\section{Introduction}

The study of fast-reaction systems plays a crucial role in many natural sciences. Examples of fast-reaction systems and their limit can be found in reaction-diffusion models \cite{murakawa2011fast, fellner2016quasi, henneke2016fast}, enzyme reactions \cite{doi:10.1063/1.1889434}, coagulation-fragmentation models \cite{doi:10.1081/PDE-120021188,carrillo2008fast} and many other bio-chemical systems \cite{guarguaglini2007fast,herrmann2014rate,de2019fast}. The important aspect of fast-reaction limits is that they can help to reduce the complexity of a system by eliminating the fast evolving dynamics of the system. One approach, that stems from the modeling of chemical reactions, is to regard the fast-reaction limit as a quasi-steady-state approximation (QSSA) of the system \cite{zbMATH06088800, eilertsen2021hunting}. The mathematical study of fast-reactions in the context of reaction-diffusion dates back at least to the seminal work by Hilhorst et al. \cite{hilhorst1996fast}. A more recent review of the topic can be found in \cite{iida2018review}. Other examples of significant contributions in this research area include \cite{crooks2007fast,doi:10.1080/03605300903225396,hausberg2018well,daus2020cross}.

One important question that arises in the study of fast-reaction systems is to rigorously prove the convergence of the system and its solutions towards the fast-reaction limit. Different approaches such as energy and entropy methods \cite{Bothe201, skrzeczkowski2022fast}, variational methods \cite{disser2018evolutionary,mielke2021edp, stephan2021edp} or homogenization techniques \cite{meier2010two,cesaroni2017homogenization} have been proposed in recent years. Our approach follows a recent geometric viewpoint using dynamical systems methods as introduced in \cite{hummel2022slow} for fast-slow systems. Here, the idea is to generalize the theory developed by Fenichel \cite{fenichel1971persistence,fenichel1979geometric} for ODE systems and combine it with techniques for infinite-dimensional Banach spaces cf.~\cite{chow1988invariant,debussche1991inertial, henry2006geometric} and the references therein. The challenges and also several applications of this approach can be found in \cite{engel2020blow,engel2021connecting, hummel2022slow}. Yet, the case of fast-reaction system is not covered by the results in~\cite{hummel2022slow} and here we cover this case. More precisely, the fast-reaction evolution equations analyzed in this work are the following
\begin{align}\label{fast-reaction}
    \begin{split}
        \partial_t u^\varepsilon &=A u^\varepsilon +\frac{1}{\varepsilon} f(u^\varepsilon, v^\varepsilon),\\
        \partial_t v^\varepsilon &= B v^\varepsilon + g(u^\varepsilon, v^\varepsilon),
    \end{split}
\end{align}
where $\varepsilon\geq 0$ is a small parameter, $A$ and $B$ are linear and densely defined operators on Banach spaces $X$ and $Y$ of scalar-valued functions respectively. Moreover, $f$ and $g$ are sufficiently regular nonlinear functions and $(u^\varepsilon(t),v^\varepsilon(t))=(u^\varepsilon,v^\varepsilon)\in X\times Y$ are the unknown functions for some $t\in[0,T]$ and a final time $T>0$. We see that for $\varepsilon>0$ the fast-reaction system~\eqref{fast-reaction} is equivalent to a fast-slow system of the form
\begin{align}\label{fast-slow}
    \begin{split}
       \varepsilon \partial_t u^\varepsilon &=\varepsilon A u^\varepsilon +f(u^\varepsilon, v^\varepsilon),\\
        \partial_t v^\varepsilon &= B v^\varepsilon + g(u^\varepsilon, v^\varepsilon).
    \end{split}
\end{align}
The parameter $\varepsilon$ gives rise to a formal time-scale separation between the fast variable $u^\varepsilon$ and the slow variable $v^\varepsilon$. Letting $\varepsilon \to 0$ in either the fast-reaction equations \eqref{fast-reaction} or the fast-slow system \eqref{fast-slow}
 yields the following slow subsystem
 \begin{align}\label{slow system}
    \begin{split}
        0 &=f(u^0, v^0),\\
        \partial_t v^0 &= B v^0 + g(u^0, v^0),
    \end{split}
\end{align}
 which is an differential-algebraic equation defined on a critical set
 \begin{align} 
 \label{critical manifold}
     S_0:=\{(u^0,v^0)\in X\times Y\,:\, f(u^0,v^0)=0\},
 \end{align}
where we refer to $S_0$ as critical manifold since we shall later on make a precise technical assumption involving the implicit function theorem to ensure that it is actually a Banach manifold. \\

The main result of our work is a generalization of Fenichel's theory for the fast-reaction system \eqref{fast-reaction}.
The two key assumptions are that the critical manifold is attracting, which is encoded in suitable technical attraction conditions and that the operator $B$ allows a splitting of the space $Y$ into a fast and slow component.
This splitting is denoted by a small parameter $\zeta >0$, depending on $\varepsilon$.
Under these assumptions we obtain several results that together give a generalization of Fenichel's theory to infinite dimensions for the case of fast-reactions. The first result is that there exists a locally invariant and continuously differentiable manifold for the fast-reaction system \eqref{fast-reaction} denoted by $S_\varepsilon$.
Secondly, this slow manifold $S_\varepsilon$ converges to the critical manifold $S_0$ as $\varepsilon\to 0$.
Lastly, we show that the semi-flow on the slow manifold converges to the slow flow on a subset of the critical manifold $S_0$. The precise technical statements of these results can be found below in Propositions \ref{Prop 4.2}-\ref{Prop 4.9}. Now, let us briefly outline the structure of this work.
\begin{itemize}
    \item We motivate the generalization of Fenichel's theorem by studying some ODE systems and highlight two different approaches for an infinite-dimensional extension (see Section \ref{section 2}).
    \item We show that the fast-reaction systems are well-posed and prove the convergence of solutions as the parameter         $\varepsilon\to 0$ (see Section \ref{section 3}).
    \item We generalize Fenichel's theorem for fast-reaction systems in infinite-dimensional Banach spaces.
          Under suitable assumptions we prove the existence of a $C^1$-regular slow manifold that is close to the critical manifold in a suitable sense.
          Moreover, we show that the critical manifold attracts solutions and that the flows on the slow manifold converge to the flow on the critical manifold.
          To conclude this part we apply the theory to an example (see Section \ref{section 4}).
    \item Lastly we present a brief overview on the theory of $C_0$ semigroups and their application to abstract Cauchy problems (see Appendix \ref{appendix A}).
\end{itemize}   


\section{Motivation}\label{section 2}

\subsection{Fenichel Theory for fast-slow ODEs}

Here we present a brief overview of the perturbation theory of normally hyperbolic invariant manifolds introduced
by Fenichel \cite{fenichel1971persistence}.
One important result of Fenichel’s works is that its provide a suitable
framework for the treatment of fast-slow systems \cite{fenichel1979geometric,jones1995geometric} of the form
\begin{align}\label{fast-slow ode}\begin{split}
    \varepsilon\frac{\textnormal{d}}{\textnormal{d}t} u^\varepsilon &=\varepsilon A u^\varepsilon + f(u^\varepsilon,v^\varepsilon),\\
    \frac{\textnormal{d}}{\textnormal{d}t} v^\varepsilon &=B v^\varepsilon + g(u^\varepsilon,v^\varepsilon)
\end{split}
\end{align}
where $ \varepsilon>0$ is a small parameter, $u^\varepsilon(t)\in \mathbb{R}^n$, $v^\varepsilon(t)\in \mathbb{R}^m$, $A,\,B$ are square matrices in the respective spaces and $f,\,g$ are differentiable
nonlinearities. 
The unknown functions $u^\varepsilon(t)$ and $v^\varepsilon(t)$ are called the fast and the slow variable,
respectively.

\begin{theorem}
[Fenichel’s theorem fast–slow version \cite{fenichel1979geometric,jones1995geometric}]
Suppose $S_0$ is a compact normally hyperbolic submanifold of the critical manifold $C_0$ of \eqref{fast-slow ode}
and that $f,\,g \in C^r$ for $r\geq 1$. Then for $\varepsilon>0$ sufficiently small, the following
hold:
\begin{itemize}
    \item[(F1)] There exists a locally invariant manifold $S_\varepsilon$ called slow manifold diffeomorphic to $S_0$.
    \item[(F2)] $S_\varepsilon$ has Hausdorff distance of order $\mathcal{O}(\varepsilon)$ from $S_0$.
    \item[(F3)] The flow on $S_\varepsilon$ converges to the slow flow as $\varepsilon\to 0$.
    \item[(F4)] $S_\varepsilon$ is $C^r$-smooth.
    \item[(F5)] $S_\varepsilon$ is normally hyperbolic and has the same stability properties with respect
                to the fast variables as has 
                $S_0$.
\end{itemize}
\end{theorem}
For more details and the exact definitions and notions of convergence we refer to \cite{kuehn2015multiple}.

\subsection{Examples from ODEs}

One challenge in transferring the result by Fenichel to general Banach spaces is to control the $\frac{1}{\varepsilon}$-term arising from the fast equation that dominates the right hand-side. A naive approach would lead to a failure in the estimates for the approximation of the critical manifold by the slow manifold. Hence, to obtain an idea how the fast-reaction affects the system and the difficulties that occur in infinite dimensions, we first study the fast-reaction system in a low dimensional ODE setting. The focus will be on the effect of the nonlinearity $f(u,v)$ and the interplay between the linear operator and the fast-reaction part. To this end, we study the following planar system
\begin{align}
    \begin{split}
        \frac{\textnormal{d}}{\textnormal{d} t}\begin{pmatrix}u^\varepsilon\\v^\varepsilon \end{pmatrix}= \begin{pmatrix} -k^2 &0\\0&-1 \end{pmatrix} \begin{pmatrix} u^\varepsilon\\v^\varepsilon\end{pmatrix} +\begin{pmatrix} \frac{1}{\varepsilon} f(u^\varepsilon,v^\varepsilon)\\ 0\end{pmatrix},
    \end{split}
\end{align}
where $k > 0$ is a fixed constant and $\varepsilon\geq 0$ is the parameter relating to the separation of time scales.\\
For the first example let $f(u,v)= -u+v^2$.
Then, the ODE system has the explicit solution
\begin{align}
u^\varepsilon(t)=c_1 \textnormal{e}^{-t(k^2+\varepsilon^{-1})} + \frac{c_2^2 \textnormal{e}^{-2t}} {(-2+k^2+\varepsilon^{-1})\varepsilon}, \qquad v^\varepsilon(t)=c_2 \textnormal{e}^{-t}.
\end{align}
 Here, we can observe a different behavior of the solution in $u$ for different ratios between $k^2$ and $\varepsilon^{-1}$.
 Assuming that $k^2\ll \varepsilon^{-1}$, i.e., the fast-reaction term is dominating, we obtain that $u^\varepsilon(t)\approx v^\varepsilon(t)^2 +\mathcal{O}(\varepsilon)$. One easily checks that that in this regime the solution $u^\varepsilon$ converges to the equilibrium state satisfying $f(u^0,v^0)=0$.
Assuming on the other hand that $k^2\gg \varepsilon^{-1}$, i.e. the linear (diffusion) term dominates and we have $f(u^\varepsilon,v^\varepsilon)\neq 0$. In particular, the leading-order term of the fast variable vector field is now just $-k^2u^\varepsilon$ and the critical manifold is now given by $\{(u,v)\in\mathbb{R}^2:u=0,v\in\mathbb{R}\}$ if one just sets the leading-order term to zero so it has nothing to do with the solutions of $f(u,v)=0$ except at the single point $u=0=v$.

\begin{remark}
The choice $f(u,v)= u-v^2$ would also work in finite dimensions. In this case the system is repelling in the fast variable. However, using the property of time reversal symmetry for ODEs, this case coincides with the first one and the Fenichel theory still applies.
Yet it is not possible to transfer this idea easily to infinite dimensions since the time reversal idea fails as semi-flows that are not defined in backward time.
\end{remark}

As a second example we consider $f(u,v)= -u^2 +v^2$, where we focus on the positive branches of the solution in $u$ given by $\{f(u,v)=0,u>0\}$. From the stability analysis for the system we conclude that these branches are locally attracting in the fast direction $u$. Therefore, in the regime where $k^2\ll \varepsilon^{-1}$, we obtain from Fenichel's Theorem that any solution $(u^\varepsilon,v^\varepsilon)$ staying near the slow manifold obeys an estimate of the form
\begin{align*}
    |f(u^\varepsilon,v^\varepsilon)|=\mathcal{O}(\varepsilon).
\end{align*}
Now, we summarize the observations made from these examples and try to formulate conclusions that also apply in the infinite-dimensional setting. The first observation we make is that it is natural to require that the nonlinearity has to satisfy $\partial_u f(u,v)<0$ when $k^2\ll \varepsilon^{-1}$. Otherwise there exists $\varepsilon_0$ such that for all $\varepsilon\in (0,\varepsilon_0]$ the equation 
\[-k^2+\frac{1}{\varepsilon} \partial_uf(u,v)=0\]
has at least one solution and therefore the system is no longer normally hyperbolic at this point. Hence, the Fenichel theory cannot be applied. The second observation is that the ratio between $k^2$ and $\varepsilon$ plays a crucial role in the behaviour of the solution in the ODE setting. In the fast-reaction regime, i.e. the case when $k^2\ll \varepsilon^{-1}$, selecting suitable initial conditions, a solution $u^\varepsilon$ flows locally towards this attracting critical manifold $f(u^0,v^0)=0$. Two solutions with different initial data in this regime satisfy
\begin{align*}
    |f(u_1,v_1)-f(u_2,v_2)|\leq C\varepsilon.
\end{align*}
In the other case, when $k^2\geq\varepsilon^{-1}$, the influence of linear term  and the nonlinearity can balance each other, or the linear part coming from $A$ may even dominate, and then $f(u^\varepsilon,v^\varepsilon)$ does not necessarily converge to $0$ as $\varepsilon\to 0$ for solutions near an attracting critical manifold, if it even exists.\\

Next, we take a more abstract look at the nonlinearity $f(u,v)$ and the implications it has on the fast-reaction system. The first assumption the nonlinearity must satisfy is that for all $v\in Y$ there exist an element $u^v\in X$ such that $f(u^v,v)=0$. Otherwise the limit system might not have a solution and thus there exists no critical manifold. Moreover, we note that a special case occurs when the nonlinearity contains a linear term in $u$.
In this case we can rewrite the fast reaction system as
\begin{align*}
    \begin{split}
        \partial_t u^\varepsilon &=A u^\varepsilon +\frac{1}{\varepsilon}\lambda_0 u^\varepsilon +\frac{1}{\varepsilon} \tilde f(u^\varepsilon, v^\varepsilon),\\
        \partial_t v^\varepsilon &= B v^\varepsilon + g(u^\varepsilon, v^\varepsilon),
    \end{split}
\end{align*}
and the first equation can be further simplified as follows
\begin{align*}
    \partial_t u^\varepsilon &=\frac{1}{\varepsilon}\tilde A_\varepsilon u^\varepsilon +\frac{1}{\varepsilon} \tilde f(u^\varepsilon, v^\varepsilon),
\end{align*}
where now $\tilde f(u,v)$ only contains true nonlinear terms in $u$ and the linear operator is given by  $\tilde A_\varepsilon u= \varepsilon Au + \lambda_0 u$.

