, 2012. Published April 11, 2013.} \subjclass[2000]{46B99, 47D06} \keywords{Regularizing family of operators; ill-posed evolution equation; \hfill\break\indent holomorphic semigroup; strongly elliptic operator} \begin{abstract} In this article, we consider the nonautonomous evolution problem $du/dt=a(t)Au(t), 0\leq s\leq t< T$ with initial condition $u(s)=\chi$ where $-A$ generates a holomorphic semigroup of angle $\theta \in (0,\pi/2]$ on a Banach space $X$ and $a\in C([0,T]:\mathbb{R}^+)$. The problem is generally ill-posed under such conditions, and so we employ methods to approximate known solutions of the problem. In particular, we prove the existence of a family of regularizing operators for the problem which stems from the solution of an approximate well-posed problem. In fact, depending on whether $\theta \in (0,\pi/4]$ or $\theta \in (\pi/4,\pi/2]$, we provide two separate approximations each yielding a regularizing family. The theory has applications to ill-posed partial differential equations in $L^p(\Omega)$, $10, \; t \in [s,T]\}$ of bounded linear operators on $X$ is called a \emph{family of regularizing operators for the problem} \eqref{1} if for each solution $u(t)$ of \eqref{1} with initial data $\chi \in X$, and for any $\delta>0$, there exists $\beta(\delta)>0$ such that \begin{itemize} \item[(i)] $\beta(\delta)\to 0$ as $\delta \to 0$, \item[(ii)] $\|u(t)-R_{\beta(\delta)}(t)\chi_\delta\|\to 0$ as $\delta \to 0$ for $s\leq t\leq T$ whenever $\|\chi - \chi_{\delta}\|\leq \delta$. \end{itemize} \end{definition} As in the case of regularization for the autonomous problem \eqref{ACP}, we will show that a family of regularizing operators for \eqref{1} stems from the solution of an approximate well-posed problem \label{2} \begin{gathered} \frac{dv}{dt} = f_{\beta}(t,A)v(t) \quad 0\leq s\leq t0$, the operators$f_{\beta}(t,A), 0\leq t\leq T$are defined by two different approximations of the operators$a(t)A$depending on where$\theta$lies in the interval$(0,\pi/2]$: $$\label{f_beta} f_{\beta}(t,A) = \begin{cases} a(t)A-\beta A^{\sigma} & \text{if } \theta \in (0,\pi/4] \\ a(t)A(I+\beta A)^{-1} & \text{if } \theta \in (\pi/4,\pi/2] \end{cases}$$ where$\sigma>1$when$\theta \in (0,\pi/4]$. Each approximation in \eqref{f_beta} yields a well-posed problem \eqref{2}, and also continuous dependence on modeling for the ill-posed problem \eqref{1} in the sense that as$\beta \to 0$, the operators$f_{\beta}(t,A)$approach the operators$a(t)A$, and given solutions$u(t)$and$v_{\beta}(t)$of \eqref{1} and \eqref{2} respectively, we have $$\label{CDMsortof} \|u(t)-v_{\beta}(t)\| \to 0 \quad \text{as } \beta \to 0$$ for each$t\in [s,T]$. We use \eqref{CDMsortof} to establish the main result of the paper, that the family$\{V_{\beta}(t,s) : \beta>0, \; t \in [s,T]\}$is a family of regularizing operators for the ill-posed problem \eqref{1} where$V_{\beta}(t,s),0\leq s\leq t\leq T$is an evolution system associated with the well-posed problem \eqref{2} satisfying$V_{\beta}(t,s)\chi =v_{\beta}(t)$. In other words, given a small change in the initial data$\|\chi-\chi_{\delta}\|\leq \delta$(which, since \eqref{1} is ill-posed, could yield a very large difference in solutions), there exists$\beta>0$so that$\beta \to 0$as$\delta\to 0$, and$\|u(t)-V_{\beta}(t,s)\chi_{\delta}\|\to 0$as$\delta \to 0$for$s\leq t\leq T$. Hence, although$u(t)$may not be close" to the solution of \eqref{1} with initial data$\chi_{\delta}$, we can still approximate$u(t)$by utilizing the well-posed problem \eqref{2} with regularization parameter$\beta>0$. The use of the two approximations in \eqref{f_beta} extends results from previous works in which the approximations$A-\beta A^{\sigma}$and$A(I+\beta A)^{-1}$are used to obtain regularization for the autonomous problem \eqref{ACP} where$-A$generates a holomorphic semigroup of angle$\theta$on$X$(cf. \cite{AmesandHughes,HuangZheng2, HuangZheng,Melnikova,MelnikovaandFilinkov}). For instance, in \cite{HuangZheng2}, Huang and Zheng obtain regularization for \eqref{ACP} using the quasi-reversibility method, first introduced by Lattes and Lions \cite{LandL}, which involves the approximation$A-\beta A^{\sigma}$of the operator$A$. Here, the requirements that$\sigma>1$and$\sigma (\pi/2-\theta)<\pi/2$are crucial in order for$A-\beta A^{\sigma}$to generate a semigroup (so as to yield an approximate well-posed problem). Hence, if$\theta \in (0,\pi/4]$, these requirements force$1<\sigma<2$whence the use of the fractional power$A^{\sigma}$is in order. In light of definition \eqref{f_beta}, we will adopt the same requirements in the current paper for the extension$a(t)A-\beta A^{\sigma}$. The second approximation$A(I+\beta A)^{-1}$, introduced by Showalter \cite{Showalter}, is applied by Ames and Hughes \cite{AmesandHughes} and Huang and Zheng \cite{HuangZheng} but only in the case where$\theta \in (\pi/4, \pi/2]$because the perturbation methods used to establish regularization in these papers (and in the current paper) are not applicable when$\theta \in (0,\pi/4]$(cf. \cite[pp. 3011--3012]{HuangZheng}). Note, if$\theta \in (\pi/2,\pi/4]$, the approximation$a(t)A-\beta A^{\sigma}$may still be used, but it is standard and easier in this case to let$\sigma =2$(cf. \cite{AmesandHughes,Fury,LandL,Melnikova,MelnikovaandFilinkov, Miller1,Payne1,Payne2}). In this regard, the current paper also furthers results from \cite{Fury} where the author uses the approximation$\sum_{j=1}^ka_j(t)A^j-\beta A^{k+1}$to obtain regularization for the nonautonomous problem \begin{gather*} \frac{du}{dt} = \sum_{j=1}^ka_j(t)A^ju(t) \quad 0\leq s\leq t0$ on a Banach space $X$ is called a \emph{bounded holomorphic semigroup of angle} $\theta$ if the following conditions are satisfied: \begin{itemize} \item[(i)] $T(t)$ is the restriction to the positive real axis of an analytic family of operators $T(z)$ in the open sector $S_{\theta}=\{re^{i\theta'} : r> 0, \; |\theta'| <\theta\}$ satisfying $T(z+w)=T(z)T(w)$ for all $z,w\in S_{\theta}$. \item[(ii)] For each $\theta_1< \theta$, $T(z)x\to x$ as $z\to 0$ in $S_{\theta_1}$ for all $x\in X$. \item[(iii)] For each $\theta_1< \theta$, $T(z)$ is uniformly bounded in the sector $S_{\theta_1}$. \end{itemize} More generally, a strongly continuous semigroup $T(t)$ on $X$ is called a \emph{holomorphic semigroup of angle} $\theta$ if $T(t)$ satisfies all the properties of a bounded holomorphic semigroup of angle $\theta$ with the exception of (iii). \end{definition} \begin{theorem}[{\cite[Theorem~X.52]{ReedandSimon}}] \label{generator_holomorphic} Let $A$ be a closed operator on a Banach space $X$. Then $-A$ is the infinitesimal generator of a bounded holomorphic semigroup of angle $\theta$ if and only if for each $\theta_1<\theta$ there exists a constant $M_1>0$ such that if $w \not \in \bar{S}_{\pi/2-\theta_1}$, then $w \in \rho(A)$ and $$\label{resolvent} \| (w -A)^{-1}\|\leq \frac{M_1}{\operatorname{dist}(w,\bar{S}_{\pi/2-\theta_1})}.