\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
{\em Electronic Journal of Differential Equations},
Conference 19 (2010),  pp. 99--121.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document} \setcounter{page}{99}
\title[\hfilneg EJDE-2010/Conf/19/\hfil Continuous dependence of solutions]
{Continuous dependence of solutions for ill-posed evolution problems}

\author[M. Fury, R. J. Hughes\hfil EJDE/Conf/19 \hfilneg]
{Matthew Fury, Rhonda J. Hughes}  % in alphabetical order

\address{Matthew Fury \newline
Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA}
\email{mfury@brynmawr.edu}

\address{Rhonda J. Hughes \newline
Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, USA}
\email{rhughes@brynmawr.edu}

\thanks{Published September 25, 2010.}
\subjclass[2000]{47A52, 42C40}
\keywords{Continuous dependence on modelling; time-dependent problems;
\hfill\break\indent  Ill-posed problems}

\begin{abstract}
 We prove H\"older-continuous dependence results for the difference
 between certain ill-posed and well-posed evolution problems in
 a Hilbert space.  Specifically, given a positive self-adjoint
 operator $D$ in a Hilbert space, we consider the ill-posed evolution
 problem
 \begin{gather*}
   \frac{du(t)}{dt} = A(t,D)u(t) \quad  0\leq t<T \\
   u(0) = \chi.
 \end{gather*}
 We determine functions $f:[0,T]\times [0,\infty)\to \mathbb{R}$
 for which solutions of the well-posed problem
 \begin{gather*}
    \frac{dv(t)}{dt} = f(t,D)v(t) \quad  0\leq t<T \\
    v(0) = \chi
 \end{gather*}
 approximate known solutions of the original ill-posed problem,
 thereby establishing continuous dependence on modelling for
 the problems under consideration.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Let $D$ be a positive self-adjoint operator in a Hilbert space $H$,
and consider the evolution problem
\begin{equation} \label{e1}
\begin{gathered}
    \frac{du(t)}{dt} = A(t,D)u(t) \quad   0\leq t< T  \\
                u(0) = \chi
\end{gathered}
\end{equation}
where $\chi$ is an arbitrary element of $H$ and
$$
A(t,D)= \sum_{j=1}^ka_j(t)D^j,
$$
with $a_j:[0,T]\to \mathbb{R}$ continuous and nonnegative for
 each $1\leq j\leq k$.  In general, \eqref{e1} is ill-posed with formal
solution given by
$$
u(t)=\text{exp}\Big\{\sum_{j=1}^k\Big(\int_0^ta_j(s)ds\Big)D^j\Big\}\chi.
$$
 Now let $f:[0,T]\times [0,\infty)\to \mathbb{R}$ be a function
continuous in $t$ and Borel in $\lambda$, and consider the evolution
problem
\begin{equation} \label{e2}
\begin{gathered}
    \frac{dv(t)}{dt} = f(t,D)v(t) \quad   0\leq t< T  \\
                v(0) = \chi\,,
\end{gathered}
\end{equation}
where for each $t\in [0,T]$, $f(t,D)$ is defined by means of the
functional calculus for self-adjoint operators in $H$.
In particular, since $D$ is positive, self-adjoint, the spectrum
$\sigma(D)$ of $D$ is contained in $[0,\infty)$ and for each
 $t\in [0,T]$,
$$
f(t,D)x=\int_0^{\infty}f(t,\lambda)dE(\lambda)x,
$$
for $x\in \operatorname{Dom}(f(t,D))=\{x\in H :
 \int_0^{\infty}|f(t,\lambda)|^2d(E(\lambda)x,x)<\infty\}$,
where $\{E(\cdot)\}$ denotes the resolution of the identity for
the self-adjoint operator $D$.

We determine conditions on $f$ so that \eqref{e2} is well-posed
and  such that solutions of \eqref{e2} approximate known solutions
of \eqref{e1}. In this way, we illustrate how we might stabilize
problems against errors that arise when formulating mathematical
models such as \eqref{e1} in attempts to describe some physical
process. Ames and Hughes \cite{a3} established such structural stability
results for the problem in the autonomous case, that is when
$A(t)=A$ is a positive self-adjoint operator in $H$, independent
of $t$.  This paper generalizes such work and yields a comparable
result in the time-dependent case.  Namely, if $u(t)$ and $v(t)$
are solutions of \eqref{e1} and \eqref{e2} respectively, we prove
the H\"{o}lder-continuous approximation
$$
\|u(t)-v(t)\|\leq C\beta^{1-\frac{t}{T}}M^{t/T},
$$ where
$0<\beta<1$, and $C$ and $M$ are constants independent of $\beta$.
Our approximation establishes continuous dependence on modelling,
meaning ``small" changes to our model yield a ``small" change in
the corresponding solution.

In Section 2, we establish conditions under which \eqref{e2} is
well-posed using stable families of generators of semigroups and
Kato's stability conditions \cite{k1,p1}; our work also
utilizes Tanaka's results on evolution problems \cite{t1}.  In
Section 3, we present our approximation theorem which achieves
H\"{o}lder-continuous dependence on modelling.  Finally, Section 4
demonstrates our theorem with examples.

Below, for a closed operator $A$ in a Banach space $X$, $\rho(A)$
will denote the resolvent set of $A$, and for $\lambda \in
\rho(A)$, $R(\lambda;A)$ will denote the resolvent operator
$R(\lambda;A)=(\lambda I - A)^{-1}$ in $X$.

\section{The Well-Posed Evolution Problem}

We first clarify what we mean by solutions of evolution problems
as well as the notions of ill-posed and well-posed evolution
problems.

\begin{definition}[\cite{p1}] \label{def1} \rm
 Let $X$ be a Banach space and for every $t\in [0,T]$, let
$A(t)$ be a linear operator in $X$.  The initial value problem
\begin{equation} \label{star}
\begin{gathered}
    \frac{du(t)}{dt} = A(t)u(t) \quad  0\leq s\leq t< T\\
    u(s) = x
\end{gathered}
\end{equation}
 is called an \emph{evolution problem}.  An $X$-valued function
$u:[s,T]\to X$ is a \emph{classical solution} of \eqref{star} if
$u$ is continuous on $[s,T]$, $u(t)\in \operatorname{Dom}(A(t))$
for $s< t< T$, $u$ is continuously differentiable on $(s,T)$, and
$u$ satisfies \eqref{star}.
\end{definition}

\begin{theorem}[{\cite[Theorem 5.1.1]{p1}}] \label{thm1}
 Let $X$ be a Banach space and for every $t\in [0,T]$, let
$A(t)$ be a bounded linear operator on $X$.  If the function
$t\mapsto A(t)$ is continuous in the uniform operator topology
then for every $x\in X$, the initial value problem \eqref{star}
has a unique classical solution $u$.
\end{theorem}

The proof of Theorem \ref{thm1}
 (cf. \cite[Theorem 5.1.1]{p1}) shows that the
mapping $S:C([s,T]:X)\to C([s,T]:X)$ defined by
$$
(Su)(t)=x+\int_s^tA(\tau)u(\tau)d\tau
$$
is a well-defined mapping with a unique fixed point $u$.  It is
easily shown that $u$ is then a unique classical solution of
\eqref{star}.  In this case we define the \emph{solution operator}
of \eqref{star} by
$$
U(t,s)x=u(t) \quad \text{for } 0\leq s\leq t\leq T.
$$

\begin{theorem}[{\cite[Theorem 5.1.2]{p1}}] \label{thm2}
 Let $U(t,s)$ be the solution operator associated with
\eqref{star} where $A(t)$ is a bounded linear operator on $X$ for
each $t\in [0,T]$ and $t\mapsto A(t)$ is continuous in the uniform
operator topology.  Then for every $0\leq s\leq t\leq  T$,
$U(t,s)$ is a bounded linear operator such that
\begin{itemize}
\item[(i)] $\|U(t,s)\| \leq e^{\int_s^t\|A(\tau )\|d\tau} $.

\item[(ii)]  $U(t,t) = I, \;\; U(t,s) = U(t,r)U(r,s)$ for $0\leq
s\leq r\leq t\leq  T$.

\item[(iii)]  $(t,s)\mapsto U(t,s)$ is continuous in the uniform
operator topology for $0\leq s\leq t\leq  T$.

\item[(iv)]  $\frac{\partial U(t,s)}{\partial t} = A(t)U(t,s)$ for
$0\leq s\leq t\leq T$.

\item[(v)]  $\frac{\partial U(t,s)}{\partial s} = -U(t,s)A(s)$ for
$0\leq s\leq t\leq T$.
\end{itemize}
\end{theorem}

We now turn to the notions of well-posedness and ill-posedness
for evolution problems.

\begin{definition}[{\cite[Definition 7.1]{f1}}] \label{def2} \rm
 The evolution problem \eqref{star} is called
\emph{well-posed} in $0\leq t< T$ if the following two assumptions
are satisfied:
\begin{itemize}
\item[(i)] (Existence of solutions for sufficiently many initial
data) There exists a dense subspace $Y$ of $X$ such that for every
$s\in [0,T)$ and every $x\in Y$, there exists a classical solution
$u(t)$ of \eqref{star}.

\item[(ii)] (Continuous dependence of solutions on their initial
data) There exists a strongly continuous $B(X)$-valued function
$U(t,s)$ defined in $0\leq s\leq t\leq T$ such that if $u(t)$ is a
classical solution of \eqref{star}, then
$$
u(t)=U(t,s)x.
$$
\end{itemize}
Equation \eqref{star} is called \emph{ill-posed} if it is not
well-posed.
\end{definition}

It is clear that under the hypotheses of Theorem \ref{thm1}, that Equation
\eqref{star} is well-posed with unique classical solution given by
$u(t)=U(t,s)\chi$ where $U(t,s), 0\leq s\leq t\leq T$, is the
solution operator given by Theorem \ref{thm1}.  Otherwise, we use the
construction of an evolution system and the theory of stable
families of operators to obtain well-posedness of \eqref{star}. We
explore stability conditions for \eqref{star} first developed by
Kato \cite{k1,p1}, and later by Tanaka  \cite{t1}.

\begin{definition}[{\cite[Definition 5.1.3]{p1}}] \label{def3} \rm
A two parameter family of bounded linear operators $U(t,s)$,
$0\leq s\leq t\leq T$, on a Banach space $X$ is called an
\emph{evolution system} if the following two conditions are
satisfied:
\begin{itemize}
\item[(i)] $U(s,s)=I, \; U(t,r)U(r,s)=U(t,s)$ for $0\leq s\leq r\leq
t\leq T$.

\item[(ii)] $(t,s)\mapsto U(t,s)$ is strongly continuous for $0\leq
s\leq t\leq T$.
\end{itemize}
\end{definition}

\begin{definition}[{\cite[Def. 5.2.1]{p1}}] \label{def4} \rm
Let $X$ be a Banach space.  A family $\{A(t)\}_{t\in [0,T]}$ of
infinitesimal generators of $C_0$ semigroups on $X$ is called
\emph{stable} if there are constants $M\geq 1$ and $\omega$
(called the stability constants) such that
$$
\rho(A(t))\supseteq (\omega,\infty) \quad \text{for }  t\in[0,T]
$$
and
$$
\|\prod_{j=1}^kR(\lambda;A(t_j))\|\leq
M(\lambda-\omega)^{-k} \quad \text{for }  \lambda>\omega
$$
and every finite sequence $0\leq t_1\leq t_2,\dots ,t_k\leq T$,
$k=1,2,\dots $.
\end{definition}

\noindent \textbf{Remark.} If for $t\in [0,T]$, $A(t)$ is the
infinitesimal generator  of a $C_0$ semigroup $\{S_t(s)\}_{s\geq
0}$ satisfying $\|S_t(s)\|\leq e^{\omega s}$, then by the
Hille-Yosida theorem (cf. \cite{p1}), the family  $\{A(t)\}_{t\in
[0,T]}$ is stable with constants $M=1$ and $\omega$.