\subsection{Two Approaches in extending the result to infinite dimensions}

From the examples in the finite setting we propose two approaches dealing with infinite-dimensional fast reaction systems.
\\

The first approach is to introduce a splitting in the fast variable into high and low frequencies, which represent the fast and slow components, to utilize the effects observed in the ODE examples.
The idea is that the fast component corresponds to the high frequencies of the operator whereas the slow component corresponds to the case where the fast-reaction dominates the right-hand side.

\begin{example}
Let $X$ be a Banach space and suppose $A:X \supset D(A)=X_1\to X$ is a generator of a $C_0$-semigroup $(T_A(t))_{t\geq 0}\subset \mathcal{B}(X)$ satisfying
\begin{align*}
     \|\textnormal{e}^{tA} x\|_{X_1}\leq M_A \textnormal{e}^{ \omega_A t}\|x\|_{X_1},
\end{align*}
where $\omega_A < 0$ and $M_A>0$. We focus on the fast-reaction equation
\begin{align}\label{one fast reaction}
    \partial_t u^\varepsilon= A u^\varepsilon +\frac{1}{\varepsilon} f(u^\varepsilon),
\end{align}
where the nonlinear function $f$ is sufficiently regular and allows for a global Lipschitz constant $L_f$.
The well-posedness of the equation \eqref{one fast reaction} follows from Proposition \ref{Prop 3.1} and we refer to \cite{lunardi2012analytic} for more details. Motivated by the examples from the finite-dimensional ODE one might consider to introduce a splitting of the form
\begin{align*}
    X=X^\xi_F \oplus X^\xi_S
\end{align*}
into a fast part $X^\xi_F$, i.e. the case where the linear operator dominates the effect of the nonlinearity, and a slow part $X^\xi_S$, i.e. the case where the fast reaction term drives the flow of the solution, and where $\xi>0$ is a small parameter encoding the splitting of the Banach space, such that the following holds
 \begin{itemize}
   
    \item The spaces $X^\xi_S\cap X_1$ and $X^\xi_F\cap X_1$ are closed subspaces in $X$ and will be endowed with the norm $\|\cdot\|_{X_1}$.
    \item The realization of the operator $A$ in $X^\xi_S$ is given by 
         $A_{X^\xi_S}: X^\xi_S\supset D(X^\xi_S)\to X^\xi_S$
    with $D(A_{X^\xi_S})=\{u_0\in X^\xi_S\cap D(A):\, Au_0\in X^\xi_S\}$ and similar for the realization of $A$ in $X^\xi_F$.
 \end{itemize}
 In addition, we make the following assumptions on the nonlinearity and the splitting based on the observations from the finite-dimensional examples.
 \begin{itemize}
      \item[(A1)] The space $X^\xi_F$ contains the parts of $X_1$ that decay under the semigroup $(\textnormal{e}^{tA})_{t\geq 0}$ satisfying the following estimate
    \begin{align*}
        \|\textnormal{e}^{tA} x_F\|_{X_1}\leq M_A \textnormal{e}^{\varepsilon^{-1} \omega_A t} \|x_F\|_{X_1},
    \end{align*}
  where we observe a much faster decay when compared with the general estimate.
    \item[(A2)] The nonlinearity $f$ satisfies
    \begin{align*}
        \|\text{pr}_{X^\xi_S} [f(u)-f( \tilde{u})]\|_{X_1}\leq   \xi(\varepsilon) L_f\|u-\tilde u\|_{X_1},
    \end{align*}
    where $\xi(\varepsilon)$ is of order $\mathcal{O}(\varepsilon)$ and $\text{pr}_{X^\xi_S}$ denotes the projection onto $X^\xi_S$.
 \end{itemize}
 
Now, we want to see if the solution $u^\varepsilon$ converges to the limit $u^0$, which satisfies $f(u^0)=0$, in the appropriate Banach space.
This convergence result is a key assumption for the existence of a slow manifold and the convergence of said slow manifold to the critical manifold (see Section \ref{section 4}). To simplify the calculation in this example, we assume that $u^0\equiv 0$.
Using the above splitting we can estimate
\begin{align*}
    \|u^\varepsilon(t)\|_{X_1}&\leq \bigg\| \textnormal{e}^{tA} u_0+ \frac{1}{\varepsilon} \int_0^t \textnormal{e}^{(t-s)A} f(u^\varepsilon(s))\, \textnormal{d}s\bigg\|_{X_1}\\
    &\leq  M_A \textnormal{e}^{\omega_A t} \|u_0\|_{X_1}+  M_A L_f\int_0^t \bigg(\frac{\textnormal{e}^{\varepsilon^{-1}(t-s)\omega_A}}{\varepsilon } \|u^\varepsilon(s)\|_{X_1}  +\textnormal{e}^{(t-s)\omega_A}\|u^\varepsilon(s)\|_{X_1} \bigg)  \, \textnormal{d}s
\end{align*}
By applying a version of Gronwall's inequality (see \cite[Lemma 2.10]{hummel2022slow}) we obtain
\begin{align*}
    \|u^\varepsilon(t)\|_{X_1}&\leq C M_A \|u_0\|_{X_1}\textnormal{e}^{(\omega_A +M_AL_F C)t},
\end{align*}
where the constant $C$ depends only on $\omega_A,\, M_A$ and $L_f$.
However, since this estimate is independent of $\varepsilon$ we can no longer recover the convergence $u^\varepsilon\to 0$ as $\varepsilon\to 0$. The reason for this is the lack of control of the initial data and the operator $A$ in terms of $\varepsilon$.
Hence, this approach, although inspired from the finite dimensional examples, does not yield the desired result.
\end{example}


\begin{example}
The second approach is based on the observation that in the case the nonlinearity has a linear term, this term can be attributed/moved to the operator. Hence, we can exploit this by adding and subtracting a linear term to the system that mimics the behavior of the linear case. 

Another way to motivate this idea is to think of a Taylor expansion of the nonlinearity close to the critical manifold $f(u^0)=0$ where we formally have
\begin{align*}
    f(u)= f(u^0) + \textnormal{D} f(u^0)(u-u^0) + \mathcal{O}((u-u^0)^2)
\end{align*}
and we note that since $f(u^0)=0$ the linear term of the derivative drives the fast-reaction.
To this end, we assume that in a neighborhood of $f(u^0)=0$ the function $f$ is twice continuously differentiable and allows a Taylor series expansion.
Moreover, we assume that the linear operator $\textnormal{D} f(u)$ has a negative spectrum such that the largest eigenvalue satisfies $\lambda_1\leq\lambda_0<0$.
In addition, we assume that the growth bound $\omega_A$ of the semigroup generated by the operator $A$ satisfies $\varepsilon\omega_A +\lambda_0<0$.

Then, going back to the example from above we write
\begin{align}\label{one fast reaction modified}
    \partial_t u^\varepsilon= A u^\varepsilon -\frac{1}{\varepsilon}\textnormal{D} f(u^0) u^\varepsilon +\frac{1}{\varepsilon} \tilde f(u^\varepsilon)
\end{align}
where $\tilde f(u)$ contains all the remainder terms in the Taylor series starting with the second order one.
From the definition of $\tilde f$ we observe that the Lipschitz constants satisfy $L_{\tilde f}\leq L_f$.
The well-posedness of this equation follows from the same reasoning as before. Again, we are interested in the convergence of $u^\varepsilon\to u^0$ as $\varepsilon \to 0$ and again for simplicity we assume that $u^0\equiv 0$.
Hence, we estimate
\begin{align*}
     \|u^\varepsilon(t)\|_{X_1}&\leq \bigg\| \textnormal{e}^{\varepsilon^{-1}t(\varepsilon A+\textnormal{D}f(u^0))} u_0+ \frac{1}{\varepsilon} \int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s)(\varepsilon A+ \textnormal{D}f(u^0))}\tilde f(u^\varepsilon(s))\, \textnormal{d}s\bigg\|_{X_1}\\
    &\leq  M_A \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A +\lambda_0)t} \|u_0\|_{X_1}+  M_A L_{\tilde f}\int_0^t \frac{\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A +\lambda_0)(t-s)}}{\varepsilon }  \|u^\varepsilon(s)\|_{X_1}  \, \textnormal{d}s.
\end{align*}
Applying Gronwall's inequality yields
\begin{align*}
     \|u^\varepsilon(t)\|_{X_1}&\leq M_A \textnormal{e}^{\varepsilon^{-1}( \varepsilon\omega_A +\lambda_0)t}\|u_0\|_{X_1} + M_A^2 L_{\tilde f} \|u_0\|_{X_1} \varepsilon^{-1} \textnormal{e}^{\varepsilon^{-1}( \varepsilon\omega_A +\lambda_0)t}\int_0^t  \exp\bigg( \int_s^t \frac{\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A +\lambda_0)(t-\tau)}}{\varepsilon } \textnormal{d}\tau\bigg)\textnormal{d}s\\
     &\leq M_A \textnormal{e}^{\varepsilon^{-1}( \varepsilon\omega_A +\lambda_0)t}\|u_0\|_{X_1}\big( 1 + M_A L_{\tilde f}  \varepsilon^{-1}\big)
\end{align*}
Here, we observe that $\|u^\varepsilon\|_{X_1}$ is bounded in terms of $\varepsilon$ and thus we obtain $u^\varepsilon \to 0$ as $\varepsilon\to 0$.
Thus, this approach allows to recover the convergence of the solutions, which is a first step in the generalization of Fenichel's theory.
\end{example}

Going forward, we will only consider the second approach in proving a generalization of Fenichel's theorem for fast-reaction systems in infinite dimensions with attracting critical manifold.

\section{Fast Reaction Systems}\label{section 3}
In this section we analyse the general fast reaction system
\begin{align}\label{fast-slow 2}
    \begin{split}
        \partial_t u^\varepsilon &= A u^\varepsilon +\frac{1}{\varepsilon} f(u^\varepsilon, v^\varepsilon),\\
        \partial_t v^\varepsilon &= B v^\varepsilon + g(u^\varepsilon, v^\varepsilon),
    \end{split}
\end{align}
for $\varepsilon\geq 0$ and show the convergence of the semi-flow of the solution of the $\varepsilon$-system to the semi-flow of the solution of the limit system as $\varepsilon\to 0$.

\subsection{Function spaces and assumptions}\label{sect 3.1}

We structure the assumptions into several parts. First of, we assume that
\begin{itemize}
    \item  Let $0\leq \varepsilon\leq 1$ be a parameter that encompasses the separation of time scales, where in the limit as $\varepsilon\to 0$ the PDE system  \eqref{fast-slow 2} turns into an algebraic-differential equation.
    \item Let $T>0$ be the final time of existence of solutions which is determined by the nonlinearity and the implicit function theorem.
    \item $X,Y$ are Banach spaces and $u_0\in X_1= D(A)$, $v_0\in Y_1=D(B)$ are initial values.
    If $\varepsilon=0$, then the initial values satisfy $f(u_0,v_0)=0$.
\end{itemize}
For the operators $A$ and $B$ we make the following assumptions
\begin{itemize}
    \item $A, B$ are closed linear operators, where $A:X\supset D(A)\to X$ generates an exponentially stable $C_0$-semigroup $(\textnormal{e}^{tA})_{t\geq 0}\subset \mathcal{B}(X)$ and $B:Y\supset D(B)\to Y$ generates an exponentially stable $C_0$-semigroup $(\textnormal{e}^{tB})_{t\geq 0}\subset \mathcal{B}(Y)$, respectively.
    \item The interpolation-extrapolation scales generated by $(X,A)$ and $(Y,B)$ are given by $(X_\alpha)_{\alpha\in [-1,\infty)}$ and $(Y_\alpha)_{\alpha\in [-1,\infty)}$ (for details see Appendix \ref{appendix A} or \cite{amann1995linear}).
    \item Let $\gamma\in (0,1]$ if $(\textnormal{e}^{tA})_{t\geq 0}$ is holomorphic and $\gamma=1$ otherwise.
    Similarly, we introduce $\delta \in (0,1]$ if $(\textnormal{e}^{tB})_{t\geq 0}$ is holomorphic and set $\delta=1$ otherwise.
     \item There are constants $C_A,C_B, M_A, M_B>0$, $\omega_A \in \mathbb{R}$ and $\omega_B\in \mathbb{R}$ such that
    \begin{align*}
        \|\textnormal{e}^{tA}\|_{\mathcal{B}(X_1)}&\leq M_A \textnormal{e}^{\omega_A t},\qquad \|\textnormal{e}^{tA}\|_{\mathcal{B}(X_\gamma,X_1)}\leq C_At^{\gamma-1}\textnormal{e}^{\omega_A t},\\
        \|\textnormal{e}^{tB}\|_{\mathcal{B}(Y_1)}&\leq M_B \textnormal{e}^{\omega_B t},\qquad \|\textnormal{e}^{tB}\|_{\mathcal{B}(Y_\delta,Y_1)}\leq C_B t^{\delta-1} \textnormal{e}^{\omega_B t}
    \end{align*}
    hold for all $t\in [0,T]$.
\end{itemize}
Next, we consider the nonlinear functions $f$ and $g$, where we assume that
\begin{itemize}
    \item The nonlinearity $f:X_1\times Y_1\to X_\gamma$ is two-times continuously differentiable and $g:X_1\times Y_1\to Y_\delta$ is continuously differentiable. Moreover, there are constants $L_f, L_g>0$ such that
    \begin{align*}
        \|f(x_1,y_1)-f(x_2,y_2)\|_{X_\gamma}&\leq L_f( \|x_1-x_2\|_{X_1}+\|y_1-y_2\|_{Y_1}),\\
        \|f(u_1,v_1)-f(u_2,v_2)\|_{C^1([0,T];X)}&\leq L_f(\|u_1-u_2\|_{C^1([0,T];X)}+\|v_1-v_2\|_{C^1([0,T];Y)}),\\
        \|g(x_1,y_1)-g(x_2,y_2)\|_{Y_\delta}&\leq L_g( \|x_1-x_2\|_{X_1}+\|y_1-y_2\|_{Y_1})
    \end{align*}
    for all $x_1,x_2\in X_1$, $y_1,y_2\in Y_1$ and all $u_1,u_2\in C^1([0,T];X)$, $v_1,v_2\in C^1([0,T];Y)$.
    \item We assume that $f(0,0)=0$ and $g(0,0)=0$.
\end{itemize}
The next set of assumptions deals with the specific restrictions on the fast-reaction nonlinearity $f$. 
\begin{itemize}
    \item Let $A\subset X\times Y$ be an open set with $f(x_0,y_0)=0$ for all $(x_0,y_0)\in A$.
    Moreover, assume that for all $(x_0,y_0)\in A$ the operator $\textnormal{D}_x f(x_0,y_0)\equiv \textnormal{D}  f(\cdot,y_0)x_0: X\to X_\gamma$ is a Banach space isomorphism, i.e. a bounded invertible linear operator.
   \end{itemize}
\begin{remark}
As a consequence of the last assumption the implicit function theorem holds for each $(x_0,y_0)\in A$. 
That is, there exists an open neighborhood $U_0$ of $y_0$ such that for all connected neighborhoods $U\subset U_0$ there exists a unique continuous function $h^0:U\to X$ such that $h^0(y_0)=x$ and $f(h^0(y),y)=0$ for all $y\in U$.
Moreover $h^0$ has the same regularity as $f$.
\end{remark}

\begin{itemize} 
    \item Select all the points $(x_0,y_0)\in A$ such that
    each neighborhood $U_{y_0}$, given by the implicit function theorem for $f$, contains a small ball with fixed radius $\rho>0$, i.e. $B(y_0,\rho)\subset U_{y_0}$. We denote this set by $A_\rho\subset A$.
    \item Assume that there exists a non-empty subset $\tilde A\subset A_\rho$ such that for all $(x_0,y_0)\in \tilde A$ the operator $\textnormal{D}_x f(x_0,y_0)$ has negative eigenvalues and that there is a uniform upper bound for the largest eigenvalue $\lambda_1$, i.e. $\textnormal{Re} \lambda_1\leq \lambda_0<0$.
    Moreover, assume that for all $(x_0,y_0)\in \tilde A$ the neighborhoods $U_{y_0}$ given by the implicit function theorem form a connected domain denoted by $\tilde U\subset Y$.
\end{itemize} 

\begin{remark}
    The last assumption guarantees that there exists a unique  $C^2$-function $h^0:\tilde U\to X$ such that $h^0(y_0)=x$ and $f(h^0(y),y)=0$ for all $y\in \tilde U$. 
\end{remark}
   