$$ \end{theorem} For this paper, we first assume that $-A$ generates a \emph{bounded} holomorphic semigroup of angle $\theta$. In fact, for most of the paper, we will make this assumption for convenience, but then generalize our results at the end for holomorphic semigroups for which only conditions (i) and (ii) of Definition~\ref{holomorphic_semigroup} hold. Since $-A$ generates a bounded holomorphic semigroup of angle $\theta$, by Theorem~\ref{generator_holomorphic}, it follows that the spectrum $\sigma (A)$ of $A$ is contained in $\bar{S}_{\pi/2-\theta}=\{re^{i\theta'} : r\geq 0, \; |\theta'|\leq \pi/2-\theta\}$. Further, for $t>0$, $T(t)$ is given by the Cauchy integral formula $$\label{T(t)} T(t)=\frac{1}{2\pi i}\int_{\Gamma_{\phi}}e^{-tw}(w-A)^{-1}dw$$ where $\pi/2>\phi>\pi/2-\theta$ and $\Gamma_{\phi}$ is a curve in $\rho(A)$ consisting of three pieces: $\Gamma_1=\{re^{i\phi} : r\geq 1\}$, $\Gamma_2=\{e^{i\theta'} : \phi \leq \theta' \leq 2\pi -\phi\}$, and $\Gamma_3=\{re^{-i\phi} : r\geq 1\}$; $\Gamma_{\phi}$ is oriented so that it runs from $\infty e^{i\phi}$ to $\infty e^{-i\phi}$ (see Figure~\ref{fig:Gamma}). Similarly, for $z\in S_{\theta}$, $T(z)=\frac{1}{2\pi i}\int_{\Gamma_{\phi}}e^{-zw}(w-A)^{-1}dw.$ \begin{figure}[ht] \begin{center} \includegraphics[height=9cm]{fig1} % newgamma.pdf} \put(-60,242){\small$\pi/2-\theta$} \put(-120,203){\small$\Gamma_1$} \put(-170,95){\small$\Gamma_2$} \put(-114,40){\small$\Gamma_3$} \put(-18,201){\small$\theta$} \put(-142,140){\small$1$} \put(-93, 245){\small$\phi$} \end{center} \caption{$\Gamma_{\phi}$} \label{fig:Gamma} \end{figure} We will first prove that the approximate problem \eqref{2} is well-posed in the case that $\theta \in (0,\pi/4]$ and $f_{\beta}(t,A), 0\leq t\leq T$ is defined by $f_{\beta}(t,A)=a(t)A-\beta A^{\sigma}$ (Proposition~\ref{well-posed_approx1} below). The idea in this case is to construct an evolution system $V_{\beta}(t,s)$ which will be defined similarly as in \eqref{T(t)}. For this, we will need to choose an appropriate value for $\phi$ in a contour similar to $\Gamma_{\phi}$. In particular, we will require that $\sigma>1$ and $\sigma (\pi/2-\theta)<\pi/2$ in order to allow $\pi/2\sigma>\phi>\pi/2-\theta$. As noted in the introduction, since $\theta \in (0,\pi/4]$, these requirements force $1<\sigma<2$ and so we will need to make sense of the operator $A^{\sigma}$ which is defined by the fractional power. To this end, we will require the assumption that $0\in \rho(A)$ (see Definition~\ref{fractional_power} below). \begin{definition}[{\cite[Definition~2.4]{HuangZheng2}}] \label{fractional_power} \rm Let $-A$ be the infinitesimal generator of a bounded holomorphic semigroup of angle $\theta$, and let $0\in \rho (A)$. For $\sigma >0$, the \emph{fractional power of} $A$ is defined as follows: \begin{eqnarray} \label{Asigma} A^{-\sigma}=\frac{1}{2\pi i}\int_{\Gamma}w^{-\sigma}(w-A)^{-1}dw, \end{eqnarray} where $w^{-\sigma}$ is defined by the principal branch, and $\Gamma$ is a path running from $\infty e^{i\phi}$ to $\infty e^{-i\phi}$ with $\pi>\phi>\pi/2-\theta$ while avoiding the negative real axis and the origin. Define $A^{\sigma}=(A^{-\sigma})^{-1}$ (see Lemma \ref{fractional_props} (i) below) and $A^0=I$. \end{definition} Note, in Definition~\ref{fractional_power}, the definition of $A^{\sigma}$ relies on the fact that the operator in \eqref{Asigma} is invertible which follows from the following properties of the fractional power. \begin{lemma}[{\cite[Lemma~2.5]{HuangZheng2}, \cite[Lemma~2.6.6, Theorem~2.6.8]{Pazy}}] \label{fractional_props} Let $-A$ be the infinitesimal generator of a bounded holomorphic semigroup of angle $\theta$, and let $0\in \rho (A)$. Then \begin{itemize} \item[(i)] $A^{-\sigma}$ is a bounded, injective operator for $\sigma>0$. \item[(ii)] $A^{\sigma}$ is a closed operator, and $\operatorname{Dom}(A^{\sigma})\subseteq \operatorname{Dom}(A^{\sigma'})$ for $\sigma > \sigma'>0$. \item[(iii)] $\operatorname{Dom}(A^{\sigma})$ is dense in $X$ for every $\sigma\geq 0$. \item[(iv)] $A^{\sigma_1+\sigma_2}x=A^{\sigma_1}A^{\sigma_2}x$ for every $\sigma_1$, $\sigma_2 \in \mathbb{R}$ and $x\in \operatorname{Dom}(A^{\sigma})$ where $\sigma = \max \{\sigma_1, \sigma_2, \sigma_1+\sigma_2\}$. \end{itemize} \end{lemma} \begin{proposition}\label{well-posed_approx1} Let $-A$ be the infinitesimal generator of a bounded holomorphic semigroup of angle $\theta \in (0,\pi/4]$, and let $0\in \rho(A)$. Let $0<\beta<1$ and assume $\sigma$ satisfies $\sigma>1$ and $\sigma(\pi/2-\theta)<\pi/2$. Define the family of operators $f_{\beta}(t,A), 0\leq t\leq T$ by $f_{\beta}(t,A)=a(t)A-\beta A^{\sigma}.$ Then $\eqref{2}$ is well-posed with unique classical solution $v_{\beta}(t)=V_{\beta}(t,s)\chi$ for each $\chi \in X$ where V_{\beta}(t,s) = \begin{cases} \frac{1}{2\pi i}\int_{\Gamma_{\phi}} e^{\int_s^tf_{\beta}(\tau,w)d\tau}(w-A)^{-1}\,dw & 0\leq s\phi>\pi/2-\theta but avoids the negative real axis and the origin. \end{proposition} \begin{proof} Notice our choice for \phi is valid by the assumption \sigma(\pi/2-\theta)<\pi/2. We first show that V_{\beta}(t,s) is uniformly bounded for 0\leq s\leq t\leq T. Following \cite[Proof of Theorem~3.1]{HuangZheng2}, we will show this in two cases. Let 0\leq s\pi/2-\theta_1>\pi/2-\theta. We have \operatorname{dist}(w,\bar{S}_{\pi/2-\theta_1})= |w| \sin (\phi-(\pi/2-\theta_1)) (cf. \cite[Figure~2]{Fury}) so that by Theorem~\ref{generator_holomorphic}, $$\label{dist} \|(w-A)^{-1}\| \leq \frac{M_1}{|w|\;\sin (\phi-(\pi/2-\theta_1))}.$$ Set M_1'=M_1/\sin (\phi-(\pi/2-\theta_1)) and B=\max_{t\in [0,T]}|a(t)|. Then \begin{align*} \big\|\int_{\Gamma^1\cup \Gamma^3}\big\| &\leq M_1' \int_{\Gamma^1\cup \Gamma^3}\big| e^{\int_s^t(a(\tau)w-\beta w^{\sigma})\; d\tau}\big|\; |w|^{-1}|dw| \\ &= 2M_1'\int_{(t-s)^{-1/\sigma}}^{\infty}e^{\int_s^t(a(\tau) r\cos \phi -\beta r^{\sigma}\cos\sigma \phi)\; d\tau} r^{-1}dr \\ &\leq 2M_1'\int_{(t-s)^{-1/\sigma}}^{\infty}e^{B(t-s)r\cos \phi -\beta (t-s)r^{\sigma}\cos\sigma \phi} r^{-1}dr \\ &= 2M_1'\int_1^{\infty}e^{B(t-s)^{1-1/\sigma}x\cos \phi -\beta x^{\sigma}\cos\sigma \phi} x^{-1}dx \\ &\leq 2M_1'\int_1^{\infty}e^{BT^{1-1/\sigma}x\cos \phi -\beta x^{\sigma}\cos\sigma \phi} dx \leq K \end{align*} where K is a constant independent of t and s since \sigma>1 and since \pi/2\sigma>\phi>\pi/2-\theta implies 0< \phi< \sigma \phi <\pi/2 so that \cos \phi>0 and \cos (\sigma \phi)>0. Also, for w \in \Gamma^2, we have \begin{align*} \big\|\int_{\Gamma^2}\big\| &\leq M_d\int_{\Gamma^2} \big|e^{\int_s^t(a(\tau)w-\beta w^{\sigma})d\tau}\big| |dw| \\ &= M_d\int_{-\phi}^{\phi} e^{\int_s^t(a(\tau)(t-s)^{-1/\sigma}\cos\theta' -\beta (t-s)^{-1}\cos\sigma \theta')d\tau}(t-s)^{-1/\sigma}d\theta' \\ &\leq dM_d\int_{-\phi}^{\phi} e^{B(t-s)^{1-1/\sigma}\cos\theta' -\beta \cos\sigma \theta'}d\theta' \\ &\leq dM_d\int_{-\phi}^{\phi} e^{BT^{1-1/\sigma}\cos\theta'}d\theta' \\ &\leq dM_d\;e^{BT^{1-1/\sigma}}2\phi \end{align*} where we have set M_d=\max_{|w|\leq d}\|(w-A)^{-1}\| since w\to (w-A)^{-1} is continuous on the interior of \rho(A). Hence, V_{\beta}(t,s) is bounded uniformly for 0\leq s\leq t\leq T in the first case. For the second case, if (t-s)^{-1/\sigma}>d, then we shift \Gamma_{\phi} to the contour (see Figure~\ref{fig:Gammaavoid}) consisting of the seven pieces: \begin{gather*} \Gamma_1 = \{re^{i\phi} : r\geq (t-s)^{-1/\sigma}\}, \quad \Gamma_2 = \{(t-s)^{-1/\sigma}e^{i\theta'} : \phi \leq \theta' \leq \pi\}, \\ \Gamma_3 = \{re^{i\pi} : d\leq r\leq (t-s)^{-1/\sigma} \} \quad \Gamma_4 = \{de^{-i\theta'} : -\pi \leq \theta' \leq \pi\}, \\ \Gamma_5 = \{re^{-i\pi} : d\leq r\leq (t-s)^{-1/\sigma} \} \quad \Gamma_6 = \{(t-s)^{-1/\sigma}e^{i\theta'} : -\pi \leq \theta' \leq -\phi\}, \\ \Gamma_7 = \{re^{-i\phi} : r\geq (t-s)^{-1/\sigma}\}. \end{gather*} \begin{figure}[ht] \begin{center} \includegraphics[height=9cm]{fig3} %gammaavoid.pdf \put(-65,205){\small\Gamma_1} \put(-180,185){\small\Gamma_2} \put(-172,140){\small\Gamma_3} \put(-113,147){\small\Gamma_4} \put(-172,108){\small\Gamma_5} \put(-178,67){\small\Gamma_6} \put(-65,48){\small\Gamma_7} \put(-53, 252){\small\phi} \put(-70,115){\smallt'} \put(-140,134){\smalld} \end{center} \caption{t':=(t-s)^{-1/\sigma}> d} \label{fig:Gammaavoid} \end{figure} First, since \Gamma_1=\Gamma^1 and \Gamma_7=\Gamma^3, we have \|\int_{\Gamma_1\cup \Gamma_7}\|=\|\int_{\Gamma^1\cup \Gamma^3}\|\leq K as before. Next, note that \eqref{dist} holds for w\in \Gamma_2 since these w satisfy the inequality \operatorname{dist}(w,\bar{S}_{\pi/2-\theta_1})\geq \operatorname{dist}((t-s)^{-1/\sigma}e^{i\phi}, \bar{S}_{\pi/2-\theta_1}). Then \begin{align*} \big\|\int_{\Gamma_2}\big\| &\leq M_1'\int_{\Gamma_2} \big|e^{\int_s^t(a(\tau)w-\beta w^{\sigma})d\tau}\big| |w|^{-1}|dw| \\ &= M_1'\int_{\phi}^{\pi} e^{\int_s^t(a(\tau)(t-s)^{-1/\sigma}\cos\theta' -\beta (t-s)^{-1}\cos\sigma \theta')d\tau}d\theta' \\ &\leq M_1'\int_{\phi}^{\pi} e^{BT^{1-1/\sigma}\cos\phi -\beta \cos\sigma \theta'}d\theta' \\ &\leq M_1'\int_{\phi}^{\pi} e^{1+BT^{1-1/\sigma}\cos\phi}d\theta' \\ &= M_1'e^{1+BT^{1-1/\sigma}\cos\phi}(\pi-\phi) \end{align*} since 0<\beta<1. The same estimate holds for \|\int_{\Gamma_6}\|. Next, using \eqref{dist}, \begin{align*} & \big\|\int_{\Gamma_3}+\int_{\Gamma_5}\big\| \\ &= \big\| \int_d^{(t-s)^{-1/\sigma}} \Big(e^{\int_s^t(-a(\tau)r-\beta r^{\sigma}e^{-i\pi \sigma})d\tau} -e^{\int_s^t(-a(\tau)r-\beta r^{\sigma}e^{i\pi \sigma})d\tau}\Big) (-r-A)^{-1}dr\big\| \\ &\leq M_1'\int_d^{(t-s)^{-1/\sigma}} \Big|e^{-(\int_s^ta(\tau)d\tau)r} \Big(e^{-\beta (t-s)r^{\sigma}e^{-i\pi \sigma }} -e^{-\beta (t-s)r^{\sigma}e^{i\pi \sigma}}\Big)\Big|r^{-1}dr \\ &= M_1'\int_d^{(t-s)^{-1/\sigma}}e^{-\left(\int_s^ta(\tau)d\tau\right)r} \left|e^{-\beta (t-s)r^{\sigma}\cos\sigma \pi}2i \sin (\beta (t-s)r^{\sigma}\sin \sigma\pi)\right|r^{-1}dr \\ &\leq M_1'\int_d^{(t-s)^{-1/\sigma}}e^{-\beta (t-s)r^{\sigma}\cos\sigma \pi}2| \sin (\beta (t-s)r^{\sigma}\sin \sigma\pi)|r^{-1}dr \\ &= M_1'\int_{(t-s)^{1/\sigma}d}^1e^{-\beta x^{\sigma}\cos\sigma \pi}2| \sin (\beta x^{\sigma}\sin \sigma\pi)|x^{-1}dx \\ &= M_1'\int_{(t-s)^{1/\sigma}d}^1x^{-1/2}e^{-\beta x^{\sigma}\cos\sigma \pi} \left\{4x^{-1}\sin ^2(\beta x^{\sigma}\sin \sigma\pi)\right\}^{1/2}dx \\ &= M_1'\int_{(t-s)^{1/\sigma}d}^1x^{-1/2}e^{-\beta x^{\sigma}\cos\sigma \pi} \left\{2x^{-1}(1-\cos (2\beta x^{\sigma}\sin \sigma\pi))\right\}^{1/2}dx. \end{align*} It is easily shown by L'Hospital's Rule that \[ 2x^{-1}(1-\cos (2\beta x^{\sigma}\sin \sigma \pi))\to 0 \quad \text{as } x \to 0. Hence, we have for a possibly different constant $M_1'$ independent of $\beta$, $\|\int_{\Gamma_3}+\int_{\Gamma_5}\| \leq M_1'\int_0^1x^{-1/2}e^{-\beta x^{\sigma}\cos\sigma \pi}dx \leq M_1'e\int_0^1x^{-1/2}dx = M_1'2e$ since $0<\beta<1$. Finally, \begin{align*} \|\int_{\Gamma_4}\| &\leq M_d\int_{\Gamma_4} \big|e^{\int_s^t(a(\tau)w-\beta w^{\sigma})d\tau}\big| |dw| \\ &= dM_d\int_{-\pi}^{\pi} e^{\int_s^t(a(\tau)d \cos\theta'-\beta d^{\sigma}\cos\sigma \theta')d\tau}d\theta' \\ &\leq dM_d\int_{-\pi}^{\pi} e^{BTd}e^{-\beta (t-s)d^{\sigma}\cos\sigma \theta'}d\theta' \\ &\leq dM_d e^{BTd}(1+e^{Td^{\sigma}})2\pi \end{align*} where $M_d=\max_{|w|\leq d}\|(w-A)^{-1}\|$ as before. Thus we have shown that in both cases, each term may be bounded independently of $t$ and $s$, and so $V_{\beta}(t,s)$ is uniformly bounded on $0\leq s\leq t\leq T$. Next, we show that $(t,s)\mapsto V_{\beta}(t,s)$ is strongly continuous for $0\leq s\leq t\leq T$. It follows from \eqref{Asigma} and by a standard argument using Cauchy's Integral Formula that $%\label{strong_cont} V_{\beta}(t,s)A^{-\sigma} =\frac{1}{2\pi i}\int_{\Gamma_{\phi}}w^{-\sigma} e^{\int_s^tf_{\beta}(\tau,w)d\tau}(w-A)^{-1}dw$ (cf. \cite[p. 46]{HuangZheng2}). Then since $t \mapsto f_{\beta}(t,w)$ is continuous, using the above calculations for $\|V_{\beta}(t,s)\|$, it follows by a dominated convergence argument that $\|V_{\beta}(t,s)A^{-\sigma}-V_{\beta}(t_0,s_0)A^{-\sigma}\|\to 0$ as $(t,s) \to (t_0,s_0)$. Then, for $x\in \operatorname{Dom}(A^{\sigma})$, we have \begin{align*} \|V_{\beta}(t,s)x-V_{\beta}(t_0,s_0)x\| &\leq \|V_{\beta}(t,s)A^{-\sigma}-V_{\beta}(t_0,s_0)A^{-\sigma}\|\|A^{\sigma}x\| \\ &\to 0 \quad \text{as } (t,s) \to (t_0,s_0). \end{align*} Strong continuity of $V_{\beta}(t,s)$ then follows since $\operatorname{Dom}(A^{\sigma})$ is dense in $X$ (Lemma~\ref{fractional_props} (iii)) and $V_{\beta}(t,s)$ is uniformly bounded. Now, we show that the mapping $[s,T]\to X$ given by $t\mapsto V_{\beta}(t,s)\chi$ is a classical solution of \eqref{2} for $\chi \in X$. We have already established that $t\mapsto V_{\beta}(t,s)\chi$ is continuous on $[s,T]$. Next, we show that $\frac{\partial}{\partial t}V_{\beta}(t,s)\chi =f_{\beta}(t,A)V_{\beta}(t,s)\chi$ for $t \in (s,T)$. We have \begin{align} \label{C1f} \frac{\partial}{\partial t}V_{\beta}(t,s)\chi &= \frac{1}{2\pi i}\int_{\Gamma_{\phi}} \Big(\frac{\partial}{\partial t}e^{\int_s^tf_{\beta}(\tau,w)d\tau}\Big) (w-A)^{-1}\chi\,dw \\ &= \frac{1}{2\pi i}\int_{\Gamma_{\phi}}e^{\int_s^tf_{\beta} (\tau,w)d\tau}f_{\beta}(t,w)(w-A)^{-1}\chi\,dw \\ \label{eleven} &= \frac{1}{2\pi i}\int_{\Gamma_{\phi}}e^{\int_s^tf_{\beta}(\tau,w)d\tau} a(t)w(w-A)^{-1}\chi \,dw \\ \label{twelve} & \quad+ \frac{1}{2\pi i}\int_{\Gamma_{\phi}} e^{\int_s^tf_{\beta}(\tau,w)d\tau} (-\beta w^{\sigma})(w-A)^{-1}\chi \; dw. \end{align} Now, \begin{align*} \text{Expression \eqref{eleven}} &= a(t)\frac{1}{2\pi i}\int_{\Gamma_{\phi}}e^{\int_s^tf_{\beta} (\tau,w)d\tau}((w-A)+A)(w-A)^{-1}\chi\,dw \\ &= \Big(a(t)\frac{1}{2\pi i}\int_{\Gamma_{\phi}}e^{\int_s^tf_{\beta} (\tau,w)d\tau}dw\Big)\chi \\ &\quad + a(t)\frac{1}{2\pi i}\int_{\Gamma_{\phi}}e^{\int_s^tf_{\beta}(\tau,w)d\tau}A(w-A)^{-1}\chi\,dw \\ &= a(t)AV_{\beta}(t,s)\chi \end{align*} where we have used Cauchy's Theorem since $w\mapsto e^{\int_s^tf_{\beta}(\tau,w)d\tau}$ is analytic, and also the fact that $A$ is a closed operator. Next, fix $t\in (s,T)$ and set $G=\frac{1}{2\pi i}\int_{\Gamma_{\phi}}w^{\sigma} e^{\int_s^tf_{\beta}(\tau,w)d\tau}(w-A)^{-1} dw$. It is clear that $G$ is a bounded operator on $X$ by calculations similarly used to calculate $\|V_{\beta}(t,s)\|$. Also, by \eqref{fractional_power} and a standard argument using Cauchy's Integral Formula (cf. \cite[Equation IX.1.52]{Kato}), it follows that $A^{-\sigma}G=V_{\beta}(t,s)$. Hence, by the fact that $A^{\sigma}=(A^{-\sigma})^{-1}$, we have $\operatorname{Ran}(V_{\beta}(t,s))\subseteq \operatorname{Ran}(A^{-\sigma}) = \operatorname{Dom}(A^{\sigma})$ and $G=A^{\sigma}V_{\beta}(t,s)$. Hence $\eqref{twelve}=-\beta G\chi=-\beta A^{\sigma}V_{\beta}(t,s)\chi$, and altogether we have shown $\frac{\partial}{\partial t}V_{\beta}(t,s) =a(t)AV_{\beta}(t,s)\chi-\beta A^{\sigma}V_{\beta}(t,s)\chi =f_{\beta}(t,A)V_{\beta}(t,s)\chi$ for $t\in (s,T)$. Also by definition, $V_{\beta}(s,s)\chi=\chi$. Thus, $t\mapsto V_{\beta}(t,s)\chi$ satisfies \eqref{2}. Finally, calculation \eqref{C1f}--\eqref{twelve} shows that $t\mapsto f_{\beta}(t,A)V_{\beta}(t,s)\chi$ is continuous on $(s,T)$ since $t\mapsto e^{\int_s^tf_{\beta}(\tau,w)d\tau}f_{\beta}(t,w)$ is continuous. Therefore, we have that $t\mapsto V_{\beta}(t,s)\chi$ is continuously differentiable on $(s,T)$, and so we have shown altogether that $t\mapsto V_{\beta}(t,s)\chi$ is a classical solution of \eqref{2}. It follows that problem \eqref{2} is well-posed due to uniqueness of the solution $t\mapsto V_{\beta}(t,s)\chi$ and continuous dependence of solutions on initial data, both of which are proved by standard arguments (see e.g. \cite[Proof of Proposition~2.3]{Fury}). \end{proof} \begin{corollary}\label{approx1bound} Let $0<\beta<1$ and let the operators $f_{\beta}(t,A), 0\leq t\leq T$ and $V_{\beta}(t,s),0\leq s\leq t\leq T$ be defined under the hypotheses of Proposition~$\ref{well-posed_approx1}$. Then for small $\beta$, $\|V_{\beta}(t,s)\|\leq K'e^{K\beta^{-1/(\sigma-1)}}$ for all $0\leq s\leq t\leq T$ where $K$ and $K'$ are constants independent of $\beta$, $t$, and $s$. \end{corollary} \begin{proof} Let $0\leq s0$ and let $\alpha>1$ satisfy $\alpha (\pi/2-\theta)<\pi/2$. Then $e^{-\epsilon A^{\alpha}},\epsilon> 0$ defined by $$\label{C_epsilon} e^{-\epsilon A^{\alpha}} = \frac{1}{2\pi i}\int_{\Gamma_{\phi}}e^{-\epsilon w^{\alpha}}(w-A)^{-1}dw$$ is a strongly continuous holomorphic semigroup generated by the fractional power $-A^{\alpha}$ where $\Gamma_{\phi}$ is similar to the contour described in Proposition~\ref{well-posed_approx1} but with $\pi/2\alpha>\phi>\pi/2-\theta$ (cf. \cite[Definition~3.4]{deL1}). For $\epsilon>0$, set $C_{\epsilon}=e^{-\epsilon A^{\alpha}}$. It follows that $C_{\epsilon}$ is injective for $\epsilon>0$ (cf. \cite[Lemma 3.1]{deL1}). We construct $C_{\epsilon}$-regularized evolution systems as follows. \begin{proposition} \label{1_epsilon} Let $\epsilon>0$ and let $\alpha >1$ satisfy $\alpha (\pi/2-\theta)<\pi/2$. For every $\chi\in X$, the evolution problem $$\label{1epsilon} \begin{gathered} \frac{du}{dt} = a(t)Au(t) \quad 0\leq s\leq t< T \\ u(s) = C_{\epsilon}\chi \end{gathered}$$ has a unique classical solution $u(t)=U_{\epsilon}(t,s)\chi$ where $U_{\epsilon}(t,s)=\frac{1}{2\pi i}\int_{\Gamma_{\phi}} e^{-\epsilon w^{\alpha}}e^{\left(\int_s^ta(\tau)d\tau\right)w}(w-A)^{-1}dw$ for all $0\leq s\leq t\leq T$ and $\Gamma_{\phi}$ is similar to the contour described in Proposition~$\ref{well-posed_approx1}$ with $\pi/2\alpha>\phi>\pi/2-\theta$. \end{proposition} \begin{proof} The proof is similar to that of Proposition~\ref{well-posed_approx1}. In particular, $U_{\epsilon}(t,s)$ is a uniformly bounded operator on $X$ for $0\leq s\leq t\leq T$ by the assumptions on $\alpha$. Also, the function $t\mapsto U_{\epsilon}(t,s)\chi$ is a unique classical solution of \eqref{1epsilon} since $\frac{\partial}{\partial t}U_{\epsilon}(t,s)\chi =a(t)AU_{\epsilon}(t,s)\chi$ for $t\in (s,T)$, and by equation \eqref{C_epsilon}, $U_{\epsilon}(s,s)\chi = \frac{1}{2\pi i}\int_{\Gamma_{\phi}} e^{-\epsilon w^{\sigma}}(w-A)^{-1}\chi\,dw = e^{-\epsilon A^{\sigma}}\chi = C_{\epsilon}\chi.$ \end{proof} \begin{lemma} \label{C_epsilonu(t)} Let $\chi \in X$. If $u(t)$ is a classical solution of problem $\eqref{1}$, then $C_{\epsilon}u(t)=U_{\epsilon}(t,s)\chi \quad \text{for all } t\in [s,T].$ \end{lemma} \begin{proof} Since $C_{\epsilon}\in B(X)$ and $C_{\epsilon}$ commutes with $A$, it is easily shown that $C_{\epsilon}u(t)$ is a classical solution of \eqref{1epsilon}. The uniqueness of solutions from Proposition~\ref{1_epsilon} then yields the desired result. \end{proof} To establish regularization, we will make use of the nature in which the operators $f_{\beta}(t,A)$ approximate the operators $a(t)A$. Motivated by the approximation condition, Condition A of Ames and Hughes (cf. \cite[Definition~1]{AmesandHughes}), we demonstrate the following property. \begin{lemma} \label{Condition_Ap} Let $-A$ be the infinitesimal generator of a bounded holomorphic semigroup of angle $\theta$, and let $0 \in \rho (A)$. Let $0<\beta<1$ and let the family of operators $f_{\beta}(t,A), 0\leq t\leq T$ be defined by $\eqref{f_beta}$. Then there exist positive constants $R$ and $\kappa$ each independent of $\beta$ and $t$ such that $\operatorname{Dom}(A^{1+\kappa})\subseteq \operatorname{Dom}(f_{\beta}(t,A))$ and $$\label{Ap} \|(-a(t)A+f_{\beta}(t,A))\psi\|\leq R\beta\|A^{1+\kappa}\psi\|$$ for all $t\in [0,T]$ and for all $\psi\in \operatorname{Dom}(A^{1+\kappa})$. \end{lemma} Note that in the statement of the lemma we use implicitly that $\operatorname{Dom}(A^{1+\kappa})\subseteq \operatorname{Dom}(A)$ which follows from Lemma~\ref{fractional_props} (ii). \begin{proof} First, assume $\theta \in (0,\pi/4]$ so that $f_{\beta}(t,A)$ is defined as in Proposition~\ref{well-posed_approx1} where $\sigma$ satisfies $\sigma>1$ and $\sigma (\pi/2-\theta)<\pi/2$. Then for $\psi \in \operatorname{Dom}(A^{\sigma})$ and $t\in [0,T]$, we have $\psi \in \operatorname{Dom}(f_{\beta}(t,A))$ and $\|(-a(t)A+f_{\beta}(t,A))\psi\| = \|(-a(t)A+(a(t)A-\beta A^{\sigma}))\psi\| = \beta \|A^{\sigma}\psi\|.