We now use the theory of stable families of operators to gain
well-posedness of \eqref{star} in the following way.  Let $X$ and
$Y$ be Banach spaces with norms $\|\cdot\|$ and $\|\cdot\|_Y$
respectively.  Assume that $Y$ is densely and continuously
imbedded in $X$, that is $Y$ is a dense subspace of $X$ and there
is a constant $C$ such that
$$
\|y\|\leq C\|y\|_Y \quad \text{for } y\in Y.
$$
For each $t\in [0,T]$, let $A(t)$ be the
infinitesimal generator of a $C_0$ semigroup
$\{S_t(s)\}_{s\geq 0}$ on $X$.  Assume the following
conditions (cf. \cite{k1,p1}):
\begin{itemize}
\item[(H1)] $\{A(t)\}_{t\in [0,T]}$ is a stable family with
stability constants $M$, $\omega$.

\item[(H2)] For each $t\in [0,T]$, $Y$ is an invariant subspace
of $S_t(s),s\geq 0$, the restriction $\tilde{S}_t(s)$ of $S_t(s)$
 to $Y$ is a $C_0$ semigroup in $Y$, and the family
$\{\tilde{A}(t)\}_{t\in [0,T]}$ of parts $\tilde{A}(t)$ of $A(t)$
in $Y$, is a stable family in $Y$.

\item[(H3)] For $t\in [0,T]$,
$\operatorname{Dom}(A(t))\supseteq Y$, $A(t)$ is a bounded
operator from $Y$ into $X$, and $t\mapsto A(t)$ is continuous
in the $B(Y,X)$ norm $\|\cdot\|_{Y\to X}$.
\end{itemize}

\begin{theorem}[{\cite[Theorem 4.1]{k1}, \cite[Theorem 5.3.1]{p1}}]
\label{thm3}
For each $t\in [0,T]$, let $A(t)$ be the infinitesimal generator
of  a $C_0$ semigroup $\{S_t(s)\}_{s\geq 0}$ on $X$.
If the family $\{A(t)\}_{t\in [0,T]}$ satisfies conditions
{\rm (H1)--(H3)}, then there exists a unique evolution system
$U(t,s)$, $0\leq s\leq t\leq T$, in $X$ satisfying
\begin{itemize}
\item[(E1)] $\|U(t,s)\|\leq Me^{\omega (t-s)}$ for
$0\leq s\leq t\leq T$.

\item[(E2)] $\frac{\partial^+}{\partial t}U(t,s)y\big|_{t=s}=A(s)y$
for $y\in Y$, $0\leq s\leq T$.

\item[(E3)] $\frac{\partial}{\partial s}U(t,s)y=-U(t,s)A(s)y$
for $y\in Y$, $0\leq s\leq t\leq T$,

\end{itemize}
where the derivative from the right in (E2) and the derivative
in (E3) are in the strong sense in $X$.
\end{theorem}

This theorem  will help in obtaining a certain kind of classical
solution of \eqref{star} in the case where the family
$\{A(t)\}_{t\in [0,T]}$ of infinitesimal generators of $C_0$
semigroups on $X$ satisfies conditions  (H1)--(H3).

\begin{definition}[{\cite[Definition 5.4.1]{p1}}] \label{def5} \rm
 Let $X$ and $Y$ be Banach spaces such
that $Y$ is densely and continuously imbedded in $X$ and let
$\{A(t)\}_{t\in [0,T]}$ be a family of infinitesimal generators of
$C_0$ semigroups on $X$ satisfying the assumptions (H1)--(H3).
A function $u\in C([s,T]:Y)$ is a \emph{Y-valued solution} of
\eqref{star} if $u\in C^1((s,T):X)$ and $u$ satisfies \eqref{star}
in $X$.
\end{definition}

\noindent \textbf{Remark.}  A $Y$-valued solution $u$ of
\eqref{star} is a classical solution of \eqref{star} such that
$u(t)\in Y\subseteq \operatorname{Dom}(A(t))$ for $t\in [s,T]$ and
$u(t)$ is continuous in the stronger $Y$-norm rather than merely
in the $X$-norm.

\begin{theorem}[{\cite[Thm. 5.4.3]{p1}}] \label{thm4}
Let $\{A(t)\}_{t\in [0,T]}$ satisfy the conditions of Theorem
\ref{thm3} and let $U(t,s)$, $0\leq s\leq t\leq T$ be the
evolution system given in Theorem \ref{thm3}.  If
\begin{itemize}
\item[(E4)] $U(t,s)Y\subseteq Y$ for $0\leq s\leq t\leq T$ and

\item[(E5)] For $x\in Y$, $U(t,s)x$ is continuous in $Y$ for
$0\leq s\leq t\leq T$,

\end{itemize} then for every $x\in Y$, $U(t,s)x$ is the unique
$Y$-valued solution of \eqref{star}.
\end{theorem}

We now use the above theory of stable families of generators to
give criteria for well-posedness of the evolution problem
\eqref{e2}.  Let \eqref{e2'} denote the initial value problem
\eqref{e2} with $0$ replaced by $s$ for $s\in [0,T)$; i.e.,
\begin{equation}
\begin{gathered}   \label{e2'}
\frac{dv(t)}{dt}
= f(t,D)v(t) \quad   0\leq s\leq t< T   \\
                v(s) = \chi.
\end{gathered}
\end{equation}
We determine conditions on $f$ so that the family of operators
$\{f(t,D)\}_{t\in [0,T]}$ is stable and such that \eqref{e2'} is
well-posed.

\begin{proposition} \label{prop1}
Let $f:[0,T]\times [0,\infty)\to \mathbb{R}$ be continuous in $t$
and Borel in $\lambda$.  Assume there exist $\omega\in \mathbb{R}$
such that $f(t,\lambda)\leq \omega$ for all $(t,\lambda)\in
[0,T]\times [0,\infty)$ and a Borel function $r:[0,\infty)\to
[0,\infty)$ such that $|f(t,\lambda)|\leq r(\lambda)$ and
$\operatorname{Dom}(f(t,D))=\operatorname{Dom}(r(D))$ for all
$t\in [0,T]$.  Set $Y=\operatorname{Dom}(r(D))$ and let $\|\cdot
\|_Y$ denote the graph norm associated with the operator $r(D)$.
Further, assume $t\mapsto f(t,D)$ is continuous in the $B(Y,H)$
norm $\|\cdot \|_{Y\to H}$.  Then \eqref{e2'} is well-posed and
for $\chi \in Y$, $V(t,s)\chi=e^{\int_s^tf(\tau,D)d\tau}\chi$ is a
unique $Y$-valued solution of \eqref{e2'}.
\end{proposition}

\begin{proof}
  By \cite[Theorem XII.2.6]{d1}, $r(D)$ is a closed operator in $H$
with dense domain.  Set $Y=\operatorname{Dom}(r(D))$ and endow
$Y$ with the graph norm $\|\cdot\|_Y$ given by
$$
\|y\|_Y=\|y\|+\|r(D)y\|
$$
for all $y\in Y$.  Since $r(D)$ is a closed operator, it follows
that $(Y,\|\cdot\|_Y)$ is a Banach space.  It is also clear
that $Y$ is densely and continuously imbedded in $H$.
Since $f(t,\lambda)\leq \omega$ for all  $(t,\lambda)\in
[0,T]\times [0,\infty)$, we have that for each $t\in [0,T]$,
$f(t,D)$ is the infinitesimal generator of the $C_0$ semigroup
$\{S_t(s)\}_{s\geq 0}$ on $H$ given by $S_t(s)=e^{sf(t,D)}$.  We
show that the family $\{f(t,D)\}_{t\in [0,T]}$ satisfies
conditions (H1)--(H3).

Let $t\in [0,T]$, $x\in H$.  Then
\[
    \|e^{sf(t,D)}x\|^2
= \int_0^{\infty}|e^{sf(t,\lambda)}|^2d(E(\lambda)x,x)
\leq  (e^{s\omega})^2\int_0^{\infty}d(E(\lambda)x,x)
= (e^{s\omega})^2\|x\|^2,
\]
showing that $\|S_t(s)\|=\|e^{sf(t,D)}\|\leq e^{\omega s}$. Thus,
$\{f(t,D)\}_{t\in [0,T]}$ is a stable family with stability
constants $M=1$ and $\omega$, and so (H1) is satisfied.

Next, let $t\in [0,T]$, $y\in Y$.  For any $s\geq 0$, since $y\in
Y=\operatorname{Dom}(r(D))$, we have
$$
\int_0^{\infty}|r(\lambda)e^{sf(t,\lambda)}|^2d(E(\lambda)y,y)\leq
(e^{s\omega})^2\int_0^{\infty}|r(\lambda)|^2d(E(\lambda)y,y)<\infty.
$$
Thus, $S_t(s)y\in \operatorname{Dom}(r(D))$ and so $Y$ is an
invariant subspace of $S_t(s)$.  Let $\tilde{S}_t(s)$ be the
restriction of $S_t(s)$ to $Y$.  For any positive constant $c$,
for $0\leq s\leq c$,
$$
|r(\lambda)(e^{sf(t,\lambda)}-1)|^2 \leq
|r(\lambda)|^2(e^{c\omega}+1)^2\in L^1(E(\cdot)y,y).
$$
Therefore, by Lebesgue's Dominated Convergence Theorem,
\begin{align*}
    \lim_{s\to 0^+}\|r(D)(S_t(s)-I)y\|^2
&=  \lim_{s\to 0^+}\int_0^{\infty}|r(\lambda)(e^{sf(t,\lambda)}-1)
 |^2d(E(\lambda)y,y) \\
&=  \int_0^{\infty}\lim_{s\to 0^+}|r(\lambda)(e^{sf(t,\lambda)}-1)
|^2d(E(\lambda)y,y)
=  0,
\end{align*}
and so
\begin{align*}
    \|\tilde{S}_t(s)y-y\|_Y
&=  \|\tilde{S}_t(s)y-y\|+\|r(D)(\tilde{S}_t(s)y-y)\| \\
&=  \|S_t(s)y-y\|+\|r(D)(S_t(s)-I)y\| \\
&\to  0 \quad \text{as }  s\to 0^+.
\end{align*}
Thus, $\tilde{S}_t(s)$ is a $C_0$ semigroup on $Y$.

 Next, consider the family $\{\tilde{f}(t,D)\}_{t\in
[0,T]}$ of parts $\tilde{f}(t,D)$ of $f(t,D)$ in $Y$.  For each
$t\in [0,T]$, $\tilde{f}(t,D)$ is defined by
$$
\operatorname{Dom}(\tilde{f}(t,D))=\{x\in
\operatorname{Dom}(f(t,D))\cap Y : f(t,D)x\in Y\}
$$
and
$$
\tilde{f}(t,D)x=f(t,D)x \quad \text{for }  x\in
\operatorname{Dom}(\tilde{f}(t,D)).
$$
It is seen \cite[Theorem 4.5.5]{p1} that $\tilde{f}(t,D)$ is the
infinitesimal generator of the $C_0$ semigroup $\tilde{S}_t(s)$.
Moreover, for
$y\in Y$,
\begin{align*}
    \|\tilde{S}_t(s)y\|_Y
&=  \|\tilde{S}_t(s)y\| +\|r(D)\tilde{S}_t(s)y\| \\
&=  \|S_t(s)y\| +\|r(D)S_t(s)y\| \\
&\leq  e^{s\omega}\|y\| + e^{s\omega}\|r(D)y\| \\
&=  e^{s\omega}\|y\|_Y.
\end{align*}
Thus, $\|\tilde{S}_t(s)\|_Y\leq e^{\omega s}$ for all $t\in [0,T]$
and so the family $\{\tilde{f}(t,D)\}_{t\in [0,T]}$ is stable with
stability constants $\tilde{M}=1$ and $\omega$.  We have shown
that (H2) is satisfied.