   
   
   
   
   
The last set of assumptions allows us to write the nonlinearity $f$ as a Taylor approximation close to the critical set $\tilde A$ 
\begin{align*}
    f(u,v)=\underbrace{f(u^0,v^0)}_{=0}+ \textnormal{D}_x f(u^0,v^0)(u-u^0)+ \textnormal{D}_y f(u^0,v^0)(v-v^0)+ \mathcal{O}((u-u^0,v-v^0)^2).
\end{align*}
Then, we can introduce the function $\tilde f$ as $\tilde f(u,v):=f(u,v)-\textnormal{D}_x f(u^0,v^0)u$ and we can rewrite the fast-reaction system in a modified form
\begin{align}\label{modified fast reaction}\begin{split}
    \partial_t u^\varepsilon&= \frac{1}{\varepsilon}\tilde A_\varepsilon u^\varepsilon +\frac{1}{\varepsilon}\tilde f(u^\varepsilon,v^\varepsilon),\\
    \partial_t v^\varepsilon&= B v^\varepsilon+ g(u^\varepsilon,v^\varepsilon),
\end{split}
\end{align}
where $\tilde A_\varepsilon:= \varepsilon A+ \textnormal{D}_x f(u^0,v^0)$.


    With the above assumptions and a perturbation result for semigroups (see Appendix \ref{appendix A}) we obtain that the operator $\tilde A_\varepsilon$ satisfies the following estimates
    \begin{align*}
         \|\textnormal{e}^{t \tilde A_\varepsilon}\|_{\mathcal{B}(X_1)}&\leq M_{\tilde A} \textnormal{e}^{(\varepsilon\omega_A+\lambda_0) t},\qquad \|\textnormal{e}^{t \tilde A_\varepsilon}\|_{\mathcal{B}(X_\gamma,X_1)}\leq C_{\tilde A}t^{\gamma-1}\textnormal{e}^{(\varepsilon\omega_A+\lambda_0) t}
    \end{align*}
    for some constants $M_{\tilde A},\, C_{\tilde A}>0$.
    Moreover, we obtain that $\tilde f$ has the same regularity as $f$ and the Lipschitz constant satisfies $L_{\tilde f} \leq L_f +\|\textnormal{D}_x f(u^0,v^0)\|_{\mathcal{B}(X_1)}$.
    
For this new system we require to the following to hold
\begin{itemize}
    \item Assume that the growth bound for the semigroup generated by the operator $\tilde A_\varepsilon$ satisfies $\varepsilon\omega_A+\lambda_0<0$.
    
\end{itemize}



\begin{remark}
   To illustrate the behaviour of $\tilde f$ we assume that the function $f$ allows a Taylor-expansion at $(u^0,v^0)$ with radius of convergence $\sigma>0$.
   Then, in a neighborhood $U\subset B_\sigma((u^0,v^0))$ the nonlinear function $f$ can be written as $f(u,v)= f(u^0,v^0)+ \textnormal{D}_x f(u^0,v^0)(u-u^0)+ \textnormal{D}_y f(u^0,v^0)(v-v^0)+ \mathcal{O}((u-u^0,v-v^0)^2)$ and thus $\tilde f(u,v)= \textnormal{D}_x f(u^0,v^0)u^0+ \textnormal{D}_y f(u^0,v^0)(v-v^0) +\mathcal{O}((u-u^0,v-v^0)^2)$, where we observe that the leading order term in $u$ is at least quadratic. 
   Hence, in a neighborhood of the critical manifold the linear term $\textnormal{D}_x f(u^0,v^0)u$ dominates the function $\tilde f$.
   
\end{remark}





\subsection{Existence of solutions}
In this section we show the well-posedness of the fast reaction system for $\varepsilon>0$ 
\begin{align}\begin{split}\label{full eq}
    \partial_t u^\varepsilon(t) &=A u^\varepsilon(t) + \frac{1}{\varepsilon} f( u^\varepsilon(t),v^\varepsilon(t)),\\
    \partial_t v^\varepsilon (t) &
    =  B v^\varepsilon(t) +g(u^\varepsilon(t),v^\varepsilon(t)),\\
    u^\varepsilon(0)&=  u_0,\quad v^\varepsilon(0)=v_0.
\end{split}
\end{align}
and the limit system when $\varepsilon=0$
\begin{align}\begin{split}\label{full eq limit}
  0&=  f(u^0(t),v^0(t)),\\
    \partial_t v^0 (t) &
    =  B v^0(t) +g(u^0(t),v^0(t)),\\
    u^0(0)&= u_0^0,\qquad v^0(0)=v_0^0.
\end{split}
\end{align}

\begin{prop}\label{Prop 3.1}
Let $\varepsilon>0$ be small and let $T>0$. Then, there exists a unique strict solution $(u^\varepsilon,v^\varepsilon)$ of the system  \eqref{full eq}, i.e the solution satisfies $(u^\varepsilon,v^\varepsilon)\in C^1([0,T];X\times Y) \cap C([0,T];X_1\times Y_1)$.
\end{prop}

\begin{proof}
The idea of the proof is to construct a contracting solution operator and apply Banach's fixed point theorem on a suitable space. 
The space we consider is $C_b([0,\infty), \textnormal{e}^{\eta t};X_1\times Y_1)$, i.e. the space of all $(u,v)\in C([0,\infty); X_1\times Y_1)$ such that
\begin{align*}
    \|(u,v)\|_{C_b([0,\infty), \textnormal{e}^{\eta t};X_1\times Y_1)}:= \sup_{t\geq 0} \textnormal{e}^{-\eta t} \big(\|u(t)\|_{X^1}+\|v(t)\|_{Y_1}\big)<\infty,
\end{align*}
where $\eta\in \mathbb{R}$.
On this space we now aim to show that the operator
\begin{align*}
    \mathcal{L}(u,v):= \begin{pmatrix}  
        \textnormal{e}^{  t A} u_0 & \quad +\varepsilon^{-1} \int_0^t \textnormal{e}^{ (t-s) A}  f(u(s),v(s)) \, \textnormal{d} s\\
     \textnormal{e}^{tB}v_0 &+ \int_0^t \textnormal{e}^{(t-s) B} g(u(s),v(s))\, \textnormal{d} s.
    \end{pmatrix}
\end{align*}
is a contraction.
Thus, estimating $\|\mathcal{L}(u,v)-\mathcal{L}(\tilde u,\tilde v)\|_{X_1\times Y_1}$ yields
\begin{align*}
     &\sup_{t\geq 0} \textnormal{e}^{-\eta t}\bigg\|\varepsilon^{-1} \int_{-\infty}^t \textnormal{e}^{(t-s)A}  [f(u(s), v(s))- f(\tilde u(s), \tilde v(s))]\, \textnormal{d} s\bigg\|_{X_1}\leq \\
     \leq & L_fC_A \varepsilon^{-1}\int_{-\infty}^t \frac{\textnormal{e}^{(t-s)(\omega_A-\eta)}}{(t-s)^{1-\gamma_X}}\, \textnormal{d}s \|(u(s), v(s))-(\tilde u(s), \tilde v(s))\|_{C_b([0,\infty), \textnormal{e}^{\eta t};X_1\times Y_1)}\\
     \leq &  L_fC_A \varepsilon^{-1}\frac{\Gamma(\gamma_X)}{(\eta -\omega_A)^{\gamma_X}} \|(u(s), v(s))-(\tilde u(s), \tilde v(s))\|_{C_b([0,\infty), \textnormal{e}^{\eta t};X_1\times Y_1)}
\end{align*}
and 
\begin{align*}
     &\sup_{t\geq 0} \textnormal{e}^{-\eta t}\bigg\| \int_{0}^t \textnormal{e}^{(t-s)B}  [g(u(s), v(s))- g(\tilde u(s), \tilde v(s))]\, \textnormal{d} s\bigg\|_{Y_1}\leq \\
     \leq & L_g C_B \int_{0}^t \frac{\textnormal{e}^{(t-s)(\omega_B-\eta)}}{(t-s)^{1-\gamma_Y}}\, \textnormal{d}s \|(u(s), v(s))-(\tilde u(s), \tilde v(s))\|_{C_b([0,\infty), \textnormal{e}^{\eta t};X_1\times Y_1)}\\
     \leq &  L_g C_B \frac{\Gamma(\gamma_Y)}{(\eta -\omega_B)^{\gamma_Y}} \|(u(s), v(s))-(\tilde u(s), \tilde v(s))\|_{C_b([0,\infty), \textnormal{e}^{\eta t};X_1\times Y_1)}.
\end{align*}
For $\eta$ large enough we have that $\mathcal{L}$ is a contraction and thus there exists a unique fixed point 
\[(u^\varepsilon_*, v^\varepsilon_*)\in C_b([0,\infty), \textnormal{e}^{\eta t}; X_1\times Y_1).\]
Let $(u^\varepsilon,v^\varepsilon)$ be this fixed point. 
Then, we have that
\begin{align*}
    u^\varepsilon(t)&= \textnormal{e}^{tA}u_0+ \frac{1}{\varepsilon}\int_0^t \textnormal{e}^{(t-s)A} f(u^\varepsilon(s),v^\varepsilon(s))\, \textnormal{d}s,\\
    v^\varepsilon(t)&= \textnormal{e}^{tB}v_0+ \int_0^t \textnormal{e}^{(t-s)B} g(u^\varepsilon(s),v^\varepsilon(s))\, \textnormal{d}s
\end{align*}
which implies that
\begin{align*}
    u^\varepsilon(t) &= u_0 + A\int_0^t u^\varepsilon(s)\, \textnormal{d}s +\frac{1}{\varepsilon} \int_0^t f( u^\varepsilon(s),v^\varepsilon(s))\, \textnormal{d}s,\\
    v^\varepsilon(t) &= v_0 + B\int_0^t v^\varepsilon(s)\, \textnormal{d}s + \int_0^t g( u^\varepsilon(s),v^\varepsilon(s))\, \textnormal{d}s
\end{align*}
see \cite{lunardi2012analytic} for details.
Thus, for all $t\in [0,T]$ we conclude that
\begin{align*}
    \partial_t u^\varepsilon(t)=\lim_{h\to 0} \frac{u^\varepsilon(t+h) -u^\varepsilon(t)}{h}&=\lim_{h\to 0}\bigg[ \int_t^{t+h} A u^\varepsilon(s)\, \textnormal{d}s +\frac{1}{\varepsilon} \int_t^{t+h} f( u^\varepsilon(s),v^\varepsilon(s))\, \textnormal{d}s \bigg]\\
\intertext{is well-defined and in the limit}
 \partial_t u^\varepsilon(t)&=A u^\varepsilon(t) + \frac{1}{\varepsilon} f(u^\varepsilon(t),v^\varepsilon(t)),
\end{align*}
where the convergence holds in $X$ as $A u^\varepsilon,\, f(u^\varepsilon,v^\varepsilon)\in C([0,T];X)$.
With the same idea we can show the well-posedness of $\partial_t v^\varepsilon\in C([0,T];Y)$.
Hence, we have shown that
\begin{align*}
    (u^\varepsilon, v^\varepsilon)\in C([0,\infty); X_1\times Y_1)\cap C^1([0,\infty); X\times Y)
\end{align*}
is the unique solution to \eqref{full eq} for $\varepsilon>0$.
\end{proof} 


\begin{prop}\label{Prop 3.2}
    Let $\varepsilon=0$. Then under the assumptions of Section \ref{sect 3.1} there exists a unique strict solution $(u^0,v^0)\in C^1([0,T];X\times Y)\cap C([0,T];X_1\times Y_1)$ to equation \eqref{full eq limit}.
    The final time $T$ is given by $T<\inf\{ \tau \geq 0: f(u(\tau),v(\tau))\neq 0 \textnormal{ or } \textnormal{D}_x f(u(\tau),v(\tau))=0\}$.
\end{prop}
  
\begin{proof}
Using the assumptions on the nonlinearity $f$ the implicit function theorem applied to the level set $f(u^0(t),v^0(t))=0$ implies that there exists a Fr\'{e}chet differentiable function $h^0:U_{Y_1}\subset Y_1\to X_1$ such that $h^0(y)=x$ for each $t\in [0,T]$.
Thus, the mapping $g: Y_1\to Y_{\delta_Y},\quad y \mapsto g(h^0(y),y)$ is continuous and satisfies 
\begin{align*}
\|g(h^0(y_1),y_1)-g(h^0(y_2),y_2)\|_{Y_1} \leq L_g\|h^0(y_1)-h^0(y_2)\|+ L_g\|y_1-y_2\| \leq L_g(1+L_h)\|y_1-y_2\|.
\end{align*}
Then, there exists a unique strict solution $v^0\in C^1([0,T];Y)\cap C([0,T];Y_1) $ to 
\begin{align*}
\partial_t v^0= B v^0 + g(h^0(v^0),v^0),\qquad v^0(0)=v_0.
\end{align*}
Indeed, using variation by constants and a suitable weighted space, similar to the one in Proposition \ref{Prop 3.1}, yields that the operator 
\begin{align*}
    \mathcal{L}_{B}(v^0)= \textnormal{e}^{Bt} v_0+ \int_0^t \textnormal{e}^{B(t-s)} g(h^0(v^0),v^0)\, \textnormal{d}s
\end{align*}
has a fixed point.
Now, setting $u^0(t):= h^0(v^0(t))$ we obtain that $u^0\in C^1([0,T];X)\cap C([0,T];X_1)$.
Thus, $(u^0,v^0)\in C^1([0,T];X\times Y)\cap C([0,T];X_1\times Y_1)$ is the unique strict solution of the limit system \eqref{full eq limit} with $\varepsilon=0$.
\end{proof}