$ Hence, \eqref{Ap} is satisfied with $R=1$ and $\kappa=\sigma-1$. Next, we assume that $\theta \in (\pi/4,\pi/2]$ in which case $f_{\beta}(t,A)$ is defined as in Proposition~\ref{well-posed_approx2}. Then $f_{\beta}(t,A)$ is a bounded, everywhere defined operator and so $\operatorname{Dom}(f_{\beta}(t,A)) = X$ for each $t\in [0,T]$. Further, for $\psi \in \operatorname{Dom}(A^2)$, \begin{align*} \|(-a(t)A+f_{\beta}(t,A))\psi\| &= \|(-a(t)A+a(t)A(I+\beta A)^{-1})\psi\| \\ &= \|-a(t)A(I-(I+\beta A)^{-1})\psi\| \\ &= \|-a(t)A(\beta A(I+\beta A)^{-1})\psi\| \\ &= \|-a(t)\beta (I+\beta A)^{-1}A^2\psi\| \\ &\leq B \beta \|(I+\beta A)^{-1}\| \|A^2\psi\| \\ &\leq B C \beta \|A^2 \psi\|, \end{align*} where $B= \max_{t\in [0,T]}|a(t)|$ and $C$ is as in the proof of Proposition~\ref{well-posed_approx2}. Hence, \eqref{Ap} is satisfied with $R=BC$ and $\kappa =1$. \end{proof} In light of Lemma~\ref{Condition_Ap}, for each $t\in [0,T]$, we define the operator $g_{\beta}(t,A)$ in $X$ by $$\label{g_beta} g_{\beta}(t,A)x=-a(t)Ax+f_{\beta}(t,A)x$$ for $x\in \operatorname{Dom}(A)\cap \operatorname{Dom}(f_{\beta}(t,A))$. Properties of the operators $g_{\beta}(t,A), 0\leq t\leq T$ and associated evolutions systems will be used heavily in proving H\"{o}lder-continuous dependence on modeling, those of which we provide now in the following proposition. \begin{proposition} \label{W_beta} Let $-A$ be the infinitesimal generator of a bounded holomorphic semigroup of angle $\theta$, and let $0\in \rho (A)$. For $0<\beta<1$, let the operators $f_{\beta}(t,A),0\leq t\leq T$ and $g_{\beta}(t,A)$, $0\leq t\leq T$ be defined by $\eqref{f_beta}$ and $\eqref{g_beta}$ respectively. Then there exists an evolution system $W_{\beta}(t,s), 0\leq s\leq t\leq T$ associated with the family $g_{\beta}(t,A), 0\leq t\leq T$ satisfying the following properties: \begin{itemize} \item[(i)] $\|W_{\beta}(t,s)\| \leq L$ for all $0\leq s\leq t\leq T$ where $L$ is a constant independent of $t$, $s$, and $\beta$. \item[(ii)] $\frac{\partial}{\partial t}W_{\beta}(t,s)\chi =g_{\beta}(t,A)W_{\beta}(t,s)\chi$ for $0\leq s d$ where $d$ is the radius of the disk contained in $\rho(A)$ as in Figure~\ref{fig:Gammadisk} and Figure~\ref{fig:Gammaavoid}. For the pieces \begin{gather*} \Gamma^1 = \Gamma_1 = \{re^{i\phi} : r\geq \beta^{-1/\sigma}(t-s)^{-1/\sigma}\}, \\ \Gamma^3 = \Gamma_7 = \{re^{-i\phi} : r\geq \beta^{-1/\sigma}(t-s)^{-1/\sigma}\}, \end{gather*} we have the calculation \begin{align*} \|\int_{\Gamma_1\cup \Gamma_7}\| = \|\int_{\Gamma^1\cup \Gamma^3}\| &\leq M_1' \int_{\Gamma^1\cup \Gamma^3}\big|e^{-\beta (t-s)w^{\sigma}}\big|\; |w|^{-1}|dw| \\ &= 2M_1'\int_{\beta^{-1/\sigma}(t-s)^{-1/\sigma}}^{\infty} e^{-\beta (t-s)r^{\sigma}\cos\sigma \phi} r^{-1}dr \\ &= 2M_1'\int_1^{\infty}e^{-x^{\sigma}\cos\sigma \phi} x^{-1}dx \\ &\leq 2M_1'\int_1^{\infty}e^{-x^{\sigma}\cos\sigma \phi} dx \leq K \end{align*} where $K$ is a constant independent of $t$, $s$, and $\beta$ since $\sigma>1$ and $0< \sigma \phi <\pi/2$ because $\pi/2\sigma>\phi>\pi/2-\theta$. Also, as in the proof of Proposition~\ref{well-posed_approx1}, in either of the two cases, the remaining pieces of the contour may be bounded independently of $t$, $s$, and $\beta$. Hence (i)--(iii) are satisfied and the proposition is proved when $\theta \in (0,\pi/4]$. If, on the other hand, $\theta \in (\pi/4,\pi/2]$ as in Proposition~\ref{well-posed_approx2}, then $g_{\beta}(t,A)=-a(t)A+a(t)A(I+\beta A)^{-1}$ and in this case, we use perturbation theory to construct an evolution system $W_{\beta}(t,s), 0\leq s\leq t\leq T$ satisfying (i)--(iii). We've seen so far that $A(I+\beta A)^{-1}$ is a bounded operator on $X$. Then since $-A$ generates a bounded holomorphic semigroup of angle $\theta$, it follows that $-(A-A(I+\beta A)^{-1})$ is also the infinitesimal generator of a holomorphic semigroup of the same angle (cf. \cite[Corollary~3.2.2]{Pazy}). Set $G_{\beta}=A-A(I+\beta A)^{-1}$. It is shown in \cite{HuangZheng} that $\mathbb{C}\backslash S_{\pi-2\theta} \subseteq \rho(G_{\beta})$ where $S_{\pi-2\theta}=\{re^{i\theta'} : r> 0, \; |\theta'| <\pi-2\theta\}$, and $\|(w-G_{\beta})^{-1}\| \leq \frac{M}{|w|} \quad \text{for } w\in \mathbb{C}\backslash S_{\pi-2\theta}$ where $M$ is a constant independent of $\beta$ (cf. \cite[Theorem~2.1]{HuangZheng}). Hence for $0\leq s\leq t\leq T$, the operator $W_{\beta}(t,s)$ defined by $W_{\beta}(t,s) = \begin{cases} \frac{1}{2\pi i}\int_{\Gamma_{\phi}} e^{-(\int_s^ta(\tau)d\tau)w}(w-G_{\beta})^{-1}\,dw & 0\leq s\phi>\pi-2\theta, is a well-defined uniformly bounded operator satisfying \|W_{\beta}(t,s)\|\leq L for 0\leq s\leq t\leq T where L is a constant independent of \beta. Hence, (i) is satisfied. Also, similar to calculation \eqref{C1f}--\eqref{twelve}, it is standard to show that for every \chi \in X, \frac{\partial}{\partial t}W_{\beta}(t,s)\chi=-a(t)G_{\beta}W_{\beta}(t,s)\chi =g_{\beta}(t,A)W_{\beta}(t,s)\chi for 0\leq s0. Then \[ U_{\epsilon}(t,s)W_{\beta}(t,s)=C_{\epsilon}V_{\beta}(t,s) =W_{\beta}(t,s)U_{\epsilon}(t,s)$ for all $0\leq s\leq t\leq T$. \end{corollary} \begin{proof} The result follows from uniqueness of solutions as each term applied to $\chi\in X$ is a classical solution of the well-posed evolution problem \eqref{2} with initial data $C_{\epsilon}\chi$. \end{proof} \section{H\"{o}lder-continuous dependence on modeling} \label{HolderCDM} We now use the results of Section~\ref{well-posed} and Section~\ref{lemmas} to prove H\"{o}lder-continuous dependence on modeling for the problems \eqref{1} and \eqref{2}, meaning a small change in the models from \eqref{1} to \eqref{2} implies a small change in the corresponding solutions. Again, as in Section~\ref{well-posed} and Section~\ref{lemmas}, we assume that $-A$ generates a bounded holomorphic semigroup $T(t)$ of angle $\theta$ on $X$ and $0 \in \rho(A)$. For $z\in S_{\theta}$, let us denote $T(z)$ by $T(z)=e^{-zA}$ and also define $e^{-zA}$ to be the identity operator when $z=0$. Assume $u(t)$ and $v_{\beta}(t)$ are classical solutions of \eqref{1} and \eqref{2} respectively where $\chi \in X$ and let $\epsilon>0$ be arbitrary. Then since $C_{\epsilon}$ is bounded and since $e^{-zA}$ is uniformly bounded in each sector $S_{\theta_1}$, $\theta_1<\theta$ (Definition~\ref{holomorphic_semigroup} (iii)), we may define for $\theta_1 \in (0,\theta)$ and for $\zeta = t+re^{\pm i\theta_1}$ in the bent strip $S=\{\zeta = t+re^{\pm i\theta_1} : s\leq t\leq T,\; r \geq 0\}$, $\phi_{\epsilon}(\zeta) = e^{-(re^{\pm i\theta_1})A}C_{\epsilon}(u(t)-v_{\beta}(t)).