 Finally, let $t\in [0,T]$.  Since $|f(t,\lambda)|\leq r(\lambda)$,
we have for $y\in Y$,
$$
\int_0^{\infty}|f(t,\lambda)|^2d(E(\lambda)y,y)
\leq \int_0^{\infty}|r(\lambda)|^2d(E(\lambda)y,y)<\infty.
$$
Thus $\operatorname{Dom}(f(t,D))\supseteq Y$.  Also, for $y\in Y$,
\[
 \|f(t,D)y\| \leq  \|y\| + \|f(t,D)y\|
\leq  \|y\|+\|r(D)y\| =  \|y\|_Y,
\]
showing that $f(t,D)$ is a bounded operator from $Y$ into $H$.
By assumption, $t\mapsto f(t,D)$ is continuous in the $B(Y,H)$
norm $\|\cdot \|_{Y \to H}$ and so (H3) is satisfied.
By Theorem \ref{thm3}, there exists a unique evolution system $V(t,s)$,
$0\leq s\leq t\leq T$, in $H$ satisfying conditions (E1)-(E3)
with the operators $f(t,D)$, $t\in [0,T]$, and $M=1$ in the
condition (E1); that is we have
\begin{gather*}
\|V(t,s)\|\leq e^{\omega (t-s)} \quad \text{for }
 0\leq s\leq t\leq T, \\
\frac{\partial^+}{\partial t}V(t,s)y\big|_{t=s}=f(s,D)y \quad
 \text{for } y\in Y, \; 0\leq s\leq T,\\
\frac{\partial}{\partial s}V(t,s)y=-V(t,s)f(s,D)y \quad \text{for }
 y\in Y, \; 0\leq s\leq t\leq T,
\end{gather*}
where the derivatives are in the strong sense in $H$.
It can be shown using the Spectral Theorem that
$e^{\int_s^tf(\tau,D)d\tau}$ is such an evolution system,
and so by uniqueness we must have $V(t,s)=e^{\int_s^tf(\tau,D)d\tau}$.
It is also readily seen that $V(t,s)=e^{\int_s^tf(\tau,D)d\tau}$
satisfies (E4) and (E5).  Therefore, by Theorem \ref{thm4}, for every
$\chi\in Y$, $V(t,s)\chi=e^{\int_s^tf(\tau,D)d\tau}\chi$
is the unique $Y$-valued solution of \eqref{e2'}.

 Finally, suppose $v_1$ is a classical solution of
\eqref{e2'}.  Then $v_1(q)\in
\operatorname{Dom}(f(q,D))=\operatorname{Dom}(r(D))$ for
$q\in (s,T)$.  As $V(t,s), \; 0\leq s\leq t\leq T$, satisfies
condition (E3) with the operators $f(t,D)$, $t\in [0,T]$, the function
$q\mapsto V(t,q)v_1(q)$ is then differentiable and
\begin{align*}
    \frac{\partial}{\partial q} V(t,q)v_1(q)
&=  -V(t,q)f(q,D)v_1(q)+V(t,q)\frac{d}{dq}v_1(q) \\
&=  -V(t,q)f(q,D)v_1(q)+V(t,q)f(q,D)v_1(q) = 0.
\end{align*}
Thus $V(t,q)v_1(q)$ is constant for $q\in (s,t)$.  Since $v_1$ is
a classical solution, the function $V(t,q)v_1(q)$ is also continuous for
$q\in [s,t]$.  Thus we have
$$
v_1(t)=V(t,t)v_1(t)=V(t,s)v_1(s)=V(t,s)\chi.
$$
Thus condition
(ii) of Definition \ref{def2} is satisfied and we see that \eqref{e2'}
is well-posed with unique classical solution given by
$v(t)=V(t,s)\chi$.
\end{proof}

\section{The Approximation Theorem}

In order that solutions of \eqref{e2} approximate known solutions
of \eqref{e1}, we will require additional conditions on $f$.  The
following definition is inspired by results obtained by Ames and
Hughes \cite[Definition 1]{a3} for continuous dependence on
modelling in the autonomous case, that is when $A(t)=A$ is
independent of $t$.

\begin{definition} \label{def6} \rm
Let $f:[0,T]\times [0,\infty)\to \mathbb{R}$ be a function
continuous in $t$ and Borel in $\lambda $ and assume the hypotheses
of Proposition \ref{prop1}.  Then $f$ is said to satisfy the
 \emph{approximation condition with polynomial p} or simply
\emph{Condition} $(\mathcal{A},p)$ if there exist a constant
$\beta$, with $0<\beta<1$, and a nonzero polynomial $p(\lambda)$
independent of $\beta$ such that for each $t\in [0,T]$,
$\operatorname{Dom}(p(D))\subseteq \operatorname{Dom}(A(t,D))
\cap \operatorname{Dom}(f(t,D))$, and
$$
\|(-A(t,D)+f(t,D))\psi\|\leq \beta\|p(D)\psi\|,
$$
for all $\psi\in \operatorname{Dom}(p(D))$.
\end{definition}

Now assume $f$ satisfies Condition $(\mathcal{A},p)$.
For each $t\in [0,T]$, set
$$
g(t,\lambda)=-A(t,\lambda)+f(t,\lambda),
$$
and for each $n\geq |\omega|$, set
$$
e_n=\{\lambda\in [0,\infty): \max_{t\in [0,T]}|g(t,\lambda)|\leq n\}.
$$
Then
\begin{gather*}
\lambda\in e_n
\Rightarrow  \max_{t\in [0,T]}|g(t,\lambda)|\leq n \\
\Rightarrow  |g(t,\lambda)|\leq n \quad  \forall t\in [0,T] \\
\Rightarrow  A(t,\lambda)\leq n+f(t,\lambda) \quad  \forall t\in [0,T].
\end{gather*}
 Since $A(t,\lambda)\geq 0$ and $f(t,\lambda)\leq \omega$ for all
$(t,\lambda)\in [0,T]\times [0,\infty)$, we have that on $e_n$,
$$
\max_{t\in [0,T]}|A(t,\lambda)|\leq n+\omega.
$$
 Since $f(t,\lambda)=A(t,\lambda)+g(t,\lambda)$, it then follows
that on $e_n$,
$$
\max_{t\in [0,T]}|f(t,\lambda)|\leq 2n+\omega.
$$
Set $E_n=E(e_n)$ and let $\psi\in H$ be arbitrary.  Consider
the following three evolution problems:
\begin{equation} \label{e3}
\begin{gathered}
\frac{du_n(t)}{dt} =  A(t,D)E_nu_n(t) \quad 0\leq s \leq t < T  \\
    u_n(s) =  \psi,
\end{gathered}
\end{equation}
\begin{equation} \label{e4}
\begin{gathered}
 \frac{dv_n(t)}{dt} =  f(t,D)E_nv_n(t) \quad   0\leq s \leq t < T  \\
    v_n(s) =  \psi,
\end{gathered}
\end{equation}
\begin{equation} \label{e5}
\begin{gathered}
    \frac{dw_n(t)}{dt} =  g(t,D)E_nw_n(t) \quad   0\leq s \leq t < T  \\
    w_n(s) =  \psi.
\end{gathered}
\end{equation}
Problems \eqref{e3}--\eqref{e5}, as we will see, are well-posed due to
the action of $E_n$ and their solutions will aid in approximating
known solutions of the ill-posed problem \eqref{e1}.

\begin{lemma} \label{lem1}
For each $t\in [0,T]$, $A(t,D)E_n$ is a bounded operator on
$H$ such that
$$
\|A(t,D)E_n\|\leq n+\omega,
$$
and \eqref{e3} has a unique classical solution
$u_n(t)=U_n(t,s)\psi$.  The solution operator $U_n(t,s)$
is a bounded operator on $H$ with
$$
\|U_n(t,s)\|\leq e^{T(n+\omega)}
$$
for all $s,t$ such that $0\leq s\leq t\leq T$.
Furthermore, if $\psi$ is replaced by $\psi_n=E_n\psi$
in \eqref{e3}, then
$$
U_n(t,s)\psi_n=e^{\int_s^tA(\tau,D)d\tau}\psi_n.
$$
\end{lemma}

\begin{proof}
 Fix $t\in [0,T]$.  For all $x\in H$, by \cite[Theorem XII.2.6]{d1},
\begin{align*}
    \|A(t,D)E_nx\|^2
&=  \int_0^{\infty}|A(t,\lambda)|^2d(E(\lambda)E_nx,E_nx) \\
&=  \int_{e_n}|A(t,\lambda)|^2d(E(\lambda)x,x) \\
&\leq  (n+\omega)^2\int_{e_n}d(E(\lambda)x,x) \\
&\leq  (n+\omega)^2\int_0^{\infty}d(E(\lambda)x,x) \\
&=  (n+\omega)^2\|x\|^2,
\end{align*}
 showing that $A(t,D)E_n$ is a bounded operator on $H$
with $\|A(t,D)E_n\|\leq n+\omega$.

 Next, let $t_0\in [0,T]$.  Since $e_n$ is a bounded subset
 of $[0,\infty)$, we have that $D^jE_n\in B(H)$ for each
 $1\leq j\leq k$.  Then by continuity of $a_j$ for each
$1\leq j\leq k$, we have
\begin{align*}
    \|A(t,D)E_n-A(t_0,D)E_n\|
&=  \|\sum_{j=1}^k(a_j(t)-a_j(t_0))D^jE_n\| \\
&\leq  \sum_{j=1}^k|a_j(t)-a_j(t_0)|\;\|D^jE_n\|
    \to  0 \quad \text{as} \;\; t\to t_0,
\end{align*}
showing that $t\mapsto A(t,D)E_n$ is continuous in the uniform
operator topology.  It follows from Theorem \ref{thm1} that \eqref{e3}
has a unique classical solution $u_n(t)=U_n(t,s)\psi$.  That
$$
\|U_n(t,s)\|\leq e^{T(n+\omega)}
$$
follows directly from Theorem \ref{thm2} (i) and the fact that
$\|A(t,D)E_n\|\leq n+\omega$ for all $t\in [0,T]$.