We note that the two above propositions also hold for the modified system \eqref{modified fast reaction} and we conclude this section with two observations.
\begin{remark}
The mapping 
\[h^0: Y_1\to X_1,\qquad y\mapsto h^0(x),\] 
satisfying $f(h^0(y),y)=0$, introduced in the previous proof describes the critical manifold $S_0$ over $Y_1$ given by
\[S_0:=\{ (h^0(y),y):\, y\in Y_1\}\subset X\times Y.\]

\end{remark}
\begin{remark}
  We note that both systems \eqref{full eq} and \eqref{full eq limit} are autonomous.
 Therefore, the solutions $(u^0,v^0)$ for $\varepsilon=0$ and $(u^\varepsilon,v^\varepsilon)$ for $\varepsilon>0$ are given by semiflows, i.e. continuous mappings of the form
 \[T_0:[0,T]\times S_0\to S_0,\qquad T_\varepsilon:[0,T]\times X_1\times Y_1\to X_1\times Y_1\]
and we write
\begin{align*}
    \begin{pmatrix}
    u^0(t)\\v^0(t)
    \end{pmatrix} =T_0(t)\begin{pmatrix}
    h(v_0)\\v_0
    \end{pmatrix},\qquad \begin{pmatrix}
    u^\varepsilon(t)\\v^\varepsilon(t)
    \end{pmatrix}=T_\varepsilon (t)\begin{pmatrix}
    u_0\\v_0
    \end{pmatrix}.
\end{align*}
\end{remark}




\subsection{Approximation by slow flow}

The main result of this subsection is to show the convergence of the solutions of the full system \eqref{modified fast reaction} to the solutions of the limit system \eqref{full eq limit} as $\varepsilon\to 0$.

\begin{theorem}\label{Thm 3.6}
There are constants $c,\,C>0$ such that
\begin{align*}
    \bigg\| T_\varepsilon(t)\begin{pmatrix}
    u_0\\v_0
    \end{pmatrix}- T_0(t)\begin{pmatrix}
    h^0(v_0)\\v_0
    \end{pmatrix}\bigg \|_{X_1\times Y_1}& \leq C\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\|u_0- h^0(v_0)\|_{X_1} (1 +{\textnormal{e}^{\textnormal{e}^{\omega_{\small{B}} t}+ct}})(M_A +  \varepsilon^{-\gamma}t^\gamma)\\
    &\quad+  \varepsilon C\|v_0\|_{Y_1}( \textnormal{e}^{\omega_B t}  +  {\textnormal{e}^{\textnormal{e}^{\omega_{\small{B}} t}+ct}} t^\delta )
\end{align*}
holds for all $(u_0,v_0)\in X_1\times Y_1$, $t\in [0,T]$ and all $\varepsilon>0$.
\end{theorem}
\begin{proof}
In the following we use the classical notion of the solution rather than the semi-flow version, i.e. we have
\begin{align*}
    \begin{pmatrix}
    u^\varepsilon(t)\\v^\varepsilon(t)
    \end{pmatrix}=T_\varepsilon(t)\begin{pmatrix}
    u_0\\v_0
    \end{pmatrix},\quad \begin{pmatrix}
    h^0(v^0)(t)\\v^0(t)
    \end{pmatrix}=T_{0}(t)\begin{pmatrix}
    h^0(v_0)\\v_0
    \end{pmatrix}.
\end{align*}
Moreover, to control the $\varepsilon$-dependent terms we will work with the modified system \eqref{modified fast reaction}.

From the existence of solutions we obtain that the following equations are well-posed
\begin{align*}
    u^\varepsilon &= \textnormal{e}^{\varepsilon^{-1} t\tilde A_\varepsilon} u_0+ \varepsilon^{-1} \int_{0}^t \textnormal{e}^{\varepsilon^{-1}(t-s)\tilde A_\varepsilon} f(u^\varepsilon, v^\varepsilon(s)) \, \textnormal{d} s,\\
   0&=- \textnormal{D}_x f(h^0(v^0),v^0) h^0(v^0) + \tilde f(h^0(v^0),v^0)
    \intertext{and similar for the second part}
    v^\varepsilon(t)&= \textnormal{e}^{tB}v_0+ \int_0^t \textnormal{e}^{(t-s)B} g(u^\varepsilon(s),v^\varepsilon(s))\, \textnormal{d}s,\\
    v^0(t)&= \textnormal{e}^{tB}v_0+ \int_0^t \textnormal{e}^{(t-s)B} g(h(v^0(s)),v^0(s))\, \textnormal{d}s.
\end{align*}

In addition, the well-posedness of the limit system \eqref{full eq limit} implies that
\begin{align*}
    \|v^0(t)\|_{Y_1}\leq M_B \textnormal{e}^{\omega_B t} \|v_0\|_{Y_1}+ C_B L_g \int_0^t \frac{\textnormal{e}^{(t-s)\omega_B}}{(t-s)^{1-\delta}} (1+L_h)\|v^0(s)\|_{Y_1}  \, \textnormal{d} s
\end{align*}
which can be estimated by applying Gronwall's inequality
\begin{align}\label{ineq v0}
     \|v^0(t)\|_{Y_1}\leq M_B \textnormal{e}^{\omega_B t} \textnormal{e}^{C_BL_g(1+L_h)\Gamma(\delta)} \|v_0\|_{Y_1}.
\end{align}
In a similar fashion we obtain that $\partial_t v^0$ satisfies
\begin{align*}
    \partial_t  v^0(t)= \textnormal{e}^{Bt}\big(Bv_0+ g(h^0(v_0),v_0)\big) +\int_0^t \textnormal{e}^{B(t-s)}\partial_s g(h^0(v^0(s)),v^0(s))\, \textnormal{d}s 
\end{align*}
and we estimate
\begin{align*}
    \|\partial_t  v^0\|_Y&\leq M_B \textnormal{e}^{\omega_B t} \big(\|Bv_0\|_Y+ L_g(L_h+1)\|v_0\|_Y\big) +M_B\int_0^t \textnormal{e}^{\omega_B(t-s)}\|D g(h^0(v^0),v^0)\|_{\mathcal{B}(X_1\times Y_1,Y)} \|\partial_s v^0\|_Y \, \textnormal{d}s 
    \end{align*}
   which yields, assuming that the total derivative of $g$ is bounded by $L_g$,
   \begin{align}\label{ineq partial t v0}
    \|\partial_t  v^0\|_Y&\leq C M_B \textnormal{e}^{\omega_B t} \big(\|Bv_0\|_Y+ L_g(L_h+1)\|v_0\|_Y\big).
\end{align}

To estimate the fast component we introduce two approximate systems in $u$, namely $\tilde u^\varepsilon$ and $u^{\varepsilon,0}$, where $\tilde u^\varepsilon$ is the solution of
\begin{align*}
    \partial_t \tilde u^\varepsilon =\frac{1}{\varepsilon} \tilde A_\varepsilon u^\varepsilon +\frac{1}{\varepsilon}\tilde f(\tilde u^\varepsilon, v^0)
\end{align*}
and $u^{\varepsilon,0}$ is the solution of
\begin{align*}
    \partial_t u^{\varepsilon,0}= \frac{1}{\varepsilon} \tilde A_\varepsilon u^{\varepsilon,0}+\frac{1}{\varepsilon} \tilde f(u^{\varepsilon,0},v^0)+\partial_t \big(\tilde{A}_\varepsilon\big)^{-1}\tilde f(h^0(v^0),v^0)
\end{align*}
and where in both cases $v^0$ satisfies the second equation of the system \eqref{full eq limit}.
The expression $\big(\tilde{A}_\varepsilon\big)^{-1}$ is well defined since the resolvent of $A$ is positive and the largest eigenvalue $\lambda_1$ of $\textnormal{D}_x f(h^0(v^0),v^0)$  has a negative real part and satisfies $\textnormal{Re}\lambda_1\leq \lambda_0 <0$ .
The existence of solutions follows from adapting the proofs of Propositions \ref{Prop 3.1} and \ref{Prop 3.2}.

We start with the estimates for the $u$ component, where we apply the triangle inequality to split the estimate into three parts
\begin{align*}
    \|u^\varepsilon(t) -h^0(v^0(t))\|_{X_1}\leq \underbrace{\|u^\varepsilon(t) -\tilde u^\varepsilon(t)\|_{X_1} }_{=\mathcal{ I}}+ \underbrace{\|\tilde u^\varepsilon(t) -u^{\varepsilon,0}(t)\|_{X_1}}_{=\mathcal{II}}+\underbrace{\|u^{\varepsilon,0}(t) -h^0(v^0(t))\|_{X_1}}_{=\mathcal{III}}.
\end{align*}
For $\mathcal{I}$ we estimate  
\begin{align*}
    \|u^\varepsilon(t) -\tilde u^\varepsilon(t)\|_{X_1}&\leq \bigg\| \varepsilon^{-1} \int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s)\tilde A_\varepsilon}( \tilde f(u^\varepsilon,v^\varepsilon)-\tilde f(\tilde u^\varepsilon, v^0))\, \textnormal{d}s\bigg\|_{X_1}\\
    &\leq L_{\tilde f} C_A \int_0^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}}\big(\|u^\varepsilon(s)-\tilde u^{\varepsilon}(s)\|_{X_1}+  \|v^\varepsilon(s)-v^0(s)\|_{Y_1}\big)\, \textnormal{d}s\\
    &\leq \frac{ L_{\tilde f} C_A  \Gamma(\gamma)}{|\varepsilon \omega_A-\lambda_0|^\gamma} \sup_{0\leq s\leq t}\|v^\varepsilon(s)-v^0(s)\|_{Y_1} + L_{\tilde f} C_A \int_0^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \|u^\varepsilon(s)-\tilde u^{\varepsilon}(s)\|_{X_1}  \, \textnormal{d}s
\end{align*}
  which by Gronwall's inequality yields
 \begin{align}\label{est u1}
   \|u^\varepsilon(t) -\tilde u^\varepsilon(t)\|_{X_1}\leq C_AL_{\tilde f} C_1 \frac{\Gamma(\gamma)}{|\varepsilon \omega_A-\lambda_0|^\gamma} \sup_{0\leq s\leq t}\|v^\varepsilon(s)-v^0(s)\|_{Y_1},
\end{align}
   where $
   C_1=\textnormal{e}^{ C_AL_{\tilde f} \frac{\Gamma(\gamma)}{|\varepsilon \omega_A-\lambda_0|^\gamma}}$.

For $\mathcal{II}$ we have
\begin{align*}
     \|\tilde u^{\varepsilon}(t)-u^{\varepsilon,0}(t)\|_{X_1}&\leq \bigg\| \varepsilon^{-1} \int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s)\tilde A_\varepsilon}\big( \tilde f(\tilde u^\varepsilon,v^0)-\tilde f( u^{\varepsilon,0}, v^0) -\varepsilon \partial_s \big(\tilde A_\varepsilon\big)^{-1}\tilde f(h^0(v^0),v^0)\big) \, \textnormal{d}s \bigg\|_{X_1}\\
    &\leq C_A L_{\tilde f}\int_0^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon \omega_A -\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma} }\|u^\varepsilon(s)-u^{\varepsilon,0}(s)\|_{X_1}\, \textnormal{d}s\\
    &\quad+ \varepsilon C_A L_{\tilde f}\int_0^t  \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon \omega_A -\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma} }\|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_1)}\|\partial_t h^0(v^0) \|_{X}\, \textnormal{d}s\\
   
    & \leq C_A L_{\tilde f}\int_0^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon \omega_A -\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma} }\|u^\varepsilon(s)-u^{\varepsilon,0}(s)\|_{X_1}\, \textnormal{d}s\\
    &\quad+  \frac{\varepsilon C_A L_{\tilde f} }{|\varepsilon\omega_A+\lambda_0|^\gamma} \|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_1)}\|\partial_t h^0(v^0) \|_{L^\infty([0,t],X)} 
    \intertext{and again applying Gronwall's inequality we get}
    \|\tilde u^{\varepsilon}(t)-u^{\varepsilon,0}(t)\|_{X_1}&\leq  C_1 \varepsilon C_A L_{\tilde f}\frac{\Gamma^(\gamma) }{|\varepsilon\omega_A+\lambda_0|^\gamma} \|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_1)}\|\partial_t h^0(v^0) \|_{L^\infty([0,t],X)} 
    \end{align*}
To improve this estimate we note that
\begin{align*}
    \|\partial_t h^0(v^0)\|_{L^\infty([0,t],X)}&\leq  \|h^0( v^0)\|_{C^1_b([0,t],X)}\\
    &\leq L_h \|v^0\|_{C^1_b([0,t],Y)}
    \intertext{and by applying inequalities \eqref{ineq v0} and \eqref{ineq partial t v0} we obtain}
     \|\partial_t h^0(v^0)\|_{L^\infty([0,t],X)}&\leq C \textnormal{e}^{\omega_B t}\big( \|v_0\|_{Y_1} +\|Bv_0\|_Y+ L_g(L_h+1)\|v_0\|_Y\big)  \leq C \textnormal{e}^{\omega_B t}  \|v_0\|_{Y_1}.
\end{align*}
Hence, we can conclude that
\begin{align}\label{est u2}
    \|\tilde u^{\varepsilon}(t)-u^{\varepsilon,0}(t)\|_{X_1}&\leq  C_1 \frac{\varepsilon C_A L_{\tilde f} \Gamma(\gamma)}{|\varepsilon\omega_A+\lambda_0|^\gamma} \|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_1)}C \textnormal{e}^{\omega_B t}  \|v_0\|_{Y_1}.
\end{align}
For $\mathcal{III}$ we have that
\begin{align*}
      \|u^{\varepsilon,0}(t)-h^0(v^0(t))\|_{X_1}&= 
      \bigg\|\textnormal{e}^{t \varepsilon^{-1} \tilde A_\varepsilon} u_0+\varepsilon^{-1}\int_0^t \textnormal{e}^{(t-s)  \varepsilon^{-1}\tilde A_\varepsilon}  \tilde f(u^{\varepsilon,0}(s) ,v^0(s))\, \textnormal{d} s\\
      &\qquad +\int_0^t \textnormal{e}^{(t-s)  \varepsilon^{-1}\tilde A_\varepsilon} \partial_s \big(\tilde A_\varepsilon\big)^{-1} \tilde f(h^0(v^0(s)),v^0(s))\, \textnormal{d}s - h^0(v^0(t))\bigg\|_{X_1}.\\
       \intertext{Integration by parts yields}
    &=       \bigg\|\textnormal{e}^{t \varepsilon^{-1} \tilde A_\varepsilon} u_0+\varepsilon^{-1}\int_0^t \textnormal{e}^{(t-s)  \varepsilon^{-1}\tilde A_\varepsilon}  \big( \tilde f(u^{\varepsilon,0}(s) ,v^0(s)) -\tilde f(h^0(v^0(s)),v^0(s)) \big) \, \textnormal{d} s\\
      &\qquad+ \big(\tilde A_\varepsilon\big)^{-1}\tilde f(h^0(v^0(t)),v^0(t))  - h^0(v^0(t)) -  \textnormal{e}^{t  \varepsilon^{-1}\tilde A_\varepsilon}\big(\tilde A_\varepsilon\big)^{-1} \tilde f(h^0(v_0),v_0) \bigg\|_{X_1}.\\
         