$ Ultimately, we will apply Carleman's Inequality (cf. \cite{Miller2}) to a function related to $\phi_{\epsilon}(\zeta)$ on the bent strip $S$. Our methods are motivated by Agmon and Nirenberg \cite{AN}. \begin{lemma} \label{u(zeta)_v(zeta)} Let $\epsilon>0$. Then $\phi_{\epsilon}(\zeta) = e^{-(re^{\pm i\theta_1})A} (U_{\epsilon}(t,s)\chi-C_{\epsilon}V_{\beta}(t,s)\chi)$ for all $\zeta = t+re^{\pm i\theta}\in S$. \end{lemma} The above lemma follows immediately from Lemma~\ref{C_epsilonu(t)} and Corollary~\ref{well-posed_both}. \begin{lemma}[{\cite[p. 148]{AN}}] \label{AN} Let $\phi(z)$ be a continuous and bounded complex function on the bent strip $S=\{z=x+\eta e^{\pm i\theta} : s\leq x \leq T, \; \eta \geq 0\}$. For $\zeta = t+re^{\pm i\theta}\in S$, define $\Phi(\zeta)=-\frac{1}{\pi}\int \int_S \phi(z) \Big(\frac{1}{z-\zeta}+\frac{1}{\bar{z}+1+\zeta}\Big)dx d\eta.$ Then $\Phi(\zeta)$ is absolutely convergent, $\bar{\partial}\Phi(\zeta)=\phi(\zeta)$ where $\bar{\partial}$ denotes the Cauchy-Riemann operator, and there exists a constant $\tilde{K}$ such that $\int_{-\infty}^{\infty}\big|\frac{1}{z-\zeta}+\frac{1}{\bar{z}+1+\zeta} \big|d\eta\leq \tilde{K}\Big(1+{\rm{log}}\frac{1}{|x-t|}\Big)$ if $x \neq t$. \end{lemma} We prove now the following theorem establishing H\"{o}lder-continuous dependence on modeling for problems \eqref{1} and \eqref{2}. We will use the results of this theorem to aid us in proving regularization in Section~\ref{reg_section}. \begin{theorem} \label{approx_thm} Let $-A$ be the infinitesimal generator of a bounded holomorphic semigroup of angle $\theta$ on a Banach space $X$ and let $0 \in \rho(A)$. For $0<\beta<1$, let the family of operators $f_{\beta}(t,A), 0\leq t\leq T$ be defined by $\eqref{f_beta}$. Let $u(t)$ and $v_{\beta}(t)$ be classical solutions of $\eqref{1}$ and $\eqref{2}$ respectively with $\chi \in X$, and assume that there exists a constant $M'\geq 0$ such that $\|A^{2+\kappa}u(t)\|\leq M'$ for all $t\in [s,T]$ where $\kappa$ is defined by Lemma~$\ref{Condition_Ap}$. Then there exist constants $\tilde{C}$ and $M$ independent of $\beta$ such that for $0\leq s\leq t 0$, $\chi \in X$, and define $\phi_{\epsilon}(\zeta) = e^{-(re^{\pm i\theta_1})A}C_{\epsilon}(u(t) -v_{\beta}(t))$ for $\zeta=t+re^{\pm i\theta_1} \in S$ as in the discussion preceding Lemma~\ref{u(zeta)_v(zeta)}. Intending to apply Lemma~\ref{AN}, we determine $\bar{\partial}\phi_{\epsilon}(\zeta)$. Since $e^{-(re^{\pm i\theta_1})A}$ is bounded for every $r\geq 0$ and since $C_{\epsilon}$ commutes with $A$, we have by Lemma~\ref{u(zeta)_v(zeta)}, \begin{align*} \frac{\partial}{\partial t}\phi_{\epsilon}(\zeta) &= \frac{\partial}{\partial t}e^{-(re^{\pm i\theta_1})A} (U_{\epsilon}(t,s)\chi - C_{\epsilon}V_{\beta}(t,s)\chi) \\ &= e^{-(re^{\pm i\theta_1})A}(\frac{\partial}{\partial t} U_{\epsilon}(t,s)\chi - C_{\epsilon}\frac{\partial}{\partial t} V_{\beta}(t,s)\chi) \\ &= e^{-(re^{\pm i\theta_1})A}(a(t)AU_{\epsilon}(t,s)\chi - f_{\beta}(t,A)C_{\epsilon}V_{\beta}(t,s)\chi) . \end{align*} Also, since $-A$ generates $e^{-zA}$ and since both $U_{\epsilon}(t,s)\chi$ and $C_{\epsilon}V_{\beta}(t,s)$ are in $\operatorname{Dom}(A)$, we have \begin{align*} \frac{\partial}{\partial r}\phi_{\epsilon}(\zeta) &= \frac{\partial}{\partial r}e^{-(re^{\pm i\theta_1})A}(U_{\epsilon}(t,s)\chi - C_{\epsilon}V_{\beta}(t,s)\chi) \\ &= e^{-(re^{\pm i\theta_1})A}(-e^{\pm i\theta_1}A)(U_{\epsilon}(t,s)\chi - C_{\epsilon}V_{\beta}(t,s)\chi). \end{align*} Therefore, by definition of the Cauchy-Riemann operator $\bar{\partial}$, \label{delbar} \begin{aligned} \bar{\partial}\phi_{\epsilon}(\zeta) &= \frac{1}{2i\;\sin (\pm \theta_1)} \Big(e^{\pm i\theta_1}\frac{\partial}{\partial t}\phi_{\epsilon}(\zeta) -\frac{\partial}{\partial r}\phi_{\epsilon}(\zeta)\Big) \\ &= \frac{e^{\pm i\theta_1}}{2i\;\sin (\pm \theta_1)} \Big[e^{-(re^{\pm i\theta_1})A}(a(t)AU_{\epsilon}(t,s)\chi - f_{\beta}(t,A)C_{\epsilon}V_{\beta}(t,s)\chi) \\ &\quad + e^{-(re^{\pm i\theta_1})A}(AU_{\epsilon}(t,s)\chi - AC_{\epsilon}V_{\beta}(t,s)\chi)\Big]. \end{aligned} Following \cite{AN}, define $\Phi_{\epsilon}(\zeta)=-\frac{1}{\pi}\iint_S \bar{\partial}\phi_{\epsilon}(z) \Big(\frac{1}{z-\zeta}+\frac{1}{\bar{z}+1+ \zeta}\Big)dx d\eta,$ where $z = x +\eta e^{\pm i \theta_1}$ and $\zeta =t+re^{\pm i\theta_1}$ are in $S$. In order to apply Lemma~\ref{AN}, we show that $\bar{\partial}\phi_{\epsilon}(z)$ is continuous and bounded on $S$. We first show that it is bounded on $S$. Let $z=x+\eta e^{\pm i \theta_1} \in S$ be arbitrary. We have from \eqref{delbar}, \begin{align*} \|\bar{\partial}\phi_{\epsilon}(z)\| \ &\leq \frac{1}{2|\sin \theta_1|}\; \|e^{-(\eta e^{\pm i\theta_1})A}\| \Big( \|a(x)AU_{\epsilon}(x,s)\chi - f_{\beta}(x,A)C_{\epsilon}V_{\beta}(x,s) \chi\| \\ &\quad + \|AU_{\epsilon}(x,s)\chi - AC_{\epsilon}V_{\beta}(x,s)\chi\| \Big) \\ &\leq \frac{\Theta}{2|\sin \theta_1|} \Big( \|a(x)AU_{\epsilon}(x,s) \chi-a(x)AC_{\epsilon}V_{\beta}(x,s)\chi\| \\ &\quad + \|a(x)AC_{\epsilon}V_{\beta}(x,s)\chi-f_{\beta}(x,A) C_{\epsilon}V_{\beta}(x,s)\chi\| \\ &\quad + \|AU_{\epsilon}(x,s)\chi-AC_{\epsilon}V_{\beta}(x,s)\chi\|\Big) \\ &\leq \frac{\Theta}{2|\sin \theta_1|} \Big((B+1)\|AU_{\epsilon}(x,s)\chi-AC_{\epsilon}V_{\beta}(x,s)\chi\| \\ & \quad + \|a(x)AC_{\epsilon}V_{\beta}(x,s)\chi-f_{\beta}(x,A)C_{\epsilon} V_{\beta}(x,s)\chi\|\Big) \end{align*} where we have set $\Theta=\max_{r \geq 0}\|e^{-(re^{\pm i\theta_1})A}\|$ and $B=\max_{t\in [0,T]}|a(t)|$. Since $U_{\epsilon}(x,s)\chi \in C_{\epsilon}(X)\subseteq \operatorname{Dom}(A^j)$ for every $j\in \mathbb{N}$ (cf. \cite[Proposition~2.10]{deL1}), it follows that $AU_{\epsilon}(x,s)\chi \in \operatorname{Dom}(A^j)$ for every $j$ as well. Therefore, we have $AU_{\epsilon}(x,s)\chi \in \operatorname{Dom}(A^{1+\kappa})$ by Lemma~\ref{fractional_props} (ii). Hence, by Corollary~\ref{factor_evsys}, Proposition~\ref{W_beta}, and Lemma~\ref{Condition_Ap}, \label{ev_sys_property} \begin{aligned} \|AU_{\epsilon}(x,s)\chi-AC_{\epsilon}V_{\beta}(x,s)\chi\| &= \|AU_{\epsilon}(x,s)\chi-AW_{\beta}(x,s)U_{\epsilon}(x,s)\chi\| \\ &= \|(I-W_{\beta}(x,s))AU_{\epsilon}(x,s)\chi\| \\ &= \big\|\int_s^{x}\frac{\partial}{\partial \tau}(W_{\beta}(x,\tau) AU_{\epsilon}(x,s)\chi)d\tau\big\| \\ &= \big\|\int_s^{x}-W_{\beta}(x,\tau)g_{\beta}(\tau,A)AU_{\epsilon}(x,s) \chi d\tau\big\| \\ &\leq \int_s^{x}L\|g_{\beta}(\tau,A)AU_{\epsilon}(x,s)\chi\|d\tau \\ &\leq T LR\beta \|A^{1+\kappa}AU_{\epsilon}(x,s)\chi\|. \end{aligned} Also, by Lemma~\ref{Condition_Ap}, \begin{align*} \|a(x)AC_{\epsilon}V_{\beta}(x,s)\chi-f_{\beta}(x,A)C_{\epsilon}V_{\beta}(x,s) \chi\| &= \|(-a(x)A+f_{\beta}(x,A))C_{\epsilon}V_{\beta}(x,s)\chi\| \\ &\leq R\beta \|A^{1+\kappa}C_{\epsilon}V_{\beta}(x,s)\chi\| \\ &= R\beta \|A^{1+\kappa}W_{\beta}(x,s)U_{\epsilon}(x,s)\chi\| \\ &= R\beta \|W_{\beta}(x,s)A^{1+\kappa}U_{\epsilon}(x,s)\chi\| \\ &\leq LR\beta \|A^{1+\kappa}U_{\epsilon}(x,s)\chi\|. \end{align*} Thus we have shown that $\|\bar{\partial}\phi_{\epsilon}(z)\| \leq \frac{\Theta (T+1)LR\beta }{2 |\sin \theta_1|} \Big((B+1)\|A^{1+\kappa}AU_{\epsilon}(x,s)\chi\|+\|A^{1+\kappa}U_{\epsilon}(x,s) \chi\|\Big).