Next, set $\psi_n=E_n\psi$ and let \eqref{e3'}
 denote the evolution problem \eqref{e3} with $\psi$ replaced
 by $\psi_n$; i.e.,
\begin{equation} \label{e3'}
\begin{gathered}
 \frac{du_n(t)}{dt} =  A(t,D)E_nu_n(t) \quad   0\leq s \leq t < T,\\
    u_n(s) =  \psi_n.
\end{gathered}
\end{equation}
Using the Spectral Theorem it can be shown that
$e^{\int_s^tA(\tau,D)d\tau}\psi_n$ is a classical solution
of \eqref{e3'}.  In particular, using properties of the projection
operator $E_n$, we have
\begin{align*}
    \frac{d}{dt}e^{\int_s^tA(\tau,D)d\tau}\psi_n
&=  A(t,D)e^{\int_s^tA(\tau,D)d\tau}\psi_n \\
&=  A(t,D)E_ne^{\int_s^tA(\tau,D)d\tau}\psi_n,
\end{align*}
and
$$
e^{\int_s^sA(\tau,D)d\tau}\psi_n=\psi_n.
$$
Therefore, by uniqueness guaranteed by Theorem \ref{thm1}, we have
\[
U_n(t,s)\psi_n=e^{\int_s^tA(\tau,D)d\tau}\psi_n.
\]
\end{proof}

\begin{lemma} \label{lem2}
For each $t\in [0,T]$, $f(t,D)E_n$ is a bounded operator on $H$
such that
$$
\|f(t,D)E_n\|\leq 2n+\omega,
$$
and
\eqref{e4} has a unique classical solution $v_n(t)=V_n(t,s)\psi$.
The solution operator $V_n(t,s)$ is a bounded operator on
$H$ with
$$
\|V_n(t,s)\|\leq e^{T(2n+\omega)}
$$
for all $s,t$ such that $0\leq s\leq t\leq T$.
 Furthermore, if $\psi$ is replaced by $\psi_n=E_n\psi$
in \eqref{e4}, then
$$
V_n(t,s)\psi_n=e^{\int_s^tf(\tau,D)d\tau}\psi_n.
$$
\end{lemma}

\begin{proof}
  Using the fact that on $e_n$,
$\max_{t\in [0,T]}|f(t,\lambda)|\leq 2n+\omega$, it is easily
shown that for each $t\in [0,T]$, $f(t,D)E_n$ is a bounded
operator on $H$ such that $\|f(t,D)E_n\|\leq 2n+\omega$.
Next, let $t_0\in [0,T]$.  Since
$E_nH\subseteq \operatorname{Dom}(f(t,D))=\operatorname{Dom}(r(D))$
for all $t\in [0,T]$, we have $r(D)E_n \in B(H)$, and so
\begin{align*}
&\|f(t,D)E_n-f(t_0,D)E_n\|\\
&=  \sup_{x\in H,\,\|x\|\leq 1} \|(f(t,D)-f(t_0,D))E_nx\| \\
&\leq \sup_{x\in H,\, \|x\|\leq 1} \|f(t,D)-f(t_0,D)
 \|_{Y\to H}\|E_nx\|_Y \\
&= \sup_{x\in H,\, \|x\|\leq 1} \|f(t,D)-f(t_0,D)\|_{Y\to H}
 (\|E_nx\|+\|r(D)E_nx\|) \\
&\leq  \|f(t,D)-f(t_0,D)\|_{Y\to H}(\|E_n\|+\|r(D)E_n\|)
\to  0 \quad \text{as } t\to t_0
\end{align*}
by the assumption that $t\mapsto f(t,D)$ is continuous in the
$B(Y,H)$ norm $\|\cdot \|_{Y\to H}$.  Therefore,
$t\mapsto f(t,D)E_n$ is continuous in the uniform operator topology.
It follows from Theorem \ref{thm1} that \eqref{e4}
 has a unique classical solution $v_n(t)=V_n(t,s)\psi$.  That
$$
\|V_n(t,s)\|\leq e^{T(2n+\omega)}
$$
follows directly from Theorem \ref{thm2} (i) and the fact that
$\|f(t,D)E_n\|\leq 2n+\omega$ for all $t\in [0,T]$.
The rest of the proof is similar to that of Lemma \ref{lem1}.
\end{proof}

\begin{lemma} \label{lem3}
For each $t\in [0,T]$, $g(t,D)E_n$ is a bounded operator
on $H$ such that $$\|g(t,D)E_n\|\leq n,$$ and \eqref{e5} has a
unique classical solution $w_n(t)=W_n(t,s)\psi$.
The solution operator $W_n(t,s)$ is a bounded operator on
$H$ with
$$
\|W_n(t,s)\|\leq e^{Tn}
$$
for all $s,t$ such that $0\leq s\leq t\leq T$.
Furthermore, if $\psi$ is replaced by $\psi_n=E_n\psi$ in \eqref{e5},
then
$$
W_n(t,s)\psi_n=e^{\int_s^tg(\tau,D)d\tau}\psi_n.
$$
\end{lemma}

\begin{proof}
  Using the fact that on $e_n$,
$\max_{t\in [0,T]}|g(t,\lambda)|\leq n$, it is easily shown that
for each $t\in [0,T]$, $g(t,D)E_n$ is a bounded operator on $H$
such that $\|g(t,D)E_n\|\leq n$.  Also, by the relation
$g(t,D)E_n=-A(t,D)E_n+f(t,D)E_n$, it follows that
$t\mapsto g(t,D)E_n$ is continuous in the uniform operator topology.
Therefore, by Theorem \ref{thm1}, \eqref{e5} has a unique classical solution
$w_n(t)=W_n(t,s)\psi$.  That
$$
\|W_n(t,s)\|\leq e^{Tn}
$$
follows directly from Theorem \ref{thm2} (i) and the fact that
$\|g(t,D)E_n\|\leq n$ for all $t\in [0,T]$.
The rest of the proof is similar to that of Lemma \ref{lem1}.
\end{proof}

\begin{corollary} \label{coro1}
Let $\psi \in H$ and $\psi_n=E_n\psi$. Then
$$
U_n(t,s)W_n(t,s)\psi_n = V_n(t,s)\psi_n
= W_n(t,s)U_n(t,s)\psi_n
$$
for all $0\leq s\leq t\leq T$.
\end{corollary}

The corollary above follows immediately from Lemmas
\ref{lem1}, \ref{lem2}, and \ref{lem3}, and
from properties of the functional calculus for unbounded
self-adjoint operators \cite[Corollary XII.2.7]{d1}.

We now have all the necessary machinery to prove our
approximation theorem.  Our strategy will be to extend the
 solutions $u_n(t)$ of \eqref{e3} with $\psi=\chi_n$,
and $v_n(t)$ of \eqref{e4} with $\psi = \chi_n$, into the complex
strip $S=\{t+i\eta:t\in [0,T],\;\eta\in \mathbb{R}\}$,
 and eventually employ Hadamard's Three Lines Theorem (cf. \cite{r1}).
To make use of such extensions we will need the following results.
Our approach is motivated by work of Agmon and Nirenberg  \cite{a1}.

\begin{definition}[{\cite[Definition 11.1]{r1}}] \label{def7} \rm
Let $\phi(\alpha)$ be a complex function defined in a plane open
set $\Omega$.  Assume all partial derivatives of $\phi$ exist
and are continuous.  Define the \emph{Cauchy-Riemann operator}
$\bar{\partial}$ as
$$
\bar{\partial}=\frac{1}{2}
\Big(\frac{\partial}{\partial t}+i\frac{\partial}{\partial \eta}\Big),
$$
where $\alpha=t+i\eta$.
\end{definition}

\begin{theorem}[{\cite[Theorem 11.2]{r1}}] \label{thm5}
Suppose $\phi(\alpha)$ is a complex function in $\Omega$ such
that all partial derivatives of $\phi$ exist and are continuous.
Then $\phi$ is analytic in $\Omega$ if and only if the Cauchy-Riemann
equation
$$
\bar{\partial}\phi(\alpha)=0
$$
holds for every $\alpha\in \Omega$.
\end{theorem}

\begin{lemma}[\cite{a1}] \label{lem4}
Let $\phi(z)$ be a complex function with $z = x+iy$.
Assume $\phi(z)$ is continuous and bounded on
$S=\{z=x+iy : x\in [0,T],y\in \mathbb{R}\}$.
 For $\alpha=t+i\eta \in S$, define
$$
\Phi(\alpha)=-\frac{1}{\pi}\int \int_S \phi(z)
\Big(\frac{1}{z-\alpha}+\frac{1}{\bar{z}+1+\alpha}\Big)\,dx\,dy.
$$
Then $\Phi(\alpha)$ is absolutely convergent,
$\bar{\partial}\Phi(\alpha)=\phi(\alpha)$, and there exists a
constant $K$ such that
$$
\int_{-\infty}^{\infty}\big|\frac{1}{z-\alpha}
+\frac{1}{\bar{z}+1+\alpha}\big|dy
\leq K\Big(1+{\rm{log}}\frac{1}{|x-t|}\Big)
$$
if $x\neq t$.
\end{lemma}

We now state and prove our approximation theorem.

\begin{theorem} \label{thm6}
Let $D$ be a positive self-adjoint operator acting on $H$ and let
$A(t,D)$ be defined as above for all $t\in [0,T]$.  Let $f$
satisfy Condition $(\mathcal{A},p)$, and assume that there exists
a constant $\gamma$, independent of $\beta$, $\omega$, and $t$
such that $g(t,\lambda)\leq \gamma$, for all $(t,\lambda)\in
[0,T]\times [0,\infty)$.  Then if $u(t)$ and $v(t)$ are classical
solutions of $\eqref{e1}$ and $\eqref{e2}$ respectively, and if
there exist constants $M',M'',M'''\geq 0$ such that $\|u(T)\|\leq
M'$, $\|p(D)\chi\|\leq M''$, and $\|p(D)A(t,D)u(T)\|\leq M'''$ for
all $t\in [0,T]$, then there exist constants $C$ and $M$
independent of $\beta$ such that for $0\leq t <T$,
$$
\|u(t)-v(t)\|\leq C\beta^{1-\frac{t}{T}}M^{t/T}.
$$
\end{theorem}