   
   
   
  \end{align*}
Recalling that $ 0 =- \textnormal{D}_x f(h^0(v^0),v^0) (h^0(v^0),v^0) + \tilde f(h^0(v^0),v^0)$ we estimate
\begin{align*}
    \|u^{\varepsilon,0}(t)-h^0(v^0(t))\|_{X_1}&\leq C_AL_{\tilde f}\int_0^t \frac{\textnormal{e}^{(t-s) \varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)}} { \varepsilon^\gamma (t-s)^{1-\gamma}}\|(u^{\varepsilon,0}(s)-h^0(v^0(s)))\|_{X_1}\, \textnormal{d}s\\
    &\quad +M_A\textnormal{e}^{\varepsilon^{-1}(\varepsilon \omega_A-\lambda_0) t}\|u_0 -  (\tilde A_\varepsilon )^{-1}\textnormal{D}_x f(h^0(v^0),v^0) h^0(v_0)\|_{X_1}\\
    &\quad +\| \big(\tilde A_\varepsilon\big)^{-1}\textnormal{D}_x f(h^0(v^0),v^0)  h^0(v^0(t)) - h^0(v^0(t))\|_{X_1}.
\end{align*}
Here, we note the additional terms in the estimate when compared with the result in \cite[Thm. 4.13]{hummel2022slow}.
This is due to the fact that $\textnormal{D}_x f(h^0(v^0),v^0)(\tilde A_\varepsilon)^{-1}$ does not equal the identity operator $I$ but a perturbation of the latter.
To be more precise, we have that
\begin{align*}
    \|\big(I- (\tilde A_\varepsilon)^{-1}\big) \textnormal{D}_x f(h^0(v^0),v^0) x\|_{X_1}& \leq \varepsilon \|  \big(\tilde A_\varepsilon\big)^{-1}A  x\|_{X_1}
\end{align*}
i.e. the difference is of order $\mathcal{O}(\varepsilon)$ and we can write 
\begin{align*}
    \textnormal{D}_x f(h^0(v^0),v^0) \big(\tilde A_\varepsilon\big)^{-1} = I -\varepsilon J,
\end{align*}
where the operator $J=  (\tilde A_\varepsilon)^{-1} A: X_1\to X_1$ is bounded. Thus,
\begin{align*}
    \|u^{\varepsilon,0}(t)-h^0(v^0(t))\|_{X_1}&  \leq C_AL_{\tilde f}\int_0^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)}} { \varepsilon^\gamma (t-s)^{1-\gamma}}\|(u^{\varepsilon,0}(s)-h^0(v^0(s)))\|_{X_1}\, \textnormal{d}s\\
      &\quad +  M_A \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\|u_0- h^0(v_0)\|_{X_1} + \varepsilon M_A L_h \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\|J\|_{\mathcal{B}(X_1)} \| v_0\|_{Y_1}\\
      &\quad+\varepsilon L_h\|J\|_{\mathcal{B}(X_1)} \| v^0(t)\|_{Y_1}.
\end{align*}
Using estimate \eqref{ineq v0} and applying Gronwall's inequality yields
\begin{align*}
     \|u^{\varepsilon,0}(t)\!-\!h^0(v^0(t))\|_{X_1}&  \leq  M_A \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\big(\|u_0\!-\! h^0(v_0)\|_{X_1} \!+ \!\varepsilon  L_h \|J\|_{\mathcal{B}(X_1)} \| v_0\|_{Y_1}\big)\! +\varepsilon C L_h\|J\|_{\mathcal{B}(X_1)} \textnormal{e}^{\omega_B t}\|v_0\|_{Y_1}\\
     &\quad +C_AL_{\tilde f} M_A \|u_0- h^0(v_0)\|_{X_1} \textnormal{e}^{\frac{\Gamma(\gamma)}{|\varepsilon \omega_A+ \lambda_0|^\gamma}} \int_0^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)}} { \varepsilon^\gamma (t-s)^{1-\gamma}} \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) s} \,\textnormal{d}s\\
     &\quad + \varepsilon C_AL_{\tilde f} M_A L_h  \|J\|_{\mathcal{B}(X_1)} \| v_0\|_{Y_1}  \textnormal{e}^{\frac{\Gamma(\gamma)}{|\varepsilon \omega_A+ \lambda_0|^\gamma}}  \int_0^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)}} { \varepsilon^\gamma (t-s)^{1-\gamma}} \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) s} \,\textnormal{d}s\\
     &\quad+ \varepsilon C_AL_{\tilde f}C L_h\|J\|_{\mathcal{B}(X_1)} \|v_0\|_{Y_1} \textnormal{e}^{\frac{\Gamma(\gamma)}{|\varepsilon \omega_A+ \lambda_0|^\gamma}}   \int_0^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)}} { \varepsilon^\gamma (t-s)^{1-\gamma}}  \textnormal{e}^{\omega_B s} \,\textnormal{d}s.
\end{align*}
This inequality can be further simplified as follows
\begin{align}\label{est u3}\begin{split}
    \|u^{\varepsilon,0}(t)-h^0(v^0(t))\|_{X_1}&  \leq  M_A \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\|u_0- h^0(v_0)\|_{X_1} +  \varepsilon C \textnormal{e}^{\omega_B t} \|v_0\|_{Y_1}\\
    &\quad+ C \varepsilon^{-\gamma}\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t} t^\gamma \|u_0- h^0(v_0)\|_{X_1},\end{split}
\end{align}
where we observe that $\lim_{\varepsilon\to 0} \varepsilon^{-\gamma}\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}=0 $.

Combining the three estimates for the $u$ component \eqref{est u1}-\eqref{est u3} yields
\begin{align}\label{est u all}\begin{split}
    \|u^\varepsilon(t)-h^0(v^0(t))\|_{X_1}&\leq C \sup_{0\leq s\leq t}\|v^\varepsilon(s)-v^0(s)\|_{Y_1}\\
    &\quad  (M_A + C \varepsilon^{-\gamma}t^\gamma)\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\|u_0- h^0(v_0)\|_{X_1} +  \varepsilon C \textnormal{e}^{\omega_B t} \|v_0\|_{Y_1},\end{split}
\end{align}
where the constants $C$ are independent of $\varepsilon$ and only depend on the nonlinearities $f$ and $g$ and on the operators $A$ and $B$.

Now, for the equation in $v$ we estimate
\begin{align*}
    \|v^\varepsilon(t)-v^0(t)\|_{Y_1}\leq L_g C_B \int_0^t \frac{\textnormal{e}^{(t-s)\omega_B }}{(t-s)^{1-\delta}} \big( \|u^\varepsilon(s) -h(v^0(s))\|_{X_1} +\|v^\varepsilon-v^0\|_{Y_1}\big)\, \textnormal{d} s.
\end{align*}
Using the previous estimate \eqref{est u all} for $u$ yields
\begin{align*}
     \|v^\varepsilon(t)-v^0(t)\|_{Y_1}&\leq L_g C_B \int_0^t \frac{\textnormal{e}^{(t-s)\omega_B }}{(t-s)^{1-\delta}} \bigg( \|v^\varepsilon(s)-v^0(s)\|_{Y_1} + \sup_{0\leq \tau \leq s}\|v^\varepsilon(\tau)-v^0(\tau)\|_{Y_1}\\
    &\quad+ (M_A + C \varepsilon^{-\gamma}s^\gamma)\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) s}\|u_0- h^0(v_0)\|_{X_1} +  \varepsilon C \textnormal{e}^{\omega_B s} \|v_0\|_{Y_1} \bigg)\, \textnormal{d} s.
    \end{align*}
    Since the right-hand side is increasing in $t$ we have
    \begin{align*}
     \sup_{0\leq s\leq t}\|v^\varepsilon(s)-v^0(s)\|_{Y_1}&\leq L_g C_B \int_0^t \frac{\textnormal{e}^{(t-s)\omega_B }}{(t-s)^{1-\delta}} \bigg(  C\sup_{0\leq\tau\leq s} \|v^\varepsilon(\tau)-v^0(\tau)\|_{Y_1} \\
     &\quad +(M_A + C \varepsilon^{-\gamma}s^\gamma)\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) s}\|u_0- h^0(v_0)\|_{X_1} +  \varepsilon C \textnormal{e}^{\omega_B s} \|v_0\|_{Y_1} \bigg)\, \textnormal{d} s.
    \intertext{Applying Gronwall's inequality yields}
     \sup_{0\leq s\leq t}\|v^\varepsilon(s)-v^0(s)\|_{Y_1}&\leq C\textnormal{e}^{\textnormal{e}^{\omega_B t}} L_g C_B \int_0^t \frac{\textnormal{e}^{(t-s)\omega_B }}{(t-s)^{1-\delta}} \bigg( (M_A + C \varepsilon^{-\gamma}s^\gamma)\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) s}\|u_0- h^0(v_0)\|_{X_1}\\
     &\qquad \qquad+  \varepsilon C \textnormal{e}^{\omega_B s} \|v_0\|_{Y_1}\bigg)\, \textnormal{d} s. 
\end{align*}
This estimate can be further improved to
\begin{align}\label{est v all}\begin{split}
     \sup_{0\leq s\leq t}\|v^\varepsilon(s)-v^0(s)\|_{Y_1}\leq& C \varepsilon {\textnormal{e}^{\textnormal{e}^{\omega_{\small{B}} t}+ct}} t^\delta  \|v_0\|_{Y_1}\\
     &+C{\textnormal{e}^{\textnormal{e}^{\omega_{\small{B}} t}+ct}}(M_A + C \varepsilon^{-\gamma}t^\gamma)\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\|u_0- h^0(v_0)\|_{X_1}. \end{split}
\end{align}
Hence, combining the estimate for the $u$ and $v$ component yields
\begin{align*}
    \|u^\varepsilon(t)-h^0(v^0(t))\|_{X_1}+\|v^\varepsilon(t)-v^0(t)\|_{Y_1} &\leq C\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\|u_0- h^0(v_0)\|_{X_1} (1 +{\textnormal{e}^{\textnormal{e}^{\omega_{\small{B}} t}+ct}})(M_A +  \varepsilon^{-\gamma}t^\gamma)\\
    &\quad+  \varepsilon C\|v_0\|_{Y_1}( \textnormal{e}^{\omega_B t}  +  {\textnormal{e}^{\textnormal{e}^{\omega_{\small{B}} t}+ct}} t^\delta ),
\end{align*}
which is the desired inequality.
\end{proof}

\begin{remark}
    Let us put this result into context. 
    For fixed $t\in [0,T]$ the difference between the solution of the fast-reaction and the limit system converges to $0$ as $\varepsilon\to 0$.
    However, when the final time $T$, which depends on the restrictions of the implicit function theorem and the on the condition $\lambda_0<0$, becomes arbitrary large the parameter $\varepsilon$ has to decrease accordingly for the convergence to still hold.
\end{remark}

\section{Slow Manifolds}\label{section 4}

In this section we generalize the Fenichel theorem to infinite dimensional systems.
Following the structure of the Fenichel theorem in finite dimension we establish the existence of a slow manifold, show that the distance between this slow manifold and the critical manifold is small in a suitable sense and that the slow flow on the critical manifold is a good approximation of the flow on the slow manifold.

One important aspect of the Fenichel theory is the notion of invariant manifold.
A manifold $M$ is invariant with respect to the semigroup $(T(t))_{t\geq 0}$ if for all $m\in M$ and all $t\geq 0$ it holds that $T(t) m\in M$.

\begin{theorem}[General Fenichel Theorem for infinite-dimensional fast-reaction systems]\label{general fenichel thm}
Suppose $S_{0,\zeta}$ is a submanifold of the critical manifold $S_0$ and let $\tilde A_\varepsilon$ be the dissipative operator of the previous section introduced in system \eqref{modified fast reaction}. 
In addition, let the technical assumptions of Sections \ref{sect 3.1} and \ref{sect 4.1} hold. 
Then, for $\varepsilon,\,\zeta>0$ small enough the following hold
\begin{itemize}
    \item[(GF1)] There exists a locally invariant manifold $S_\varepsilon$ for the system \eqref{full eq splitting} called the slow manifold. Local invariance means that trajectories can enter or leave $S_\varepsilon$ only through its boundaries;
    \item[(GF2)] The slow manifold $S_\varepsilon$ has a distance of $\mathcal{O}(\varepsilon)$ to $S_0$ in the $Y_1$-norm, i.e. the norm generated by the operator $B$;
    \item[(GF3)] The slow manifold $S_\varepsilon$ is $C^1$-regular in the topology induced by the operators $A$ and $B$;
    \item[(GF4)] The flow on $S_\varepsilon$ converges to the semi-flow on $S_{0,\zeta}$, where again the convergence is with respect to the $Y_1$-norm.
\end{itemize}
\end{theorem}
The details as well as the proofs of each of the statements follow in  the subsequent subsections.

\begin{remark}
    We want to remark on the difference between the results presented in this section and the result by Bates et al. in  \cite{bates1998existence}.
    One key assumption in the later work is that 
    \begin{align*}
        \| T_{\varepsilon}(t)-T_0(t)\|_{C^1(N, X\times Y)} \to 0 \quad\text{as } \varepsilon\to 0,
    \end{align*}
    where $N$ is a neighborhood of $S_0$.
    When we compare this assumption with Theorem \ref{Thm 3.6} we observe that we only require
    \begin{align*}
        \| T_{\varepsilon}(t)-T_0(t)\|_{C(N, X_1\times Y_1)} \to 0 \quad\text{as } \varepsilon \to 0
    \end{align*}
    for $t\in [0,T]$ to hold.
    Hence, our result applies to a larger class of fast-reaction systems.
    One way to satisfy the assumption in \cite{bates1998existence} is to assume that the operator $B$ generates a $C^0$-group.
    We, however, choose a different approach in extending the Fenichel theory to infinite dimensions.
    Lastly, in the case of the equation for the slow component is an ODE or if $B$ generates a $C^0$-group we observe that the results of Theorem \ref{general fenichel thm} coincide with the ones in \cite{bates1998existence}. 
\end{remark}

\subsection{Assumptions}\label{sect 4.1}

To show the results in this section we need additional assumptions on the operator $B$ and the Banach space $Y$.
The reason for this is that we want to express all the terms that arise in the upcoming estimates in terms of the slow variable $v$.
However, when we rewrite the fast reaction system in the slow time scale by $\tau =\varepsilon t$
\begin{align*}\begin{split}
    \partial_\tau u^\varepsilon &= \tilde A_\varepsilon u^\varepsilon + \tilde f(u^\varepsilon,v^\varepsilon),\\
    \partial_\tau v^\varepsilon&= \varepsilon \big( Bv^\varepsilon + g(u^\varepsilon,v^\varepsilon) \big)\end{split}
\end{align*}
we observe that in the limit $\varepsilon\to 0$ the expression $\varepsilon B v^\varepsilon$ might not be well-defined as $B$ is an unbounded operator. For more details and an example we refer to \cite[Section 3.3]{hummel2022slow}.