$ Now, by the assumption that $\|A^{2+\kappa}u(t)\|\leq M'$ for all $t\in [s,T]$ and by Lemma~\ref{fractional_props} (iv), we have $\|A^{1+\kappa}u(t)\|=\|A^{-1}A^{2+\kappa}u(t)\|\leq M''$ for all $t\in [s,T]$ for some constant $M''\geq 0$, where we have used the fact that $0 \in \rho(A)$. By the fact that $C_{\epsilon}=e^{-\epsilon A^{\alpha}}, \epsilon>0$ is a holomorphic semigroup, set $J= \sup_{0<\epsilon<1}\|C_{\epsilon}\|$. Then for small $\epsilon>0$, since $C_{\epsilon}$ commutes with $A$, we have from Lemma~\ref{C_epsilonu(t)}, $$\label{p(D)u(t)} \|A^{1+\kappa}U_{\epsilon}(x,s)\chi\|=\|A^{1+\kappa}C_{\epsilon}u(x)\| =\|C_{\epsilon}A^{1+\kappa}u(x)\|\leq JM''$$ and similarly $\|A^{1+\kappa}AU_{\epsilon}(x,s)\chi\|=\|A^{2+\kappa}U_{\epsilon}(x,s)\chi\| \leq JM'$. Therefore, we have shown that $$\label{delbar_bdd} \|\bar{\partial}\phi_{\epsilon}(z)\| \leq \beta C',$$ where $C'$ is a constant independent of $\epsilon$ and also of $\beta$ since $L$ is independent of $\beta$ (Proposition~\ref{W_beta} (i)). We have shown that $\bar{\partial}\phi_{\epsilon}(z)$ is bounded on $S$. It follows easily that $\bar{\partial}\phi_{\epsilon}(z)$ is also continuous on $S$. Having satisfied the hypotheses of Lemma~\ref{AN}, it follows that $\Phi_{\epsilon}(\zeta)$ is absolutely convergent, $\bar{\partial}\Phi_{\epsilon}(\zeta)=\bar{\partial}\phi_{\epsilon}(\zeta)$, and there exists a constant $\tilde{K}$ such that, for $x\neq t$, $\int_{-\infty}^{\infty}\big|\frac{1}{z-\zeta}+\frac{1}{\bar{z}+1+ \zeta} \big|d\eta \leq \tilde{K}\Big(1+\text{log}\frac{1}{|x-t|}\Big).$ We now construct a candidate to satisfy Carleman's Inequality. Define $\Psi_{\epsilon}:S\to \mathbb{C}$ by $\Psi_{\epsilon}(\zeta)=x^*(\phi_{\epsilon}(\zeta)-\Phi_{\epsilon}(\zeta))$ where $x^*\in X^*$, the dual space of $X$, is arbitrary. For $\zeta$ in the interior of $S$, using the results from Lemma~\ref{AN}, $\bar{\partial}\Psi_{\epsilon}(\zeta) = x^*(\bar{\partial}\phi_{\epsilon}(\zeta)-\bar{\partial}\Phi_{\epsilon}(\zeta)) = x^*(0)=0.$ Therefore, $\Psi_{\epsilon}$ is analytic on the interior of $S$ (cf. \cite[Theorem 11.2]{Rudin}). \\ \indent Next, we show that $\Psi_{\epsilon}$ is bounded on $S$. Similar to the calculation in \eqref{ev_sys_property}, and using \eqref{p(D)u(t)}, we have \label{phi_bdd} \begin{aligned} \|\phi_{\epsilon}(\zeta)\| &= \|e^{-(re^{\pm i\theta_1})A}(U_{\epsilon}(t,s)\chi-C_{\epsilon}V_{\beta}(t,s)\chi)\| \\ &\leq \Theta \|U_{\epsilon}(t,s)\chi-C_{\epsilon}V_{\beta}(t,s)\chi\| \\ &\leq \Theta T LR\beta \|A^{1+\kappa}U_{\epsilon}(t,s)\chi\| \leq \beta K' \end{aligned} where $K'$ is a constant independent of $\beta$, $\epsilon$, and $\zeta$. Next, from \eqref{delbar_bdd} and Lemma~\ref{AN}, \label{Phi_bdd} \begin{aligned} \|\Phi_{\epsilon}(\zeta)\| &= \Big\|-\frac{1}{\pi}\int \int_S \bar{\partial}\phi_{\epsilon}(z) \Big(\frac{1}{z-\zeta}+\frac{1}{\bar{z}+1+\zeta}\Big)dxd\eta \Big\| \\ &\leq \frac{1}{\pi}\beta C'\int_s^T \Big(\int_{-\infty}^{\infty} \big|\frac{1}{z-\zeta}+\frac{1}{\bar{z}+1+\zeta}\big| d\eta \Big) dx \\ &\leq \beta \frac{\tilde{K}}{\pi}C'\int_s^T \Big(1+\text{log}\frac{1}{|x - t|}\Big) dx \leq \beta C' \end{aligned} for a possibly different constant $C'$ independent of $\beta$, $\epsilon$, and $\zeta$. Then from \eqref{phi_bdd} and \eqref{Phi_bdd}, we have for $\zeta=t+re^{\pm i\theta_1} \in S$, \label{Psi_bdd} \begin{aligned} |\Psi_{\epsilon}(\zeta)| &= |x^*(\phi_{\epsilon}(\zeta)-\Phi_{\epsilon}(\zeta))| \\ &\leq \|x^*\|\big(\|\phi_{\epsilon}(\zeta)\|+\|\Phi_{\epsilon}(\zeta)\|\big) \\ &\leq \beta M\|x^*\| \end{aligned} where $M$ is a constant independent of $\beta$, $\epsilon$, and $\zeta$. We have shown that $\Psi_{\epsilon}$ is bounded on $S$. It is easy to show that $\Psi_{\epsilon}$ is also continuous on $S$, and we have already seen that $\Psi_{\epsilon}$ is analytic on the interior of $S$. By Carleman's Inequality (cf. \cite{Miller2}), we then obtain $$\label{3LT} |\Psi_{\epsilon}(t)| \leq M_{\epsilon}(s)^{1-h(t)}M_{\epsilon}(T)^{h(t)},$$ for $s\leq t\leq T$, where $M_{\epsilon}(t)= \sup_{r \geq 0}|\Psi_{\epsilon}(t+re^{\pm i\theta_1})|$ and $h$ is a harmonic function which is bounded and continuous on $S$ and assumes the values $0$ and $1$ respectively on the left and right hand boundary curves of $S$. Note that \begin{align*} \|\phi_{\epsilon}(s+re^{\pm i\theta_1})\| &= \|e^{-(re^{\pm i\theta_1})A}(U_{\epsilon}(s,s)\chi-C_{\epsilon}V_{\beta}(s,s)\chi)\| \\ &= \|e^{-(re^{\pm i\theta_1})A}(C_{\epsilon}\chi-C_{\epsilon}\chi)\| =0. \end{align*} Then from \eqref{Phi_bdd}, we have $|\Psi_{\epsilon}(s+re^{\pm i\theta_1})| \leq \|x^*\| \left(\|\phi_{\epsilon}(s+re^{\pm i\theta_1})\| +\|\Phi_{\epsilon}(s+re^{\pm i\theta_1})\|\right) \leq \|x^*\| \beta C',$ and so $$\label{M(0)} M_{\epsilon}(s) = \sup_{r \geq 0} |\Psi_{\epsilon}(s+re^{\pm i\theta_1})| \leq \beta C'\|x^*\|.$$ Also, from \eqref{Psi_bdd} and the fact that $0<\beta<1$, we have $$\label{M(T)} M_{\epsilon}(T) = \max_{r \geq 0} |\Psi_{\epsilon}(T+re^{\pm i\theta_1})| \leq M\|x^*\|.$$ From \eqref{3LT}, \eqref{M(0)}, and \eqref{M(T)}, it follows that for $s\leq t0,\; t\in [s,T]\} \] is a family of regularizing operators for the problem \eqref{1} where$\tilde{V}_{\beta}(t,s), 0\leq s\leq t\leq T$is the evolution system of Corollary~$\ref{well-posed_both}$corresponding to the operators$f_{\beta}(t,A-\lambda), 0\leq t\leq T$defined by $$\label{f_beta_Atilde} f_{\beta}(t,A-\lambda) = \begin{cases} a(t)(A-\lambda)-\beta (A-\lambda)^{\sigma} & \text{if } \theta \in (0,\pi/4] \\ a(t)(A-\lambda)(I+\beta (A-\lambda))^{-1} & \text{if } \theta \in (\pi/4,\pi/2] \end{cases}$$ where$\sigma>1$when$\theta \in (0,\pi/4]$. The regularization parameter$\beta$is chosen as follows: for a given perturbed initial data$\chi_{\delta}$where$\|\chi-\chi_{\delta}\|\leq \delta$, $\beta = \begin{cases} (-2K/\ln \delta)^{\sigma-1} & \text{if } \theta \in (0,\pi/4] \\ -2CT/\ln \delta & \text{if } \theta \in (\pi/4,\pi/2] \end{cases}$ where$K$and$C$are constants independent of$\delta$. \end{theorem} \begin{proof} First, in accordance with Theorem~\ref{approx_thm}, assume that$-A$generates a bounded holomorphic semigroup and that$0\in \rho(A)$. Let$u(t)$be a classical solution of \eqref{1} with initial data$\chi $and assume$u(t)$satisfies the stabilizing condition of Theorem~\ref{approx_thm}, that is$\|A^{2+\kappa}u(t)\| \leq M'$for all$t\in [s,T]$. Also, let$\|\chi-\chi_{\delta}\|\leq \delta$. Let$v_{\beta}(t)$be a solution of \eqref{2} and let$V_{\beta}(t,s), 0\leq s\leq t\leq T$be the evolution system given in Corollary~\ref{well-posed_both}. Then for$0\leq s\leq t\leq T$, we