\begin{proof}
  Let $\chi_n = E_n\chi$ and set $S=\{t+i\eta:t\in [0,T],\;
\eta\in \mathbb{R}\}$.  Letting $\psi=\chi_n$ and $s=0$
in \eqref{e3} and \eqref{e4}, we extend the solutions $u_n(t)$
and $v_n(t)$ into the complex strip $S$ in the following way.
Since $A(0,D)$ and $f(0,D)$ are self-adjoint,
$e^{i\eta A(0,D)}$ and $e^{i\eta f(0,D)}$ are bounded
operators on $H$ for all $\eta \in \mathbb{R}$, and so we
define
\begin{gather*}
u_n(\alpha)=e^{i\eta A(0,D)}U_n(t,0)\chi_n,\\
v_n(\alpha)=e^{i\eta f(0,D)}V_n(t,0)\chi_n,
\end{gather*}
for $\alpha=t+i\eta\in S$.  Finally define $\phi_n:S\to H$ by
$$
\phi_n(\alpha)=u_n(\alpha)-v_n(\alpha).
$$
 We first determine $\bar{\partial}\phi_n(\alpha)$.  Since
$e^{i\eta A(0,D)}$ and $e^{i\eta f(0,D)}$ are bounded operators
on $H$, and $A(t,D)$ and $f(t,D)$ are bounded when acting on
$E_nH$, we have
\begin{align*}
    \frac{\partial}{\partial t}\phi_n(\alpha)
&=  \frac{\partial}{\partial t}e^{i\eta A(0,D)}U_n(t,0)\chi_n
  - \frac{\partial}{\partial t}e^{i\eta f(0,D)}V_n(t,0)\chi_n \\
&=  e^{i\eta A(0,D)}\frac{d}{dt}U_n(t,0)\chi_n
  - e^{i\eta f(0,D)}\frac{d}{dt}V_n(t,0)\chi_n \\
&=  e^{i\eta A(0,D)}A(t,D)U_n(t,0)\chi_n-e^{i\eta f(0,D)}f(t,D)V_n(t,0)
   \chi_n \\
&=  A(t,D)u_n(\alpha)-f(t,D)v_n(\alpha).
\end{align*}
 Next, by standard properties of semigroups of linear operators
(cf. \cite[Theorem 1.2.4 (c)]{p1}),
since $U_n(t,0)\chi_n \in \operatorname{Dom}(A(0,D))$,
 and $V_n(t,0)\chi_n \in \operatorname{Dom}(f(0,D))$, we have
\begin{align*}
 \frac{\partial}{\partial \eta}\phi_n(\alpha)
&=  \frac{\partial}{\partial \eta}e^{i\eta A(0,D)}U_n(t,0)\chi_n - \frac{\partial}{\partial \eta}e^{i\eta f(0,D)}V_n(t,0)\chi_n \\
&=  iA(0,D)e^{i\eta A(0,D)}U_n(t,0)\chi_n - if(0,D)e^{i\eta f(0,D)}V_n(t,0)\chi_n \\
&=  i(A(0,D)u_n(\alpha) - f(0,D)v_n(\alpha)).
\end{align*}
Therefore,
\begin{equation} \label{e6}
\begin{aligned}
    \bar{\partial}\phi_n(\alpha)
&=  \frac{1}{2}\left(\frac{\partial}{\partial t}\phi_n(\alpha)
   +i\frac{\partial}{\partial \eta}\phi_n(\alpha)\right)  \\
&=  \frac{1}{2}[(A(t,D)u_n(\alpha)-f(t,D)v_n(\alpha))
    -(A(0,D)u_n(\alpha) - f(0,D)v_n(\alpha))]  \\
&=  \frac{1}{2}[(A(t,D)-A(0,D))u_n(\alpha)-(f(t,D)
    -f(0,D))v_n(\alpha)]. 
\end{aligned}
\end{equation}
Since, in general, this quantity is not identically zero,
$\phi_n$ is not analytic and so we cannot apply the Three
Lines Theorem to $\phi_n$.  To amend this, we introduce a related
function.  Define
$$
\Phi_n(\alpha)=-\frac{1}{\pi}\int \int_Se^{z^2}
\bar{\partial}\phi_n(z)\Big(\frac{1}{z-\alpha}
+\frac{1}{\bar{z}+1+\alpha}\Big)\,dx\,dy,
$$
where $z = x+iy$ and $\alpha=t+i\eta$ are in $S$.
To apply Lemma \ref{lem4}, we show that $e^{z^2} \bar{\partial}\phi_n(z)$
is bounded  and continuous on $S$.   Let $z=x+iy \in S$ be arbitrary.
We have from \eqref{e6},
\begin{align*}
\|e^{z^2} \bar{\partial}\phi_n(z)\|
&=  \frac{1}{2}|e^{z^2}|\;\|(A(x,D)-A(0,D))u_n(z)-(f(x,D)-f(0,D))v_n(z)\| \\
&\leq  \frac{1}{2}e^{T^2}(\|A(x,D)u_n(z)\|+\|A(0,D)u_n(z)\|\\
&\quad +\|f(x,D)v_n(z)\|+\|f(0,D)v_n(z)\|).
\end{align*}
Since $\|e^{iyA(0,D)}\|=1$, we have by Lemma \ref{lem1} and properties
of $E_n$,
\begin{align*}
    \|A(x,D)u_n(z)\|
&=  \|A(x,D)e^{iyA(0,D)}U_n(x,0)\chi_n\| \\
    &=  \|A(x,D)E_ne^{iyA(0,D)}U_n(x,0)\chi_n\| \\
    &\leq  (n+\omega)\|e^{iyA(0,D)}U_n(x,0)\chi_n\| \\
    &\leq  (n+\omega)\|U_n(x,0)\chi_n\| \\
    &\leq  (n+\omega)e^{T(n+\omega)}\|\chi_n\|.
\end{align*}
Note that since $x\in [0,T]$ is arbitrary, the same bound
holds for $\|A(0,D)u_n(z)\|$.
Similarly, $\|e^{iyf(0,D)}\|=1$, and using Lemma \ref{lem2},
\begin{align*}
    \|f(x,D)v_n(z)\|
&=  \|f(x,D)e^{iyf(0,D)}V_n(x,0)\chi_n\| \\
    &=  \|f(x,D)E_ne^{iyf(0,D)}V_n(x,0)\chi_n\| \\
    &\leq  (2n+\omega)\|e^{iyf(0,D)}V_n(x,0)\chi_n\| \\
    &\leq  (2n+\omega)\|V_n(x,0)\chi_n\| \\
    &\leq  (2n+\omega)e^{T(2n+\omega)}\|\chi_n\|.
\end{align*}
Again, since $x\in [0,T]$ is arbitrary, the same bound holds
for $\|f(0,D)v_n(z)\|$.  Setting
$$
C_n=(n+\omega)e^{T(n+\omega)}+(2n+\omega)e^{T(2n+\omega)},
$$
we have
\begin{equation} \label{e7}
    \|e^{z^2} \bar{\partial}\phi_n(z)\| \leq  e^{T^2}C_n\|\chi_n\|,
\end{equation}
showing that $e^{z^2} \bar{\partial}\phi_n(z)$ is indeed bounded on $S$.
It can also easily be shown using continuity of $A(t,D)E_n$ and
$f(t,D)E_n$ in the $B(H)$ norm,
continuity of $U_n(t,s)$ and $V_n(t,s)$ in the $B(H)$ norm
(Theorem \ref{thm2} (iii)), and strong continuity of the groups
$\{e^{iyA(0,D)}\}_{y\in \mathbb{R}}$ and
$\{e^{iyf(0,D)}\}_{y\in \mathbb{R}}$, that
$e^{z^2} \bar{\partial}\phi_n(z)$ is continuous on $S$.
Having satisfied the hypotheses of Lemma \ref{lem4}, it follows that
$$
\Phi_n(\alpha)=-\frac{1}{\pi}\int \int_Se^{z^2}
\bar{\partial}\phi_n(z)\Big(\frac{1}{z-\alpha}
+\frac{1}{\bar{z}+1+\alpha}\Big)\,dx\,dy
$$
is absolutely convergent,
$$
\bar{\partial}\Phi_n(\alpha)=e^{\alpha^2}\bar{\partial}\phi_n(\alpha),
$$
and there exists a constant $K$ such that
$$
\int_{-\infty}^{\infty}\big|\frac{1}{z-\alpha}+\frac{1}{\bar{z}
+1+\alpha}\big|dy\leq K\Big(1+\log \frac{1}{|x-t|}\Big)
$$
for $x\neq t$.

 We now construct a candidate for the Three Lines Theorem.
Define $\Psi_n:S\to H$ by
$$
\Psi_n(\alpha)=e^{\alpha^2}\phi_n(\alpha)-\Phi_n(\alpha).
$$
For $\alpha$ in the interior of $S$, using the product rule and
results from Lemma \ref{lem4},
\begin{align*}
    \bar{\partial}\Psi_n(\alpha)
&=  \bar{\partial}[e^{\alpha^2}\phi_n(\alpha)]
  -\bar{\partial}\Phi_n(\alpha) \\
&=  [(\bar{\partial}e^{\alpha^2})\phi_n(\alpha)
  +e^{\alpha^2}\bar{\partial}\phi_n(\alpha)]
  -\bar{\partial}\Phi_n(\alpha) \\
&=  [(0)\phi_n(\alpha)+\bar{\partial}\Phi_n(\alpha)]
  -\bar{\partial}\Phi_n(\alpha) =0.
\end{align*}
Therefore, by Theorem \ref{thm5}, $\Psi_n$ is analytic on the interior of $S$.
Next, for $\alpha=t+i\eta \in S$, from \eqref{e7} and the results
from Lemma \ref{lem4},
\begin{align*}
    \|\Phi_n(\alpha)\|
&=  \|-\frac{1}{\pi}\int \int_S e^{z^2}\bar{\partial}\phi_n(z)
 \Big(\frac{1}{z-\alpha}+\frac{1}{\bar{z}+1+\alpha}\Big)\,dx\,dy
\| \\
&\leq \frac{1}{\pi}\int_{-\infty}^{\infty}
\int_0^T e^{T^2}C_n\|\chi_n\|\big|\frac{1}{z-\alpha}+\frac{1}{\bar{z}
+1+\alpha}\big|\,dx\,dy \\
&=  \frac{1}{\pi}e^{T^2}C_n\|\chi_n\|\int_0^T
\Big(\int_{-\infty}^{\infty} \big|\frac{1}{z-\alpha}
+\frac{1}{\bar{z}+1+\alpha}\big|dy\Big)dx \\
&\leq  \frac{K}{\pi}e^{T^2}C_n\|\chi_n\|\int_0^T
\big(1+\log \frac{1}{|x-t|}\big)dx. \\
\end{align*}
Also, using Lemmas \ref{lem1} and \ref{lem2},
\begin{align*}
    \|\phi_n(\alpha)\|
&=  \|e^{i\eta A(0,D)}U_n(t,0)\chi_n-e^{i\eta f(0,D)}V_n(t,0)\chi_n\| \\
    &\leq  (\|U_n(t,0)\|+\|V_n(t,0)\|)\|\chi_n\| \\
    &\leq  (e^{T(n+\omega)}+e^{T(2n+\omega)})\|\chi_n\|.
\end{align*}
Therefore,
\begin{align*}
    \|\Psi_n(\alpha)\|
&=  \|e^{\alpha^2}\phi_n(\alpha)-\Phi_n(\alpha)\| \\
    &\leq  |e^{\alpha^2}|\;\|\phi_n(\alpha)\|+\|\Phi_n(\alpha)\| \\
    &\leq  e^{T^2}(e^{T(n+\omega)}+e^{T(2n+\omega)})\|\chi_n\|\\
&\quad +\frac{K}{\pi}e^{T^2}C_n\|\chi_n\|\big\{\max_{t\in [0,T]}
\int_0^T\big(1+\log \frac{1}{|x-t|}\big)dx\big\},
\end{align*}
proving that $\Psi_n$ is bounded on $S$.   From \eqref{e7}
 and the results from Lemma \ref{lem4}, it follows via a dominated convergence
argument that $\Phi_n$ is continuous on $S$.  It is also easily
shown that $\phi_n$ is continuous on $S$, and therefore $\Psi_n$
is continuous on $S$.