Therefore, we introduce a splitting in the slow variable space
\begin{align*}
    Y= Y^\zeta_F\otimes Y^\zeta_S
\end{align*}
into a fast part $Y^\zeta_F$ and a slow part $Y^\zeta_S$, where  $\zeta>0$ is a small parameter, such that 
\begin{itemize}
    \item The spaces $Y^\zeta_F$ and $Y^\zeta_S$ are closed in $Y$ and the projections $\text{pr}_{Y^\zeta_F}$ and $\text{pr}_{Y^\zeta_S}$ commute with $B$ on $Y_1$.
    \item The spaces $Y^\zeta_F\cap Y_1$ and $Y^\zeta_S\cap Y_1$ are closed subspaces of $Y_1$ and are endowed with the norm $\|\cdot\|_{Y_1}$.
    \item The realization of $B$ in $Y^\zeta_F$, i.e. 
    \begin{align*}
        B_{Y^\zeta_F}:Y^\zeta_F&\supset D(Y^\zeta_F)\to Y^\zeta_F,\quad v\mapsto Bv
        \intertext{with}
        D(Y^\zeta_F)&:= \{v_0\in Y^\zeta_F\,:\, Bv_0\in Y^\zeta_F\}
    \end{align*}
    has $0$ in its resolvent set.
    \item The realization of $B$ in $Y^\zeta_S$ generates a $C_0$-group $(\textnormal{e}^{tB_{Y^\zeta_S}})_{t\in \mathbb{R}}\subset \mathcal{B}((Y^\zeta_S,\|\cdot\|_{Y}) )$ which for $t\geq 0$ satisfies $\textnormal{e}^{tB_{Y^\zeta_S}}=\textnormal{e}^{tB}$ on $Y^\zeta_S$.
    \item The fast subspace $Y^\zeta_F$ contains the part of $Y_1$ that decay under the semigroup $(\textnormal{e}^{tB})_{t\geq 0}$ almost as fast as functions in the fast variable space $X_1$ under the semi-group $(\textnormal{e}^{\zeta^{-1}t\tilde A_\varepsilon})_{t\geq 0}$ generated by the modified operator $\tilde A_\varepsilon$.
    The space $Y^\zeta_S$ on the other hand contains the parts of $Y_1$ that do not decay or which decay only slowly under the semigroup $(\textnormal{e}^{tB})_{t\geq 0}$ compared to $X_1$ under $(\textnormal{e}^{\zeta^{-1}t \tilde A_\varepsilon})_{t\geq 0}$. 
    Hence, there are constants $C_B,\,M_B>0$ such that for $\zeta>0$ small enough there are constants $0\leq N_F^\zeta< N_S^\zeta$ such that for all $t\geq 0$ and $y_F\in Y^\zeta_F,\, y_S\in Y^\zeta_S$ we have 
    \begin{align*}
        \|\textnormal{e}^{tB}y_F\|_{Y_1}&\leq C_B t^{\delta -1} \textnormal{e}^{(N_F^\zeta +\zeta^{-1}(\varepsilon\omega_A+\lambda_0))t}\|y_F\|_{Y_\delta},\\
        \|\textnormal{e}^{-tB}y_S\|_{Y_1}&\leq M_B \textnormal{e}^{-(N_S\zeta +\zeta^{-1}(\varepsilon\omega_A+\lambda_0))t}\|y_S\|_{Y_1}.
    \end{align*}
    \item We assume that the the parameter $\zeta>0$ is chosen in such a way that $\varepsilon \zeta^{-1}< 1$.
\end{itemize}
\begin{remark}
The quantity $N_S^\zeta-N^\zeta_F$  can be considered the spectral gap of the operator $B$ and it relates to the parameter $\zeta$ through $N_S^\zeta-N^\zeta_F\approx C \zeta^{-\alpha}$ for some $\alpha>0$ as $\zeta\to 0$, (see Subsection \ref{subsection example} for an example).

However, this spectral gap condition may be satisfied in low dimensions only.
Recalling that the Weyl asymptotic for the eigenvalues of the Laplacian reads $\lambda_m \sim C m^{2/d}$ the above spectral gap condition (depending on the domain $\Omega\subset \mathbb{R}^d$) may only hold in 1 dimension.
Methods to overcome this problem are discussed in \cite{kostianko2021kwak}.
\end{remark}

With this splitting we can rewrite the fast reaction system in the following way
\begin{align}\label{full eq splitting}
    \begin{split}
             \partial_t u^\varepsilon(t)&=A u^\varepsilon(t)+ \frac{1}{\varepsilon} f(u^\varepsilon(t),v_F^\varepsilon(t),v_S^\varepsilon(t)),\\
           \partial_t v^\varepsilon(t)&=B v^\varepsilon(t)+ \text{pr}_{Y_F^\zeta} g(u^\varepsilon(t),v_F^\varepsilon(t),v_S^\varepsilon(t)),\\
            \partial_t v^\varepsilon(t)&=B v^\varepsilon(t)+ \text{pr}_{Y_S^\zeta} g(u^\varepsilon(t),v_F^\varepsilon(t),v_S^\varepsilon(t)),\\
            u^\varepsilon(0)&=u_0,\quad     v^\varepsilon_F(0)=\text{pr}_{Y_F^\zeta} v_0,\quad ~\, v^\varepsilon_S(0)=\text{pr}_{Y_S^\zeta} v_0.
    \end{split}
\end{align}
   

\subsection{Existence of slow manifolds}
We construct a family of slow manifolds $S_{\varepsilon,\zeta}$ as 
\begin{align*}
    S_{\varepsilon,\zeta}:=\{(h^{\varepsilon,\zeta}(v_0),v_0):\, v_0\in Y_1\cap Y_S^\zeta\},
\end{align*}
where the function $h^{\varepsilon,\zeta}$ maps
\begin{align*}
    h^{\varepsilon,\xi}: (Y_S^\zeta\cap Y_1)\to  X_1
    \times(Y_F^\zeta\cap Y_1).
\end{align*}
The next step is to construct the functions $h^{\varepsilon,\zeta}$ using a Lyapunov-Perron method (cf. \cite{chow1991smooth,henry2006geometric}).
To this end, we introduce the operator
\begin{align*}
    \mathcal{L}_{v_0,\varepsilon,\zeta}:C_\eta&\to C_\eta,\\
    \begin{pmatrix}
    u\\v_F\\v_S
    \end{pmatrix}&\mapsto \left[ t\mapsto \begin{pmatrix}
    \varepsilon^{-1}\int_{-\infty}^t \textnormal{e}^{(t-s)A}f(u(s),v_F(s),v_S(s))\, \textnormal{d}s\\ \int_{-\infty}^t \textnormal{e}^{(t-s)B}\text{pr}_{Y^\zeta_F}g(u(s),v_F(s),v_S(s))\, \textnormal{d}s\\ \textnormal{e}^{tB}v_0+\int_{0}^t \textnormal{e}^{(t-s)B}\text{pr}_{X^\zeta_S}g(u(s),v_F(s),v_S(s))\, \textnormal{d}s
    \end{pmatrix}\right],
\end{align*}
 where $v_0\in Y_1\cap Y^\zeta_S$ and 
 \[C_\eta:= C((-\infty,0],\textnormal{e}^{\eta t};Y_S^\zeta\cap Y_1)\to  X_1
    \times(Y_F^\zeta\cap Y_1)\times (Y_1\cap Y^\zeta_S)) \]
    is the space of $(u,v_F,v_S)\in C((-\infty,0];Y_S^\zeta\cap Y_1)\to X_1
    \times(Y_F^\zeta\cap Y_1)\times ( Y^\zeta_S \cap Y_1) $ such that
    \begin{align*}
        \|(u,v_F,v_S)\|_{C_\eta}:=\sup_{t\leq 0} \textnormal{e}^{-\eta t}(\|u\|_{X_1}+\|v_F\|_{Y_1}+\|v_S\|_{Y_1})<\infty.
    \end{align*}

\begin{prop} \label{Prop 4.2}
    Let $v_0\in Y_1\cap Y^\zeta_S$. Then the operator $\mathcal{L}_{v_0,\varepsilon,\zeta}$ has a unique fixed point in $C_\eta$.
\end{prop}
\begin{remark}
    The function $h^{\varepsilon,\zeta}$ used to construct the family of slow manifolds is given by the first two components of the fixed point of $\mathcal{L}_{v_0,\varepsilon,\zeta}$ evaluated at $t=0$,
    i.e. $h^{\varepsilon,\zeta}(v_0)= (u^{v_0}(0),v^{v_0}_F(0))$.
\end{remark}

\begin{proof}
As a first step we rewrite the system into the modified fast-slow system while maintaining the splitting in the slow variable $v$, where we replace the operator $A$ by $\frac{1}{\varepsilon}\tilde A_\varepsilon$ and the nonlinear function $f$ by $\tilde f$.
Next, we apply the existence result shown in \cite[Prop. 5.1]{hummel2022slow}, where we set
\begin{align*}
    \eta= \zeta^{-1}(\varepsilon\omega_A+\lambda_0) +\frac{N_S^\zeta+N_F^\zeta}{2}.
\end{align*}
\end{proof}

\begin{prop}
Let $(u^{v_0},v_F^{v_0},v_S{v_0})$ be the unique fixed point of $\mathcal{L}_{v_0,\varepsilon,\zeta}$ evaluated at $t=0$ and assume that 
\begin{align*}
  L:=  \frac{2^\gamma L_{\tilde f} C_A \Gamma(\gamma)}{(2(\varepsilon\zeta^{-1} -1)(\varepsilon\omega_A+\lambda_0) + \varepsilon (N_S^\zeta+N_F^\zeta) +2\lambda_0 )^\gamma} + \frac{2^\delta L_gC_B \Gamma(\delta)}{(N_S^\zeta-N_F^\zeta)^\delta}+\frac{2L_gM_B \Gamma(\delta)}{N_S^\zeta-N_F^\zeta}< 1.
\end{align*}
Then, the mapping
\begin{align*}
    h^{\varepsilon,\zeta}: (Y^\zeta_S\cap Y_1)\to X_1\times (Y^\zeta_F\cap Y_1),\, v_0\mapsto (u^{v_0}(0), v_F^{v_0}(0))
\end{align*}
is Lipschitz continuous with Lipschitz constant $L_\zeta= M_B/(1-L)$.
\end{prop}
\begin{proof}
This follows from the estimates in Proposition \ref{Prop 4.2}, cf. \cite[Prop 5.2]{hummel2022slow}.
\end{proof}