have$v_{\beta}(t)=V_{\beta}(t,s)\chiand \label{reg} \begin{aligned} \|u(t)-V_{\beta}(t,s)\chi_{\delta}\| &\leq \|u(t)-v_{\beta}(t)\|+\|V_{\beta}(t,s)\chi-V_{\beta}(t,s)\chi_{\delta}\| \\ &\leq \|u(t)-v_{\beta}(t)\| + \delta \|V_{\beta}(t,s)\|. \end{aligned} First consider0\leq s\leq t0, \; t\in [s,T]\}$is a family of regularizing operators for problem \eqref{1}. Now, for the general case, assume that$-A$generates a holomorphic semigroup of angle$\theta$on$X$. It is known that for$\theta' \in (0,\theta)$, then there exists$\lambda \in \mathbb{R}$such that$-A+\lambda$is the infinitesimal generator of a bounded holomorphic semigroup of angle$\theta'$on$X$and$0 \in \rho (A-\lambda)$(cf. \cite[Section~X.8, p. 253]{ReedandSimon}). Let$u(t)$be a classical solution of \eqref{1} with initial data$\chi \in X$. It is easily shown that$w(t)=e^{-(\int_s^ta(\tau)d\tau)\lambda}u(t)$is then a classical solution of the evolution problem \label{1shifted} \begin{gathered} \frac{dw}{dt} = a(t)(A-\lambda)w(t) \quad 0\leq s\leq t0, \; t\in [s,T]\}$ is a family of regularizing operators for the problem \eqref{1shifted} where $\tilde{V}_{\beta}(t,s), 0\leq s\leq t\leq T$ is the evolution system of Corollary~\ref{well-posed_both} corresponding to the family of operators $f_{\beta}(t,A-\lambda), 0\leq t\leq T$ defined by \eqref{f_beta_Atilde}. Hence, given $\delta >0$ and $\|\chi-\chi_{\delta}\|\leq \delta$, there exists $\beta>0$, such that $\beta \to 0$ as $\delta\to 0$ and \begin{align*} \|u(t)-e^{\left(\int_s^ta(\tau)d\tau\right)\lambda} \tilde{V}_{\beta}(t,s)\chi_{\delta}\| &= e^{\left(\int_s^ta(\tau)d\tau\right)\lambda}\|w(t) -\tilde{V}_{\beta}(t,s)\chi_{\delta}\| \\ &\to 0 \quad \text{as } \delta \to 0 \end{align*} for $0\leq s\leq t\leq T$, proving that $\{e^{(\int_s^ta(\tau)d\tau)\lambda}\tilde{V}_{\beta}(t,s) : \beta>0, \; t\in [s,T]\}$ is a family of regularizing operators for the problem \eqref{1}. \end{proof} \section{Examples in $L^p$ spaces} \label{ex_section} In this final section, we apply the theory of regularization in Section~\ref{reg_section} to ill-posed partial differential equations in $L^p$ spaces where $A$ is a strongly elliptic differential operator. We will use the following notation (cf. \cite[Chapter~7.1]{Pazy}). For an $n$-tuple of nonnegative integers $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_n)$ (called a multi-index), we define $|\alpha|=\sum_{i=1}^n\alpha_i$ and $x^{\alpha}=x_1^{\alpha_1}x_2^{\alpha_2}\dots x_n^{\alpha_n}$ for $x=(x_1,x_2,\dots,x_n) \in \mathbb{R}^n$. Also, denote $D_k=\partial/\partial x_k$ and $D=(D_1,D_2,\dots,D_n)$. Then $D^{\alpha}$ is defined by $D^{\alpha}=D_1^{\alpha_1}D_2^{\alpha_2}\dots D_n^{\alpha_n} =\frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}} \frac{\partial^{\alpha_2}}{\partial x_2^{\alpha_2}}\dots \frac{\partial^{\alpha_n}}{\partial x_n^{\alpha_n}}.$ Finally, for a fixed domain $\Omega$ in $\mathbb{R}^n$, $W^{m,p}(\Omega)$ will denote the Sobolev space consisting of functions $u \in L^p(\Omega)$ whose derivatives $D^{\alpha}u$, in the sense of distributions, of order $k\leq m$ are in $L^p(\Omega)$. Also, $W_0^{m,p}(\Omega)$ denotes the subspace of functions in $W^{m,p}(\Omega)$ with compact support in $\Omega$. Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$. Consider the differential operator of order $2m$, $$\label{diff_operator} P(x,D)=\sum_{|\alpha|\leq 2m}h_{\alpha}(x)D^{\alpha}$$ where the coefficients $h_{\alpha}(x)$ are sufficiently smooth complex-valued functions of $x$ in $\overline{\Omega}$, the closure of $\Omega$. \begin{definition}[{\cite[Definition~7.2.1]{Pazy}}] \rm The operator $P(x,D)$ is called \emph{strongly elliptic} if there exists a constant $c>0$ such that $\text{Re}\{(-1)^mP_{2m}(x,\xi)\}\geq c|\xi|^{2m}$ for all $x \in \overline{\Omega}$ and $\xi \in \mathbb{R}^n$, where $P_{2m}(x,\xi)=\sum_{|\alpha|=2m}h_{\alpha}(x)\xi^{\alpha}$. \end{definition} \begin{example} \label{example1} \rm Following \cite[Example~5.2]{HuangZheng2}, consider the nonautonomous problem $$\label{1_elliptic} \begin{gathered} \frac{\partial}{\partial t} u(t,x) = a(t)P(D)u(t,x), \quad (t,x)\in [s,T)\times \mathbb{R}^n \\ u(s,x) = \psi(x), \quad x\in \mathbb{R}^n \end{gathered}$$ where $a \in C([0,T]:\mathbb{R}^+)$ and $P:\mathbb{R}^n \to \mathbb{C}$ is a polynomial of order $2m$ such that $A=P(D)$ is strongly elliptic with domain $W^{2m,p}(\mathbb{R}^n)$ . Set $\mu_1= \sup_{|\xi|=1}|\text{Re}P_{2m}(\xi)|,\quad \mu_2= \sup_{|\xi|=1}|\text{Im}P_{2m}(\xi)|.$ Then, as seen in \cite{ZhengLi}, $-A=-P(D)$ is the generator of a holomorphic semigroup of angle $\theta$ on the Banach space $X=L^p(\mathbb{R}^n)$, $10, \; t\in [s,T]\}$ is a family of regularizing operators for the ill-posed problem \eqref{1_elliptic} where $\tilde{V}_{\beta}(t,s), 0\leq s\leq t\leq T$ is the evolution system of Corollary~\ref{well-posed_both} corresponding to the operators $f_{\beta}(t,P(D)-\lambda)=a(t)(P(D)-\lambda)-\beta (P(D)-\lambda)^{\sigma}.$ On the other hand, if $\mu_1> \mu_2$ or if $\mu_2=0$ so that $\theta \in (\pi/4,\pi/2]$, then for some $\lambda \in \mathbb{R}$, \eqref{2} becomes \begin{gather*} (1-\beta \lambda+\beta P(D))\frac{\partial}{\partial t} v(t,x) = a(t)(P(D)-\lambda) v(t,x) \\ \text{for } (t,x)\in [s,T)\times \mathbb{R}^n, \\ v(s,x) = \psi(x) \quad \text{for } x\in \mathbb{R}^n. \end{gather*} Again, by Theorem~\ref{reg_thm}, $\{e^{(\int_s^ta(\tau)d\tau)\lambda}\tilde{V}_{\beta}(t,s) : \beta>0, \; t\in [s,T]\}$ is a family of regularizing operators for the ill-posed problem \eqref{1_elliptic} where $\tilde{V}_{\beta}(t,s), 0\leq s\leq t\leq T$ is the evolution system of Corollary~\ref{well-posed_both}, in this case corresponding to the operators $f_{\beta}(t,P(D)-\lambda)=a(t)(P(D)-\lambda)(I+\beta (P(D) -\lambda))^{-1}$. Note, as mentioned in the introduction, the model \eqref{2_elliptic} may still be used with $\sigma =2$ if $\theta > \pi/4$. \end{example} \begin{example} \label{example2} \rm Following \cite[Chapter~7.6]{Pazy}, consider the nonautonomous problem \label{1_elliptic2} \begin{gathered} \frac{\partial}{\partial t} u(t,x) = a(t)P(x,D)u(t,x) \quad \text{for } (t,x)\in [s,T)\times \Omega \\ D^{\alpha}u(t,x) = 0, \quad |\alpha|0, \; t\in [s,T]\}\$ is a family of regularizing operators for the ill-posed problem \eqref{1_elliptic2}. \end{example} \subsection*{Acknowledgments} The author would like to thank Rhonda J. Hughes for her enthusiasm, encouragement, and willingness to offer what is always excellent advice. \begin{thebibliography}{00} \bibitem{AN} S. Agmon, L. Nirenberg; \emph{Properties of solutions of ordinary differential equations in Banach space}, Comm. Pure Appl. Math. 16 (1963) 121--151. \bibitem{AmesandHughes} K. A. Ames, R. J. 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