 We have shown that $\Psi_n$ is bounded and continuous on $S$,
and analytic on the interior of $S$.  It follows from the
Cauchy-Schwarz Inequality that for arbitrary $h\in H$,
the mapping
$$
\alpha\mapsto (\Psi_n(\alpha),h)
$$
from $S$ into $\mathbb{C}$, where $(\cdot,\cdot)$ denotes
the inner product in $H$, has the same properties.
Therefore, by the Three Lines Theorem,
$$
|(\Psi_n(\alpha),h)|\leq M(0)^{1-\frac{t}{T}}M(T)^{t/T},
$$
for $t\in [0,T]$, where $\alpha=t+i\eta$ and
$$
M(t)=\max_{\eta\in \mathbb{R}}|(\Psi_n(t+i\eta),h)|.
$$
We aim to find bounds on $M(0)$ and $M(T)$.  First, for
$\eta\in \mathbb{R}$,
\begin{align*}
    |(\Psi_n(i\eta),h)|
&\leq  \|\Psi_n(i\eta)\|\|h\| \\
    &=  \|e^{-\eta^2}\left(e^{i\eta A(0,D)}U_n(0,0)
\chi_n-e^{i\eta f(0,D)}V_n(0,0)\chi_n\right)-\Phi_n(i\eta)\|\|h\| \\
    &=  \|e^{-\eta^2}\left(e^{i\eta A(0,D)}\chi_n-e^{i\eta
  f(0,D)}\chi_n\right)-\Phi_n(i\eta)\|\|h\| \\
    &\leq  \left(e^{-\eta^2}\|e^{i\eta A(0,D)}
 \chi_n-e^{i\eta f(0,D)}\chi_n\|+\|\Phi_n(i\eta)\|\right)\|h\|.
\end{align*}
Now, since $\|e^{i\eta A(0,D)}\|=1$,
\begin{align*}
    \|e^{i\eta A(0,D)}\chi_n-e^{i\eta f(0,D)}\chi_n\|
&=  \|e^{i\eta A(0,D)}\chi_n-e^{i\eta A(0,D)}e^{i\eta g(0,D)}\chi_n\| \\
&=  \|e^{i\eta A(0,D)}(I-e^{i\eta g(0,D)})\chi_n\| \\
&\leq  \|(I-e^{i\eta g(0,D)})\chi_n\|.
\end{align*}
 For $\psi\in \operatorname{Dom}(g(0,D))$ and
$\eta \in \mathbb{R}$, we have by standard properties of
semigroups (cf. \cite[Theorem 1.2.4]{p1}) that
\[
   \|(I-e^{i\eta g(0,D)})\psi\|
 = \|-i\int_0^{\eta}e^{isg(0,D)}g(0,D)\psi ds\|
 \leq  |\eta|\|g(0,D)\psi\|.
\]
Note $\chi_n\in \operatorname{Dom}(A(0,D))\cap
\operatorname{Dom}(f(0,D))\subseteq \operatorname{Dom}(g(0,D))$
and since $e_n$ is a bounded subset of $[0,\infty)$,
$\chi_n\in \operatorname{Dom}(p(D))$.  Thus we have by Condition
$(\mathcal{A},p)$ and the above inequality that
$$
\|e^{i\eta A(0,D)}\chi_n-e^{i\eta f(0,D)}\chi_n\|
\leq \beta|\eta|\|p(D)\chi_n\|.
$$
 Next we would like a bound on $\|\Phi_n(i\eta)\|$ in terms of
$\beta$.  Let $z=x+iy\in S$.  Then from \eqref{e6},
\begin{align*}
    2\|\bar{\partial}\phi_n(z)\|
&=  \|(A(x,D)-A(0,D))u_n(z)-(f(x,D)-f(0,D))v_n(z)\| \\
    &\leq  \|A(x,D)u_n(z)-f(x,D)v_n(z)\|+\|A(0,D)u_n(z)-f(0,D)v_n(z)\|.
\end{align*}
Now,
\begin{align}
    & \|A(x,D)u_n(z)-f(x,D)v_n(z)\|  \nonumber\\
    &=  \|A(x,D)e^{iy A(0,D)}U_n(x,0)\chi_n-f(x,D)
 e^{iy f(0,D)}V_n(x,0)\chi_n\| \nonumber \\
    &\leq  \|A(x,D)e^{iy A(0,D)}U_n(x,0)\chi_n-A(x,D)
e^{iy f(0,D)}U_n(x,0)\chi_n\|  \label{e8}\\
    &\quad +  \|A(x,D)e^{iy f(0,D)}U_n(x,0)\chi_n-A(x,D)
e^{iy f(0,D)}V_n(x,0)\chi_n\|  \label{e9}\\
    &\quad +  \|A(x,D)e^{iy f(0,D)}V_n(x,0)\chi_n-f(x,D)
e^{iy f(0,D)}V_n(x,0)\chi_n\|. \label{e10}
\end{align}
 Set $\psi_n=A(x,D)U_n(x,0)\chi_n$. We note that
$\psi_n\in \operatorname{Dom}(A(t,D))\cap \operatorname{Dom}(f(t,D))
\subseteq \operatorname{Dom}(g(t,D))$ for all $t\in [0,T]$,
 and $\psi_n \in \operatorname{Dom}(p(D))$.  Then expression
\eqref{e8} is equal to
\[
\|e^{iyA(0,D)}\psi_n-e^{iyf(0,D)}\psi_n\| \\
    \leq  \beta|y|\|p(D)\psi_n\|,
\]
 as above.

 Next, using Theorem \ref{thm2}, Lemma \ref{lem3}, and the assumption that
 $g(t,\lambda)\leq \gamma$ for all
$(t,\lambda) \in [0,T]\times [0,\infty)$, we have
\begin{align*}
    \|(I-W_n(x,0))\psi_n\|
&=  \|(W_n(x,x)-W_n(x,0))\psi_n\| \\
    &=  \|\int_0^x\frac{\partial}{\partial s}W_n(x,s)\psi_n ds\| \\
    &=  \|\int_0^x(-W_n(x,s)g(s,D)E_n)\psi_n ds\| \\
    &\leq  \int_0^x\|W_n(x,s)g(s,D)\psi_n\| ds \\
    &\leq  \int_0^x(1+e^{\gamma T})\|g(s,D)\psi_n\| ds.
\end{align*}
It then follows from Corollary \ref{coro1} and Condition $(\mathcal{A},p)$ that
expression \eqref{e9} is equal to
\begin{align*}
  &\|e^{iy f(0,D)}(U_n(x,0)-V_n(x,0))A(x,D)\chi_n\| \\
    &\leq  \|(U_n(x,0)-V_n(x,0))A(x,D)\chi_n\| \\
    &=  \|(U_n(x,0)-W_n(x,0)U_n(x,0))A(x,D)\chi_n\| \\
    &=  \|(I-W_n(x,0))\psi_n\| \\
    &\leq  \int_0^x(1+e^{\gamma T})\|g(s,D)\psi_n\| ds \\
    &\leq  \int_0^x(1+e^{\gamma T})\beta\|p(D)\psi_n\| ds \\
    &\leq  \beta T(1+e^{\gamma T})\|p(D)\psi_n\|.
\end{align*}
 Finally, since $U_n(x,0)\chi_n\in \operatorname{Dom}(p(D))$,
Corollary \ref{coro1} and Condition $(\mathcal{A},p)$ imply that
expression \eqref{e10} is equal to
\begin{align*}
 & \|e^{iy f(0,D)}V_n(x,0)(-A(x,D)+f(x,D))\chi_n\| \\
    &\leq  \|V_n(x,0)(-A(x,D)+f(x,D))\chi_n\| \\
    &=  \|W_n(x,0)U_n(x,0)(-A(x,D)+f(x,D))\chi_n\| \\
    &\leq  (1+e^{\gamma T})\|(-A(x,D)+f(x,D))U_n(x,0)\chi_n\| \\
    &\leq  \beta (1+e^{\gamma T})\|p(D)U_n(x,0)\chi_n\|.
\end{align*}
Therefore, we have shown
\begin{align*}
&\|A(x,D)u_n(z)-f(x,D)v_n(z)\|\\
&\leq \beta(1+e^{\gamma T})\left((|y|+T)\|p(D)A(x,D)U_n(x,0)\chi_n\|
+\|p(D)U_n(x,0)\chi_n\|\right).
\end{align*}
Since $x$ is arbitrary, it follows similarly that
\begin{align*}
&\|A(0,D)u_n(z)-f(0,D)v_n(z)\| \\
&\leq \beta(1+e^{\gamma T})
\left((|y|+T)\|p(D)A(0,D)U_n(x,0)\chi_n\|+\|p(D)U_n(x,0)
\chi_n\|\right).
\end{align*}
 From the assumptions $\|u(T)\|\leq M'$ and
$\|p(D)A(t,D)u(T)\|\leq M'''$ for all $t\in [0,T]$, it follows
that $\|p(D)u(T)\|\leq N'$ for some constant $N'\geq 0$.
It then follows from these estimates that for $z=x+iy\in S$,
$$
\|\bar{\partial}\phi_n(z)\| \leq \beta(1+e^{\gamma T})\left((|y|+T)M'''
+N'\right),
$$
so that by Lemma \ref{lem4},
\begin{align*}
    & \|\Phi_n(i\eta)\| \\
    &=  \|-\frac{1}{\pi}\int \int_S e^{z^2}\bar{\partial}\phi_n(z)
\Big(\frac{1}{z-i\eta}+\frac{1}{\bar{z}+1+i\eta}\Big)\,dx\,dy\| \\
    &\leq  \frac{1}{\pi}\int_{-\infty}^{\infty}\int_0^T|e^{z^2}|
\|\bar{\partial}\phi_n(z)\|\big|\frac{1}{z-i\eta}+\frac{1}{\bar{z}
+1+i\eta}\big|\,dx\,dy \\
    &\leq  \frac{1}{\pi}\int_{-\infty}^{\infty}
\int_0^Te^{x^2-y^2}\beta(1+e^{\gamma T})\left((|y|+T)M'''+N'\right)\left|\frac{1}{z-i\eta}+\frac{1}{\bar{z}+1+i\eta}\right|\,dx\,dy \\
    &=  \frac{1}{\pi}\int_{-\infty}^{\infty}\int_0^Te^{x^2}
\beta(1+e^{\gamma T})\left((|y|e^{-y^2}+Te^{-y^2})M'''
+e^{-y^2}N'\right)\\
&\quad \times \big|\frac{1}{z-i\eta}+\frac{1}{\bar{z}
+1+i\eta}\big|\,dx\,dy \\
    &\leq  \beta\Big[\frac{1}{\pi}e^{T^2}(1+e^{\gamma T})
\left((1+T)M'''+N'\right)\int_0^T\Big(\int_{-\infty}^{\infty}
\big|\frac{1}{z-i\eta}+\frac{1}{\bar{z}+1+i\eta}\big|dy\Big)dx
\Big] \\
    &\leq  \beta\Big[\frac{K}{\pi}e^{T^2}(1+e^{\gamma T})
\left((1+T)M'''+ N'\right)\int_0^T\big(1+\log \frac{1}{x}\big)
dx\Big].
\end{align*}
Therefore,
\begin{equation}
\begin{aligned} \label{e11}
    & M(0)  \\
    &=  \max_{\eta\in \mathbb{R}}|(\Psi_n(i\eta),h)|  \\
    &\leq  \max_{\eta\in \mathbb{R}}
\left(e^{-\eta^2}\|e^{i\eta A(0,D)}\chi_n-e^{i\eta f(0,D)}\chi_n\|
 +\|\Phi_n(i\eta)\|\right)\|h\|  \\
    &\leq  \max_{\eta\in \mathbb{R}}\left(\beta |\eta|
e^{-\eta^2}\|p(D)\chi_n\|+\|\Phi_n(i\eta)\|\right)\|h\|  \\
    &\leq  \beta\Big(M''+\Big[\frac{K}{\pi}e^{T^2}
 (1+e^{\gamma T})\left((1+T)M'''+ N'\right)
\int_0^T\big(1+\log \frac{1}{x}\big)dx\Big] \Big)\|h\|.
\end{aligned}
\end{equation}
Next, for $\eta\in \mathbb{R}$,
\begin{align*}
&|(\Psi_n(T+i\eta),h)| \\
&\leq  \|\Psi_n(T+i\eta)\|\|h\| \\
&=  \|e^{(T+i\eta)^2}\left(e^{i\eta A(0,D)}U_n(T,0)\chi_n-e^{i\eta f(0,D)}V_n(T,0)\chi_n\right)-\Phi_n(T+i\eta)\|\|h\| \\
&\leq  \left(e^{T^2-\eta^2}\|e^{i\eta A(0,D)}U_n(T,0)\chi_n-e^{i\eta f(0,D)}V_n(T,0)\chi_n\|+\|\Phi_n(T+i\eta)\|\right)\|h\|.
\end{align*}
Using the assumption that $\|u(T)\|\leq M'$,
\begin{align*}
& \|e^{i\eta A(0,D)}U_n(T,0)\chi_n-e^{i\eta f(0,D)}V_n(T,0)\chi_n\|\\
&\leq  \|U_n(T,0)\chi_n\|+\|V_n(T,0)\chi_n\| \\
&=  \|U_n(T,0)\chi_n\|+\|W_n(T,0)U_n(T,0)\chi_n\| \\
&\leq  (1+(1+e^{\gamma T}))\|U_n(T,0)\chi_n\| \\
&\leq  (2+e^{\gamma T})M'.
\end{align*}
 Next, the assumptions $\|u(T)\|\leq M'$ and
$\|p(D)A(t,D)u(T)\|\leq M'''$ for all $t\in [0,T]$ imply that
$\|A(t,D)u(T)\|\leq N''$ for all $t\in [0,T]$, for some constant
$N''\geq 0$.  Thus, for $z = x+iy\in S$, we have
\begin{align*}
   &\|(A(x,D)-A(0,D))u_n(z)\| \\
&\leq  \|e^{iyA(0,D)}A(x,D)U_n(x,0)\chi_n\|
+\|e^{iyA(0,D)}A(0,D)U_n(x,0)\chi_n\| \\
    &\leq  \|A(x,D)U_n(x,0)\chi_n\|+\|A(0,D)U_n(x,0)\chi_n\| \\
    &\leq  2N''.
\end{align*}
 Meanwhile, using Condition $(\mathcal{A},p)$ and the fact that
$0<\beta<1$,
\begin{align*}
    &  \|(f(x,D)-f(0,D))v_n(z)\| \\
    &\leq  \|e^{iyf(0,D)}f(x,D)V_n(x,0)\chi_n\|+\|e^{iyf(0,D)}f(0,D)V_n(x,0)\chi_n\| \\
    &\leq  \|f(x,D)V_n(x,0)\chi_n\|+\|f(0,D)V_n(x,0)\chi_n\| \\
    &\leq  \|A(x,D)V_n(x,0)\chi_n\|+\|(-A(x,D)+f(x,D))V_n(x,0)\chi_n\| \\
    & \quad +  \|A(0,D)V_n(x,0)\chi_n\|+\|(-A(0,D)+f(0,D))V_n(x,0)\chi_n\| \\
    &\leq  \|V_n(x,0)A(x,D)\chi_n\|+\|V_n(x,0)A(0,D)\chi_n\|+2\beta \|p(D)V_n(x,0)\chi_n\| \\
    &\leq  (1+e^{\gamma T})(\|A(x,D)U_n(x,0)\chi_n\|+\|A(0,D)U_n(x,0)\chi_n\|+2\|p(D)U_n(x,0)\chi_n\|) \\
    &\leq  2(1+e^{\gamma T})(N''+N').
\end{align*}
Therefore, for $z=x+iy\in S$, from \eqref{e6},
\begin{align*}
    \|\bar{\partial}\phi_n(z)\|
&\leq  \frac{1}{2}\left(\|(A(x,D)-A(0,D))u_n(z)\|
+\|(f(x,D)-f(0,D))v_n(z)\|\right) \\
&\leq  N''+(1+e^{\gamma T})(N''+N')
\end{align*}
so that by Lemma \ref{lem4},
\begin{align*}
    & \|\Phi_n(T+i\eta)\| \\
    &=  \|-\frac{1}{\pi}\int \int_S e^{z^2}\bar{\partial}\phi_n(z)
 \left(\frac{1}{z-(T+i\eta)}+\frac{1}{\bar{z}+1+(T+i\eta)}\right)
 \,dx\,dy\| \\
    &\leq  \frac{1}{\pi}\int_{-\infty}^{\infty}
\int_0^T|e^{z^2}|\|\bar{\partial}\phi_n(z)\|
\big|\frac{1}{z-(T+i\eta)}+\frac{1}{\bar{z}+1+(T+i\eta)}\big|\,dx\,dy \\
    &\leq  \frac{1}{\pi}\int_{-\infty}^{\infty}\int_0^Te^{T^2}(N''
+(1+e^{\gamma T})(N''+N'))\\
&\quad\times \big|\frac{1}{z-(T+i\eta)}
+\frac{1}{\bar{z}+1+(T+i\eta)}\big|\,dx\,dy \\
    &=  \frac{1}{\pi}e^{T^2}(N''+(1+e^{\gamma T})(N''+N'))\\
&\quad\times \int_0^T\Big(\int_{-\infty}^{\infty}\big|\frac{1}{z-(T+i\eta)}
+\frac{1}{\bar{z}+1+(T+i\eta)}\big|dy\Big)dx \\
    &\leq  \frac{K}{\pi}e^{T^2}(N''+(1+e^{\gamma T})(N''+N'))
\int_0^T\big(1+\log \frac{1}{|x-T|}\big)dx.
\end{align*}
Thus,
\begin{equation} 
\begin{aligned} \label{e12}
    & M(T)  \\
    &=  \max_{\eta\in \mathbb{R}}|(\Psi_n(T+i\eta),h)|  \\
    &\leq  \max_{\eta\in \mathbb{R}}
\Big(e^{T^2-\eta^2}(2+e^{\gamma T})M'+\frac{K}{\pi}e^{T^2}
(N''+(1+e^{\gamma T})(N''+N'))\\
&\quad \times \int_0^T\big(1+\log \frac{1}{|x-T|}\big)dx\Big)\|h\|  \\
&\leq  \Big(e^{T^2}(2+e^{\gamma T})M'+\frac{K}{\pi}e^{T^2}
(N''+(1+e^{\gamma T})(N''+N'))\\
&\quad\times \int_0^T\big(1+\log \frac{1}{|x-T|}\big)dx\Big)\|h\|.
\end{aligned} 
\end{equation}
 It follows from \eqref{e11} and \eqref{e12} that there
exist constants $C'$ and $M$, independent of $\beta$, such
that for $0\leq t<T$,
\[
    |(\Psi_n(t),h)|
\leq  (C'\beta \|h\|)^{1-\frac{t}{T}}(M\|h\|)^{t/T}
=  (C'\beta)^{1-\frac{t}{T}}M^{t/T}\|h\|.
\]
Taking the supremum over all $h\in H$ with $\|h\|\leq 1$, we have
constants $C$ and $M$, independent of $\beta$, such that for
$0\leq t<T$,
$$
\|\Psi_n(t)\|\leq C\beta^{1-\frac{t}{T}}M^{t/T}.
$$
Consequently,
\begin{align*}
    \|u_n(t)-v_n(t)\|
&=  \|\phi_n(t)\| \\
    &=  e^{-t^2}\|\Psi_n(t)+\Phi_n(t)\| \\
    &\leq  \|\Psi_n(t)\|+\|\Phi_n(t)\| \\
    &\leq  C\beta^{1-\frac{t}{T}}M^{t/T}+\|\Phi_n(t)\|.
\end{align*}
It follows from an earlier estimate on $\|\Phi_n(i\eta)\|$, that
$$
\|\Phi_n(t)\|\leq \beta\Big[\frac{K}{\pi}e^{T^2}(1+e^{\gamma
T})\left((1+T)M'''+
N'\right)\int_0^T\big(1+\log \frac{1}{|x-t|}\big)dx\Big].
$$
Setting
$$
K'=\frac{K}{\pi}e^{T^2}(1+e^{\gamma T})\left((1+T)M'''+
N'\right)\Big\{\max_{t\in
[0,T]}\int_0^T\big(1+\log \frac{1}{|x-t|}\big)dx\Big\},
$$
we have
\begin{align*}
    \|u_n(t)-v_n(t)\| &\leq  C\beta^{1-\frac{t}{T}}M^{t/T}+\|\Phi_n(t)\| \\
    &\leq  C\beta^{1-\frac{t}{T}}M^{t/T}+\beta K' \\
    &=  \left(C+\beta^{t/T} K'M^{-\frac{t}{T}}\right)\beta^{1-\frac{t}{T}}M^{t/T} \\
    &\leq  C\beta^{1-\frac{t}{T}}M^{t/T},
\end{align*}
 for a possibly different constant $C$.  Letting $n\to \infty$,
we have found constants $C$ and $M$, independent of $\beta$,
 such that for $0\leq t<T$,
$$
\|u(t)-v(t)\|\leq C\beta^{1-\frac{t}{T}}M^{t/T},
$$
as desired.
\end{proof}