\subsection{Distance to critical manifold}
\begin{prop}\label{Prop 4.6}
There exists a constant $C>0$ depending on the operator $A$ and the nonlinear functions $f$ and $g$ such that for all $\varepsilon,\,\zeta>0$ small enough satisfying $\varepsilon\zeta^{-1}<1$ and all initial data $v_0 \in Y_S^\zeta$ it holds that 
\begin{align}
    \bigg\|\begin{pmatrix}
   h^{\varepsilon,\zeta}_{X_1}(v_0)- h^0(v_0)\\ h^{\varepsilon,\zeta}_{Y_F}(v_0)
    \end{pmatrix}\bigg\|_{X_1\times Y_1}&\leq C\bigg( \varepsilon^\gamma +\frac{1}{N_S^\zeta-N_F^\zeta}\bigg) \|v_0\|_{Y_1}.
\end{align}
\end{prop}
\begin{proof}
Let $(\bar u, \bar v_F, \bar v_S)\in C_\eta$ be the unique fixed point of $\mathcal{L}_{v_0,\varepsilon,\zeta}$.
That is we have 
\[(\bar u, \bar v_F, \bar v_S)=(h^{\varepsilon,\zeta}_{X_1}(\bar v_S),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S)), \bar v_S).\]
Since $(\bar u, \bar v_F, \bar v_S)$ solves the system \eqref{full eq splitting} on $(-\infty,0]$ we have that $\bar v_S\in C^1((-\infty,0],\textnormal{e}^{\eta t};Y)$.
We estimate that
\begin{align*}
  \|h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(t))\|_{Y_1}& = \bigg\|\int_{-\infty}^t \textnormal{e}^{(t-s)B}\text{pr}_{Y_F^\zeta} g(h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s)),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(s)),\bar v_S(s)) \, \textnormal{d} s\bigg\|_{Y_1}\\
  &\leq L_gC_B \textnormal{e}^{\eta t} \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s)),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(s)),\bar v_S(s)\|_{C_\eta}\int_{-\infty}^t \frac{\textnormal{e}^{(t-s)(\zeta^{-1}\omega_A +N_F^\zeta -\eta)}}{(t-s)^{1-\delta}}\, \textnormal{d}s\\
  &\leq \frac{L_\zeta L_g C_B \Gamma(\delta)\textnormal{e}^{\eta t}}{ (\eta -\zeta^{-1}(\varepsilon\omega_A+\lambda_0)- N_F^\zeta)^\delta}\|v_0\|_{Y_1}\\
  &\leq C \frac{1}{N_S^\zeta-N_F^\zeta} \|v_0\|_{Y_1} .
\end{align*}
In addition, for $t_0\leq t\leq 0$ we compute
\begin{align*}
    h^{\varepsilon,\zeta}_{X_1}(\bar v_S(t))- h^0(\bar v_S(t))&=  \varepsilon^{-1}\int_{-\infty}^t \textnormal{e}^{(t-s)\varepsilon^{-1} \tilde A_\varepsilon}  \tilde f(h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s)),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(s)),\bar v_S(s)) \, \textnormal{d} s - h^0(\bar v_S(t)) \\
    &=  \varepsilon^{-1}\int_{-\infty}^t \textnormal{e}^{(t-s)\varepsilon^{-1} \tilde A_\varepsilon}  \tilde f(h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s)),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(s)),\bar v_S(s)) \, \textnormal{d} s\\
    &\quad  +\big(\textnormal{D}_x f(h^0(v^0),v^0)\big)^{-1} \tilde f( h^0(\bar v_S),0,\bar v_S) -\big( \tilde A_\varepsilon)^{-1} \tilde f( h^0(\bar v_S),0,\bar v_S)\\
    &\quad+ \big( \tilde A_\varepsilon)^{-1} \tilde f( h^0(\bar v_S),0,\bar v_S)\\
    \intertext{Using integration by parts yields}
    &= \varepsilon^{-1}\int_{-\infty}^{t_0} \textnormal{e}^{(t-s)\varepsilon^{-1} \tilde A_\varepsilon}  \tilde f(h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s)),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(s)),\bar v_S(s)) \, \textnormal{d} s\\
    &\quad +\int_{t_0}^t  \textnormal{e}^{(t-s)\varepsilon^{-1} \tilde A_\varepsilon}\big(\tilde A_\varepsilon\big)^{-1}\partial_s \tilde f( h^0(\bar v_S(s)),0,\bar v_S(s))\, \textnormal{d}s  \\
    &\quad + \varepsilon^{-1}\int_{t_0}^{t} \textnormal{e}^{(t-s)\varepsilon^{-1} \tilde A_\varepsilon} \big( \tilde f(h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s)),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(s)),\bar v_S(s))- \tilde f( h^0(\bar v_S(s)),0,\bar v_S(s))\big)\, \textnormal{d} s\\
    &\quad +\textnormal{e}^{(t-t_0)\varepsilon^{-1} \tilde A_\varepsilon}\big(\tilde A_\varepsilon\big)^{-1} \tilde f( h^0(\bar v_S(t_0)),0,\bar v_S(t_0))\\
    &\quad+\big((\textnormal{D}_x f(h^0(v^0),v^0))^{-1}- (\tilde A_\varepsilon)^{-1}\big) \tilde f( h^0(\bar v_S),0,\bar v_S). 
\end{align*}
Again we observe the additional terms in the last line due to the modified operator $\tilde A_\varepsilon$ when comparing this proof with the one in \cite{hummel2022slow}.
Hence, we estimate the above expression as follows
\begin{align*}
   \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S)&- h^0(\bar v_S)\|_{X_1} \leq \\
    & \leq C_A L_{\tilde f} \int_{-\infty}^{t_0} \frac{\textnormal{e}^{(t-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s)),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(s)),\bar v_S(s))\|_{X_1\times Y_1\times Y_1} \, \textnormal{d} s \\
    &\quad + \varepsilon^\gamma \|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_\gamma)}\int_{t_0}^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \, \textnormal{d} s \|\tilde f (h^0(\bar v_S),\bar v_S)\|_{L^\infty((-\infty,0];X)}\\
    &\quad+ C_A L_{\tilde f} \int_{t_0}^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s))-h^0(\bar v_S(s)),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S(s))\|_{X_1\times Y_1}\, \textnormal{d}s\\
    &\quad+ M_A L_{\tilde f} \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)(t-t_0)}  \|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X_\gamma,X_1)} \| h^0(\bar v_S(t_0)),\bar v_S(t_0)\|_{X_1\times Y_1}\\
    &\quad +L_{\tilde f}\varepsilon \| \big(\textnormal{D}_x f(h^0(v^0),v^0)\big)^{-1}J\|_{\mathcal{B}(X_\gamma,X_1)}\| h^0(\bar v_S(t)),\bar v_S(t)\|_{X_1\times Y_1}\\
    & \leq  C_A L_{\tilde f}\textnormal{e}^{(t-t_0)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)+\eta t_0}  \int_{-\infty}^{t_0} \frac{\textnormal{e}^{(t_0-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0-\varepsilon\eta )}}{\varepsilon^\gamma (t-s)^{1-\gamma}}\, \textnormal{d} s  \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S),h^{\varepsilon,\zeta}_{Y_F}(\bar v_S),\bar v_S)\|_{C_\eta}\\
    &\quad + \varepsilon^\gamma L_{\tilde f}\|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_\gamma)}\int_{t_0}^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \, \textnormal{d} s \times\\
    &\qquad\times (1+L_h)\tilde C \textnormal{e}^{\omega_B t} \big(\|Bv_0\|_Y+ L_g(L_h+1)\|v_0\|_Y + \|v_0\|_{Y_1}\big)\\
    &\quad+ C_A L_{\tilde f} \int_{t_0}^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s))-h^0(\bar v_S(s))\|_{X_1}\, \textnormal{d}s\\
    &\quad  + C_A L_{\tilde f} \frac{L_\zeta L_g C_B \Gamma(\delta)}{ (\eta -\zeta^{-1}( \varepsilon\omega_A+\lambda_0)- N_F^\zeta)^\delta}\int_{t_0}^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \textnormal{e}^{\eta s} \, \textnormal{d}s \|v_0\|_{Y_1}\\
    &\quad +M_A L_{\tilde f} \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)(t-t_0)+\eta t_0}  \|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X_\gamma,X_1)} \| h^0(\bar v_S),0,\bar v_S\|_{C_\eta}\\
    &\quad +L_{\tilde f} \lambda_0^{-1}\varepsilon \|J\|_{\mathcal{B}(X_\gamma,X_1)} (L_h+1)\| \bar v_S(t)\|_{ Y_1}.
    \intertext{Using the Lipschitz continuity of $h^{\varepsilon,\zeta}$ yields}
    &\leq C_A L_{\tilde f} \frac{\textnormal{e}^{t_0\varepsilon^{-1} (\varepsilon\eta -\varepsilon\omega_A+\lambda_0)} \textnormal{e}^{t\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0) } }{-\omega_A +\varepsilon^{-1}\lambda_0 +\eta} L_\zeta\|v_0\|_{Y_1}\\
    &\quad + \varepsilon^\gamma L_{\tilde f}\|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_\gamma)}\frac{\Gamma(\gamma)}{|\varepsilon\omega_A -\lambda_0|^{\gamma}}(1+L_h)\tilde C \textnormal{e}^{\omega_B t} \big(\|Bv_0\|_Y+ L_g(L_h+1)\|v_0\|_Y + \|v_0\|_{Y_1}\big)\\
    &\quad  + C_A L_{\tilde f} \frac{L_\zeta L_g C_B \Gamma(\delta)}{ (\eta -\zeta^{-1}( \varepsilon\omega_A+\lambda_0)- N_F^\zeta)^\delta} \frac{\Gamma(\gamma)}{|\varepsilon\omega_A -\lambda_0|^{\gamma}} \textnormal{e}^{\eta t}\|v_0\|_{Y_1}\\
    &\quad +M_A L_{\tilde f} \textnormal{e}^{t_0\varepsilon^{-1}(\varepsilon\eta -\varepsilon\omega_A+\lambda_0)}  \textnormal{e}^{t \varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)} \|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X_\gamma,X_1)}L_\zeta \| v_0\|_{Y_1}\\
    &\quad +L_{\tilde f} \lambda_0^{-1}\varepsilon \|J\|_{\mathcal{B}(X_\gamma,X_1)} (L_h+1)M_B \textnormal{e}^{\omega_B t} \textnormal{e}^{C_BL_g(1+L_h)\Gamma(\delta)} \|v_0\|_{Y_1}\\
    &\quad+ C_A L_{\tilde f} \int_{t_0}^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s))-h^0(\bar v_S(s))\|_{X_1}\, \textnormal{d}s.
\end{align*}
We observe that the right-hand side of the inequality is of the form
\begin{align*}
     \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S)&- h^0(\bar v_S)\|_{X_1} \leq R(t) + C_A L_{\tilde f} \int_{t_0}^t \frac{\textnormal{e}^{(t-s)\varepsilon^{-1} (\varepsilon\omega_A+\lambda_0)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(s))-h^0(\bar v_S(s))\|_{X_1}\, \textnormal{d}s,
\end{align*}
  where $R(t)$ is non-decreasing in time and thus
 \begin{align*}
    C_1^{-1}  \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(t))&- h^0(\bar v_S(t))\|_{X_1}  \leq C_A L_{\tilde f} \frac{\textnormal{e}^{t_0\varepsilon^{-1} (\varepsilon\eta -\varepsilon\omega_A+\lambda_0)} }{-\omega_A +\varepsilon^{-1}\lambda_0 +\eta} L_\zeta\|v_0\|_{Y_1}\\
    &\quad + \varepsilon L_{\tilde f}\|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_\gamma)}\frac{\Gamma(\gamma)}{|\varepsilon\omega_A -\lambda_0|^{\gamma}}(1+L_h)\tilde C \big(\|Bv_0\|_Y+ L_g(L_h+1)\|v_0\|_Y + \|v_0\|_{Y_1}\big)\\
    &\quad  + C_A L_{\tilde f} \frac{L_\zeta L_g C_B \Gamma(\delta)}{ (\eta -\zeta^{-1}( \varepsilon\omega_A+\lambda_0)- N_F^\zeta)^\delta} \frac{\Gamma(\gamma)}{|\varepsilon\omega_A -\lambda_0|^{\gamma}} \|v_0\|_{Y_1}\\
    &\quad +M_A L_{\tilde f} \textnormal{e}^{t_0\varepsilon^{-1}(\varepsilon\eta -\varepsilon\omega_A+\lambda_0)}   \|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X_\gamma,X_1)}L_\zeta \| v_0\|_{Y_1}\\
    &\quad +L_{\tilde f} \lambda_0^{-1}\varepsilon \|J\|_{\mathcal{B}(X_\gamma,X_1)} (L_h+1)M_B \textnormal{e}^{C_BL_g(1+L_h)\Gamma(\delta)} \|v_0\|_{Y_1}.
 \end{align*}
Using the assumption that $\varepsilon \zeta^{-1}<1$ we have that $\varepsilon\eta -\varepsilon\omega_A+\lambda_0>0$. Hence, we can let $t_0\to -\infty$ and obtain
\begin{align*}
    C_1^{-1}  \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(t))&- h^0(\bar v_S(t))\|_{X_1}  \leq \\
    & \leq  \varepsilon^\gamma L_{\tilde f}\|\big(\tilde A_\varepsilon\big)^{-1}\|_{\mathcal{B}(X,X_\gamma)}\frac{\Gamma(\gamma)}{|\varepsilon\omega_A -\lambda_0|^{\gamma}}(1+L_h)\tilde C \big(\|Bv_0\|_Y+ L_g(L_h+1)\|v_0\|_Y + \|v_0\|_{Y_1}\big)\\
    &\quad  + C_A L_{\tilde f} \frac{L_\zeta L_g C_B \Gamma(\delta)}{ (\eta -\zeta^{-1}( \varepsilon\omega_A+\lambda_0)- N_F^\zeta)^\delta} \frac{\Gamma(\gamma)}{|\varepsilon\omega_A -\lambda_0|^{\gamma}} \|v_0\|_{Y_1}\\
    &\quad +L_{\tilde f} \lambda_0^{-1}\varepsilon \|J\|_{\mathcal{B}(X_\gamma,X_1)} (L_h+1)M_B \textnormal{e}^{C_BL_g(1+L_h)\Gamma(\delta)} \|v_0\|_{Y_1}.
\end{align*}
 Focusing on the terms in $\varepsilon$ and using $\eta= \zeta^{-1}(\varepsilon\omega_A+\lambda_0) +\frac{ N_F^\zeta+ N_S^\zeta}{2}$ yields the desired estimate
 \begin{align*}
      \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(t))- h^0(\bar v_S(t))\|_{X_1}  &\leq C \varepsilon^\gamma \big(\|Bv_0\|_Y+ L_g(L_h+1)\|v_0\|_Y + \|v_0\|_{Y_1}\big)+ C \frac{1}{N_S^\zeta-N_F^\zeta} \|v_0\|_{Y_1}\\
      &\leq C\bigg( \varepsilon^\gamma +\frac{1}{N_S^\zeta-N_F^\zeta}\bigg) \|v_0\|_{Y_1}.
 \end{align*}
 
\end{proof}

\begin{remark}
    If we have additional knowledge on the convergence of $N_S^\zeta-N_F^\zeta$ we can improve the above result.
    Assuming that we have the  asymptotic $N_S^\zeta-N_F^\zeta= C \zeta^{-\alpha}$, for some $\alpha>0$ then we obtain for $\varepsilon>0$ small enough
    \begin{align*}
        \|h^{\varepsilon,\zeta}_{X_1}(\bar v_S(t))- h^0(\bar v_S(t))\|_{X_1} &\leq C\bigg( \varepsilon^\gamma +\frac{1}{N_S^\zeta-N_F^\zeta}\bigg) \|v_0\|_{Y_1}\\
        &\leq C\bigg( \varepsilon^\gamma +\zeta^\alpha \bigg) \|v_0\|_{Y_1},
    \end{align*}
    where we used the condition $\varepsilon\zeta^{-1}<1$;
    for more background on the importance of double limits in singular perturbations we also refer to~\cite{KUEHNdouble}.
    \end{remark}

\subsection{Attraction of trajectories and approximation by slow flow}

One technical result is to show the differentiability of the slow manifold.
To this end we assume that the nonlinearities $f$ and $g$ are continuously differentiable such that
\begin{align*}
    \|\textnormal{D} f(x,y)\|_{\mathcal{B}(X_1\times Y_1,X_\gamma)}\leq L_f,\qquad \|\textnormal{D} g(x,y)\|_{\mathcal{B}(X_1\times Y_1,Y_\delta)}\leq L_g.
\end{align*}
\begin{prop}
Under the previous assumptions in this section the slow manifold is differentiable, i.e. the mapping
\begin{align*}
    (Y_S^\zeta,\|\cdot\|_{Y_1})\to (X_1,\|\cdot\|_{X_1})\times (Y_F^\zeta,\|\cdot\|_{Y_1}),\, v_0\mapsto  (h_{X_1}^{\varepsilon,\zeta}(v_0),h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0))
\end{align*}
is differentiable.
\end{prop}

\begin{proof}
The proof can be found in \cite[Prop 5.4]{hummel2022slow}.
\end{proof}
This will allow us to prove the attraction of trajectories and the approximation by the slow flow.

Next, let $(h_{X_1}^{\varepsilon,\zeta}(v_0),h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0),v_0) \in S_{\varepsilon,\zeta}$.
Moreover, let $(u,v_F,v_S)$ be the strict solution of \eqref{full eq splitting} with initial data $(h_{X_1}^{\varepsilon,\zeta}(v_0),h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0),v_0)$, i.e. it holds that
\begin{align*}
  \partial_t u(t)= \frac{1}{\varepsilon}\tilde A_\varepsilon u(t) +\frac{1}{\varepsilon}\tilde{f} (u(t),v_F(t),v_S(t)) 
\end{align*}
and let $(u^\varepsilon,v_F^\varepsilon,v_S^\varepsilon)$ be the strict solution to \eqref{full eq splitting} with initial data $(u_0,v_{0,F},v_{0,S})$.
From the invariance and differentiability of the slow manifold it follows that $u(t)= h_{X_1}^{\varepsilon,\zeta}(v_S(t))$ and therefore
\begin{align*}
    \partial_t u(t)&= \partial_t  h_{X_1}^{\varepsilon,\zeta}(v_S(t))= \textnormal{D} h_{X_1}^{\varepsilon,\zeta}(v_S(t))[\partial_t v_S(t)]\\
    &=\textnormal{D} h_{X_1}^{\varepsilon,\zeta}(v_S(t))[B v_S(t)+ \text{pr}_{Y_S^\zeta} g(u(t),v_F(t),v_S(t))].
\end{align*}
Combining both equations for $t=0$ yields that
\begin{align}\label{eq 4.4}
    h_{X_1}^{\varepsilon,\zeta} (v_0)= \varepsilon (\tilde A_\varepsilon)^{-1} \textnormal{D} h_{X_1}^{\varepsilon,\zeta}(v_0) \big[B v_0+ \text{pr}_{Y_S^\zeta} g(h_{X_1}^{\varepsilon,\zeta}(v_0) ,h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0),v_0)\big] - (\tilde A_\varepsilon)^{-1} f( h_{X_1}^{\varepsilon,\zeta}(v_0), h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0),v_0).
\end{align}
Similarly we obtain
\begin{align}\label{eq 4.5}
    h_{Y_F^\zeta}^{\varepsilon,\zeta} (v_0)= B_{Y_F^\zeta}^{-1} \textnormal{D} h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0) \big[B v_0+ \text{pr}_{Y_S^\zeta} g(h_{X_1}^{\varepsilon,\zeta}(v_0) ,h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0),v_0)\big] - B_{Y_F^\zeta}^{-1}\text{pr}_{Y_F^\zeta} g( h_{X_1}^{\varepsilon,\zeta}(v_0), h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0),v_0).
\end{align}
We note that these two relations hold for arbitrary $v_0\in Y_S^\zeta$.
Therefore it also holds for $ v_0= v^\varepsilon_S(t)$ for fixed $t\geq 0$ and we have
\begin{align}\label{eq 4.6}
    \partial_t  h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(t)) &= \textnormal{D}   h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(t))\big[B v^\varepsilon_S(t)+ \text{pr}_{Y_S^\zeta} g(u^\varepsilon(t),v^\varepsilon_F(t),v^\varepsilon_S(t))\big],\\ \label{eq 4.7}
     \partial_t  h_{Y_F^\zeta}^{\varepsilon,\zeta}(v^\varepsilon_S(t)) &= \textnormal{D}   h_{Y_F^\zeta}^{\varepsilon,\zeta}(v^\varepsilon_S(t))\big[B v^\varepsilon_S(t)+ \text{pr}_{Y_S^\zeta} g(u^\varepsilon(t),v^\varepsilon_F(t),v^\varepsilon_S(t))\big].
\end{align}
Thus, we can express the solution as a flow on the slow manifold.