\section{Examples}

Below, we give examples illustrating our approximation theorem.
Each is a general case of the following universal example.
Let $H=L^2(\mathbb{R}^n)$ and $D=-\Delta$ where $\Delta$ denotes
the Laplacian defined by
$$
\Delta h=\sum_{i=1}^n\frac{\partial^2h}{\partial x_i^2}.
$$
The operator $-\Delta$ is a positive self-adjoint operator
on $L^2(\mathbb{R}^n)$ and so we compare the ill-posed evolution
problem
\begin{equation} \label{e13}
\begin{gathered}
    \frac{\partial}{\partial t} u(t,x)
=  A(t,-\Delta)u(t,x), \quad  (t,x)\in [0,T)\times \mathbb{R}^n \\
    u(0,x) =  h(x), \quad  x\in \mathbb{R}^n
\end{gathered}
\end{equation}
in $L^2(\mathbb{R}^n)$ to the well-posed approximate problem
\begin{equation} \label{e14}
\begin{gathered}
    \frac{\partial}{\partial t} v(t,x) =  f(t,-\Delta)v(t,x),
\quad  (t,x)\in [0,T)\times \mathbb{R}^n \\
    v(0,x) =  h(x), \quad  x\in \mathbb{R}^n.
\end{gathered}
\end{equation}

\begin{example} \label{exa1} \rm
 As initially defined, let
$$
A(t,D)= \sum_{j=1}^ka_j(t)D^j.
$$
Let $0<\epsilon <1$ and set $B_j= \max_{t\in [0,T]}|a_j(t)|$
for each $1\leq j\leq k$.  Consider the problem
\begin{gather*}
    \frac{\partial}{\partial t} v(t,x)
=  A(t,-\Delta)v(t,x)-\epsilon(-\Delta)^{k+1}v(t,x), \quad
(t,x)\in [0,T)\times \mathbb{R}^n \\
    v(0,x) =  h(x), \quad  x\in \mathbb{R}^n.
\end{gather*}
\end{example}

Motivated by approximations used by Lattes and Lions, Miller,
and Ames and Hughes  \cite{a2,a3,l1,m1}, we define
$f:[0,T]\times [0,\infty)\to \mathbb{R}$ by
$$
f(t,\lambda)=\sum_{j=1}^ka_j(t)\lambda^j-\epsilon\lambda^{k+1}.
$$
Then for each $(t,\lambda)\in [0,T]\times [0,\infty)$,
\[
f(t,\lambda)\leq h(\lambda) :=
 \sum_{j=1}^kB_j\lambda^j-\epsilon\lambda^{k+1}.
\]
 The polynomial $h(\lambda)$ has at most $k+1$ real roots.
If $h(\lambda)$ has no real roots on $[0,\infty)$, then
$h(\lambda)<0$ for all $\lambda\geq 0$.  Otherwise, let $R$
be the maximum of all such roots of $h(\lambda)$ on $[0,\infty)$.
Then $h(\lambda)$ is bounded above on $[0,R]$ and is negative
on $(R,\infty)$.  Therefore, in any case, there exists
$\omega\in \mathbb{R}$ such that $h(\lambda)\leq \omega$
for all $\lambda \geq 0$.  Consequently,
$$
f(t,\lambda)\leq \omega
$$
for all $(t,\lambda)\in [0,T]\times [0,\infty)$.  Also
$$
|f(t,\lambda)|\leq r(\lambda):=\sum_{j=1}^kB_j\lambda^j
+\epsilon\lambda^{k+1}
$$
for all $(t,\lambda)\in [0,T]\times[0,\infty)$.
We set $Y=\operatorname{Dom}(r(D))$ and let $\|\cdot \|_Y$
denote the graph norm associated with the operator $r(D)$.
We note that $Y=\operatorname{Dom}(r(D))=\operatorname{Dom}(f(t,D))$
for all $t\in [0,T]$, and that $Y$ is the Sobolev space
$W^{2(k+1),2}(\mathbb{R}^n)$, consisting of functions
$h\in L^2(\mathbb{R}^n)$ whose derivatives, in the sense of
distributions, of order $j\leq 2(k+1)$ are in $L^2(\mathbb{R}^n)$
(cf. \cite[Chapter 7.1]{p1}).

 Now, let $t_0\in [0,T]$.  It follows from the definition of
$r(D)$ that $D^j\in B(Y,H)$ for each $1\leq j\leq k$.
Then since $a_j$ is continuous for each $1\leq j\leq k$,
\begin{align*}
    \|f(t,D)-f(t_0,D)\|_{Y\to H} &=  \|(A(t,D)-\epsilon D^{k+1})-(A(t_0,D)-\epsilon D^{k+1})\|_{Y\to H} \\
    &=  \|A(t,D)-A(t_0,D)\|_{Y\to H} \\
    &=  \|\sum_{j=1}^k(a_j(t)-a_j(t_0))D^j\|_{Y\to H} \\
    &\leq  \sum_{j=1}^k|a_j(t)-a_j(t_0)|\;\|D^j\|_{Y\to H}
    \to  0 \quad \text{as }  t\to t_0,
\end{align*}
showing that $t\mapsto f(t,D)$ is continuous in the $B(Y,H)$
norm $\|\cdot\|_{Y\to H}$.

 Next, set
$$
p(\lambda)=\lambda^{k+1}.
$$
Then $\operatorname{Dom}(p(D))\subseteq \operatorname{Dom}(D^j)$
for all $1\leq j\leq k+1$ so that
$$
\operatorname{Dom}(p(D))\subseteq \operatorname{Dom}(A(t,D))
\cap \operatorname{Dom}(f(t,D))=\operatorname{Dom}(f(t,D))
$$
for each $t\in [0,T]$.  Furthermore, for
$\psi\in \operatorname{Dom}(p(D))$ and $t\in [0,T]$,
\[
    \|(-A(t,D)+f(t,D))\psi\| =  \|(-\epsilon D^{k+1})\psi\|
    =  \epsilon \|D^{k+1}\psi\|.
\]
Thus $f$ satisfies Condition $(\mathcal{A},p)$ with
$r(\lambda)=\sum_{j=1}^kB_j\lambda^j+\epsilon\lambda^{k+1}$,
$\beta=\epsilon$, and $p(\lambda)=\lambda^{k+1}$.
Moreover, $g(t,\lambda)=-\epsilon \lambda^{k+1}\leq 0$
for all $(t,\lambda)\in [0,T]\times [0,\infty)$,
so we may choose $\gamma = 0$.  Theorem \ref{thm6} then yields
 the result
$$
\|u(t)-v(t)\|\leq C\beta^{1-\frac{t}{T}}M^{t/T}
$$
for $0\leq t<T$, where $u(t)$ and $v(t)$ are solutions
of \eqref{e13} and \eqref{e14} respectively.


\begin{example} \label{exa2} \rm
  As in Example \ref{exa1}, let
$$
A(t,D)= \sum_{j=1}^ka_j(t)D^j.
$$
 Let $0<\epsilon <1$ and set $B_j= \max_{t\in [0,T]}|a_j(t)|$
for each $1\leq j\leq k$.  Consider the problem
\begin{align*}
    \frac{\partial}{\partial t} v(t,x)-A(t,-\Delta)v(t,x)+\epsilon
(-\Delta)^k\frac{\partial }{\partial t }v(t,x) &=  0, \quad
(t,x)\in [0,T)\times \mathbb{R}^n \\
    v(0,x) &=  h(x), \quad  x\in \mathbb{R}^n.
\end{align*}
\end{example}

Motivated by work of Showalter \cite{s1}, we define
$f:[0,T]\times [0,\infty)\to \mathbb{R}$ by
$$
f(t,\lambda)=\frac{\sum_{j=1}^ka_j(t)\lambda^j}{1+\epsilon \lambda^k}.
$$
Then for each $(t,\lambda)\in [0,T]\times [0,\infty)$,
$$
f(t,\lambda)\leq \frac{\sum_{j=1}^kB_j\lambda^j}{1+\epsilon \lambda^k}.
$$
The rational function
$r(\lambda)=\frac{\sum_{j=1}^kB_j\lambda^j}{1+\epsilon \lambda^k}$
is continuous on $[0,\infty)$ and tends to $\frac{B_k}{\epsilon}$
as $\lambda \to \infty$.  Therefore, there exists
$\omega\in \mathbb{R}$ such that $r(\lambda)\leq \omega$ for all
$\lambda \geq 0$.  Consequently,
$$
f(t,\lambda)\leq \omega
$$
for all $(t,\lambda) \in [0,T]\times [0,\infty)$.
As $r(\lambda)$ is a bounded Borel function on $[0,\infty)$,
the Spectral Theorem yields that $r(D)$ is a bounded
everywhere-defined operator on $H$.  Thus, we may choose
$Y=\operatorname{Dom}(r(D))=H$.

 Now, let $t_0\in [0,T]$.  It follows from the definition
of $r(D)$ that $D^j(I+\epsilon D^k)^{-1} \in B(H)$ for each
$1\leq j\leq k$.  Then since $a_j$ is continuous for each
$1\leq j\leq k$,
\begin{align*}
    \|f(t,D)-f(t_0,D)\| &=  \|A(t,D)(I+\epsilon D^k)^{-1}-A(t_0,D)(I+\epsilon D^k)^{-1}\| \\
    &=  \|\sum_{j=1}^k(a_j(t)-a_j(t_0))D^j(I+\epsilon D^k)^{-1}\| \\
    &\leq \sum_{j=1}^k|a_j(t)-a_j(t_0)|\;\|D^j(I+\epsilon D^k)^{-1}\|
    \to  0 \quad \text{as } t\to t_0,
\end{align*}
showing that $t\mapsto f(t,D)$ is continuous in the $B(H)$ norm.

 Next, set
$$
p(\lambda)=\sum_{j=1}^kB_j\lambda^{k+j}.
$$
Then for each $t\in [0,T]$,
$\operatorname{Dom}(p(D))\subseteq \operatorname{Dom}(A(t,D))$.
Next, note $D^k$ is positive since $D$ is, and so
$\frac{1}{\epsilon} \in \rho(-D^k)$.  Therefore, in view of
the fact that
$$
(I+\epsilon D^k)^{-1} = \frac{1}{\epsilon}R
\big(\frac{1}{\epsilon};-D^k\big),
$$
we see that for each $t\in [0,T]$,
$$
\operatorname{Dom}(f(t,D))=\operatorname{Dom}(A(t,D)
(I+\epsilon D^k)^{-1})=H\supseteq \operatorname{Dom}(A(t,D)).
$$
Therefore,
$$
\operatorname{Dom}(p(D))\subseteq \operatorname{Dom}(A(t,D))
\cap \operatorname{Dom}(f(t,D))
$$
for all $t\in [0,T]$.
 Next, fix $t\in [0,T]$ and assume $\psi\in \operatorname{Dom}(p(D))$.
  Set $$y=A(t,D)(I+\epsilon D^k)^{-1}\psi.$$  Since $-D^k$
generates a $C_0$ semigroup of contractions, we have by the
Hille-Yosida Theorem \cite[Theorem 1.3.1]{p1} that
$$
\|(I+\epsilon D^k)^{-1}\|=\|\frac{1}{\epsilon}R
\Big(\frac{1}{\epsilon};-D^k\Big)\|\leq \frac{1}{\epsilon}
\Big(\frac{1}{1/\epsilon}\Big)=1.
$$
Thus,
\begin{align*}
    \|(-A(t,D)+f(t,D))\psi\|
&=  \|(-A(t,D)+A(t,D)(I+\epsilon D^k)^{-1})\psi\| \\
    &=  \|-\epsilon D^ky\| \\
    &=  \epsilon \|\Big(\sum_{j=1}^ka_j(t)D^{k+j}\Big)(I+\epsilon D^k)^{-1}\psi\| \\
    &\leq  \epsilon \|\Big(\sum_{j=1}^ka_j(t)D^{k+j}\Big)\psi\| \\
    &\leq  \epsilon \|\Big(\sum_{j=1}^kB_jD^{k+j}\Big)\psi\|.
\end{align*}
Then $f$ satisfies Condition $(\mathcal{A},p)$ with
$r(\lambda)=\frac{\sum_{j=1}^kB_j\lambda^j}{1+\epsilon \lambda^k}$,
 $\beta=\epsilon$, and $p(\lambda)=\sum_{j=1}^kB_j\lambda^{k+j}$.
Moreover,
$g(t,\lambda)=-A(t,\lambda)+A(t,\lambda)(1+\epsilon \lambda^k)^{-1}
\leq 0$ for all $(t,\lambda)\in [0,T]\times [0,\infty)$, so we
 may choose $\gamma = 0$.  Again, Theorem \ref{thm6} yields the result
$$
\|u(t)-v(t)\|\leq C\beta^{1-\frac{t}{T}}M^{t/T}
$$
for $0\leq t<T$, where $u(t)$ and $v(t)$ are solutions of \eqref{e13}
and \eqref{e14} respectively.

\begin{thebibliography}{00}

\bibitem{a1} Agmon, S. and L. Nirenberg,
\emph{Properties of solutions of ordinary differential equations
in Banach space}, Comm. Pure Appl. Math. {\bf{16}} (1963), 121-151.

\bibitem{a2} Ames, K. A.,
``Comparison Results for Related Properly and Improperly
Posed Problems, with Applications to Mechanics,"
Ph.D. Thesis, Cornell University, Ithaca, NY, 1980.

\bibitem{a3} Ames, K. A. and R. J. Hughes,
\emph{Structural Stability for Ill-posed Problems in Banach Space},
Semigroup Forum {\bf{70}} (2005), 127-145.

\bibitem{c1} Campbell Hetrick, B. and R. J. Hughes;
 \emph{Continuous Dependence Results for Inhomogeneous Ill-posed
Problems in Banach Space}, J. Math. Anal. Appl. {\bf{331}} (2007),
342-357.

\bibitem{d1} Dunford, N. and J. Schwartz,
``Linear Operators, Part II," John Wiley and Sons, Inc.,
 New York, 1957.

\bibitem{f1} Fattorini, H. O.,
``The Cauchy Problem," Addison-Wesley, Reading, 1983.

\bibitem{g1} Goldstein, J. A.,
``Semigroups of Linear Operators and Applications,"
Oxford Univ. Press, New York, 1985.

\bibitem{k1} Kato, T.,
\emph{Linear evolution equations of hyperbolic type},
 J. Fac. Sci. Univ. Tokyo {\bf{25}} (1970), 241-258.

\bibitem{l1} Lattes, R. and J. L. Lions,
"The Method of Quasireversibility, Applications to Partial
Differential Equations," Amer. Elsevier, New York, 1969.

\bibitem{m1} Miller, K.,
Stabilized quasi-reversibility and
other nearly-best-possible methods for non-well-posed problems,
"Symposium on Non-Well-Posed Problems and Logarithmic Convexity
(Heriot-Watt Univ., Edinburgh, 1972)," 161-176,
Springer Lecture Notes in Mathematics, Volume 316,
Springer, Berlin, 1973.

\bibitem{p1} Pazy, A.,
``Semigroups of Linear Operators and Applications to Partial
Differential Equations," Springer-Verlag, New York 1983.

\bibitem{r1} Rudin, W.,
``Real and Complex Analysis," 3rd ed., McGraw-Hill Inc.,
New York, 1987.

\bibitem{s1} Showalter, R. E.,
\emph{The final value problem for evolution equations},
J. Math. Anal. Appl. {\bf{47}} (1974), 563-572.

\bibitem{t1} Tanaka, N.,
\emph{Linear Evolution Equations In Banach Spaces},
Proc. Lond. Math. Soc. {\bf{63}} (1991), 657-672.

\bibitem{y1} Yosida, K., ``Functional Analysis,"
Springer-Verlag, New York, 1978.

\end{thebibliography}

\end{document}