The next step in the generalization of the Fenichel theorem is to show that the fast components of the fast-reaction system approach the solutions on the slow manifold with an exponential rate in time.
\begin{prop}
For $\varepsilon$ and $\zeta$ small enough satisfying $\varepsilon\zeta^{-1}<1$ there exist constants $C,\, c>0$ such that
\begin{align*}
    \bigg\|\begin{pmatrix}
    u^\varepsilon(t)-h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(t))\\ v^\varepsilon_F(t) -h_{Y_F^\zeta}^{\varepsilon,\zeta}(v^\varepsilon_S(t))
    \end{pmatrix} \bigg\|_{X_1\times Y_1} \leq C \textnormal{e}^{-ct} \bigg\|\begin{pmatrix}
    u_0- h_{X_1}^{\varepsilon,\zeta}(v_{0,S})\\ v_{0,F}-h_{X_1}^{\varepsilon,\zeta} (v_{0,S})
    \end{pmatrix}\bigg\|_{X_1\times Y_1}.
\end{align*}
\end{prop}
\begin{proof}
Starting with the first component we compute
\begin{align*}
    u^\varepsilon(t)-h_{X_1}^{\varepsilon,\zeta}(v_S^\varepsilon(t))&= \textnormal{e}^{\varepsilon^{-1}t \tilde A_\varepsilon}(u_0-h_{X_1}^{\varepsilon,\zeta}(v_{0,S}))+\textnormal{e}^{\varepsilon^{-1}t \tilde A_\varepsilon}h_{X_1}^{\varepsilon,\zeta}(v_{0,S}) - h_{X_1}^{\varepsilon,\zeta}(v_S^\varepsilon(t))\\
    &\quad+ \varepsilon^{-1} \int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s) \tilde A_\varepsilon} \tilde f(u^\varepsilon(s),v^\varepsilon_F(s),v^\varepsilon_S(s))\, \textnormal{d}s\\
    &= \textnormal{e}^{\varepsilon^{-1}t \tilde A_\varepsilon}(u_0-h_{X_1}^{\varepsilon,\zeta}(v_{0,S})) - \int_0^t \partial_s \big(\textnormal{e}^{\varepsilon^{-1}(t-s) \tilde A_\varepsilon}h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(s))\big) \, \textnormal{d}s\\
    &\quad+ \varepsilon^{-1} \int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s) \tilde A_\varepsilon} \tilde f(u^\varepsilon(s),v^\varepsilon_F(s),v^\varepsilon_S(s))\, \textnormal{d}s\\
    &=\textnormal{e}^{\varepsilon^{-1}t \tilde A_\varepsilon}(u_0-h_{X_1}^{\varepsilon,\zeta}(v_{0,S})) + \varepsilon^{-1}\int_0^t  \textnormal{e}^{\varepsilon^{-1}(t-s) \tilde A_\varepsilon}  \tilde A_\varepsilon h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(s)) \, \textnormal{d}s\\
    &\quad-\int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s) \tilde A_\varepsilon} \partial_s h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(s)) \, \textnormal{d}s + \varepsilon^{-1} \int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s) \tilde A_\varepsilon} \tilde f(u^\varepsilon(s),v^\varepsilon_F(s),v^\varepsilon_S(s))\, \textnormal{d}s.
    \intertext{Applying relations \eqref{eq 4.4} and \eqref{eq 4.6} yields}
    &=\textnormal{e}^{\varepsilon^{-1}t \tilde A_\varepsilon}(u_0-h_{X_1}^{\varepsilon,\zeta}(v_{0,S})) \\
    &\quad+ \int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s) \tilde A_\varepsilon} \textnormal{D}h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(s))\text{pr}_{Y_S^\zeta}\big[g(u^\varepsilon,v^\varepsilon_F,v^\varepsilon_S)-g(h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(s)),h_{Y_F^\zeta}^{\varepsilon,\zeta}(v^\varepsilon_S(s)),v^\varepsilon_S(s))\big]\, \textnormal{d}s\\
    &\quad + \varepsilon^{-1}\int_0^t \textnormal{e}^{\varepsilon^{-1}(t-s) \tilde A_\varepsilon} \big[f(u^\varepsilon,v^\varepsilon_F,v^\varepsilon_S)-f(h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(s)),h_{Y_F^\zeta}^{\varepsilon,\zeta}(v^\varepsilon_S(s)),v^\varepsilon_S(s))\big]\, \textnormal{d}s.
\end{align*}
Similarly, we obtain for the second component
\begin{align*}
    v^\varepsilon_F(t)-h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_S^\varepsilon(t))&= \textnormal{e}^{t B}(v_{0,F}-h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_{0,S})) \\
    &\quad+ \int_0^t \textnormal{e}^{(t-s) B} \textnormal{D}h_{Y_F^\zeta}^{\varepsilon,\zeta}(v^\varepsilon_S(s))\text{pr}_{Y_S^\zeta}\big[g(u^\varepsilon,v^\varepsilon_F,v^\varepsilon_S)-g(h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(s)),h_{Y_F^\zeta}^{\varepsilon,\zeta}(v^\varepsilon_S(s)),v^\varepsilon_S(s))\big]\, \textnormal{d}s\\
    &\quad + \int_0^t \textnormal{e}^{(t-s) B} \text{pr}_{Y_F^\zeta} \big[g(u^\varepsilon,v^\varepsilon_F,v^\varepsilon_S)-g(h_{X_1}^{\varepsilon,\zeta}(v^\varepsilon_S(s)),h_{Y_F^\zeta}^{\varepsilon,\zeta}(v^\varepsilon_S(s)),v^\varepsilon_S(s))\big]\, \textnormal{d}s.
\end{align*}
Setting $z(t)= \| u^\varepsilon(t)-h_{X_1}^{\varepsilon,\zeta}(v_S^\varepsilon(t))\|_{X_1}+ \| v^\varepsilon_F(t)-h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_S^\varepsilon(t))\|_{Y_1}$
we can estimate the two components as follows
\begin{align*}
    z(t)&\leq M_A \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)}  \| u^\varepsilon(t)-h_{X_1}^{\varepsilon,\zeta}(v_S^\varepsilon(t))\|_{X_1} + M_B  \textnormal{e}^{(\zeta^{-1}(\varepsilon\omega_A+\lambda_0) +N_F^\zeta)t} \| v^\varepsilon_F(t)-h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_S^\varepsilon(t))\|_{Y_1} \\
    &\quad + C_A L_f(\varepsilon L_g+1) \int_0^t \frac{\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A -\lambda_0)(t-s)}}{\varepsilon^\gamma (t-s)^{1-\gamma}} z(s)\, \textnormal{d}s\\
    &\quad + C_B L_g( L_g+1) \int_0^t \frac{\textnormal{e}^{(\zeta^{-1}(\varepsilon\omega_A+\lambda_0) +N_F^\zeta )(t-s)}}{(t-s)^{1-\delta}} z(s)\, \textnormal{d}s.
\end{align*}
Applying a version of Gronwall's inequality yields
\begin{align*}
    z(t)\leq C \textnormal{e}^{-ct}\bigg\|\begin{pmatrix}
    u_0- h_{X_1}^{\varepsilon,\zeta}(v_{0,S})\\ v_{0,F}-h_{X_1}^{\varepsilon,\zeta} (v_{0,S})
    \end{pmatrix}\bigg\|_{X_1\times Y_1}.
\end{align*}
This completes the proof.
\end{proof}

The last step in the generalization of Fenichel's theorem is to show that the semi-slow flow is a good approximation of the semi-flow of the original fast-reaction system.

So far we have measured the distance of the slow manifold to the subset $S_{0,\zeta}$ of the critical manifold given by
\begin{align*}
    S_{0,\zeta}:=\{ (h^0(v_0),v_0)\in S_0\,:\, \text{pr}_{Y_F^\zeta}v_0=0\}.
\end{align*}
However, this subset is in general not invariant under the slow flow. 
Hence, we introduce the reduced slow subsystem
\begin{align}
\label{eq. reduced slow}
\begin{split}
    0&= f(h^0(v^0_\zeta(t)),v^0_\zeta(t)),\\
    0&= \text{pr}_{Y_F^\zeta} v^0_\zeta(t),\\
    \partial_t v^0_\zeta(t)&= B v^0_\zeta(t)+ \text{pr}_{Y_S^\zeta} g(h^0(v^0_\zeta(t)),v^0_\zeta(t)),\\
    v^0_\zeta(0)&= \text{pr}_{Y_S^\zeta}v_0,
\end{split}
\end{align}
 and we observe that $S_{0,\zeta}$ is invariant under the flow generate by the slow subsystem.

\begin{prop}\label{Prop 4.9}
There exists a constant $C>0$ such that for all $t\in [0,T]$ and all $\varepsilon,\,\zeta>0$ small enough satisfying $\varepsilon\zeta^{-1}<1$ the following holds
\begin{align*}
    \|v^0(t)-v^0_\zeta(t)\|_{Y_1} \leq  C \textnormal{e}^{\omega_B t} \textnormal{e}^{\textnormal{e}^{\omega_B t}}\bigg(\|\text{pr}_{Y_F^\zeta} v_0\|_{Y_1}+ \frac{\|v_0\|_{Y_1}}{(\omega_B -\zeta^{-1}(\varepsilon\omega_A+\lambda_0) -N_F^\zeta)^\delta}\bigg).
\end{align*}
\end{prop}
\begin{proof}
The well-posedness of the slow subsystem follows from Prop \ref{Prop 3.2} and thus we can write the solution as
\begin{align*}
    v^0_\zeta(t)= \textnormal{e}^{tB} \text{pr}_{Y_S^\zeta} v_0+ \int_0^t \textnormal{e}^{(t-s)B} \text{pr}_{Y_S^\zeta} g(h^0(v^0_\zeta(s)),v^0_\zeta(s))\, \textnormal{d}s.
\end{align*}
Hence
\begin{align*}
    v^0(t)-v^0_\zeta(t)=& \textnormal{e}^{tB} \text{pr}_{Y_F^\zeta} v_0+ \int_0^t \textnormal{e}^{(t-s)B} \text{pr}_{Y_S^\zeta}\big( g(h^0(v^0(s)),v^0(s))-g(h^0(v^0_\zeta(s)),v^0_\zeta(s))\big)\, \textnormal{d}s\\
    &- \int \textnormal{e}^{(t-s)B}\text{pr}_{Y_F^\zeta} g(h^0(v^0_\zeta(s)),v^0_\zeta(s))\, \textnormal{d}s.
\end{align*}
Now, we can estimate
\begin{align*}
    \|v^0(t)-v^0_\zeta(t)\|_{Y_1} &\leq M_B \textnormal{e}^{(\zeta^{-1}(\varepsilon\omega_A+\lambda_0) +N_f^\zeta)t} \|\text{pr}_{Y_F^\zeta} v_0\|_{Y_1}\\
    &\quad + L_gC_B\int_0^t \frac{\textnormal{e}^{(\zeta^{-1}(\varepsilon\omega_A+\lambda_0) +N_f^\zeta)(t-s)}}{(t-s)^{1-\delta}}\big( \|h^0(v^0(s))\|_{X_1}+ \|v^0(s)\|_{Y_1}\big)\, \textnormal{d}s\\
    &\quad+ L_GC_B \int_0^t \frac{\textnormal{e}^{\omega_B(t-s)}}{(t-s)^{1-\delta}} \big( \|h^0(v^0(s))-h^0(v^0_\zeta(s))\|_{X_1}+ \|v^0(s)-v^0_\zeta(s)\|_{Y_1}\big)\, \textnormal{d}s\\
    &\leq M_B \textnormal{e}^{(\zeta^{-1}(\varepsilon\omega_A+\lambda_0)+N_f^\zeta)t} \|\text{pr}_{Y_F^\zeta} v_0\|_{Y_1}\\
    &\quad+  L_g(L_h+1) M_B C_B \textnormal{e}^{C_BL_g(1+L_h)\Gamma(\delta)} \textnormal{e}^{\omega_B t} \|v_0\|_{Y_1} \int_0^t \frac{\textnormal{e}^{(\zeta^{-1}(\varepsilon\omega_A+\lambda_0) +N_f^\zeta-\omega_B)(t-s)}}{(t-s)^{1-\delta}}  \, \textnormal{d}s\\
    &\quad+ L_G (L_h+1) C_B \int_0^t \frac{\textnormal{e}^{\omega_B(t-s)}}{(t-s)^{1-\delta}}  \|v^0(s)-v^0_\zeta(s)\|_{Y_1}\, \textnormal{d}s\\
    &\leq C \textnormal{e}^{\omega_B t}\bigg(\|\text{pr}_{Y_F^\zeta} v_0\|_{Y_1}+ \frac{\|v_0\|_{Y_1}}{(\omega_B -\zeta^{-1}(\varepsilon\omega_A+\lambda_0) -N_F^\zeta)^\delta}\bigg)\\
    &\quad+L_G (L_h+1) C_B \int_0^t \frac{\textnormal{e}^{\omega_B(t-s)}}{(t-s)^{1-\delta}}  \|v^0(s)-v^0_\zeta(s)\|_{Y_1}\, \textnormal{d}s.
\end{align*}
Applying Gronwall's inequality yields the desired result.
\end{proof}

\begin{remark}
    In the case that the operator $B$ is dissipative, i.e. $\| \textnormal{e}^{tB}\|_{\mathcal{B}(Y_1)}\leq \textnormal{e}^{\omega_B t}$ with $\omega_B<0$ the above result reduces to 
    \begin{align*}
        \|v^0(t)-v^0_\zeta(t)\|_{Y_1} \leq  C \textnormal{e}^{\omega_B t} \bigg(\|\text{pr}_{Y_F^\zeta} v_0\|_{Y_1}+ \frac{\|v_0\|_{Y_1}}{|\omega_B -\zeta^{-1}(\varepsilon\omega_A+\lambda_0) -N_F^\zeta|^\delta}\bigg).
    \end{align*}
\end{remark}

\begin{prop}
There exists a constant $C>0$ such that for all $t\in [0,T]$ and all $\varepsilon,\,\zeta>0$ small enough satisfying $\varepsilon\zeta^{-1}<1$ it holds that
\begin{align*}
    \bigg\| \begin{pmatrix}
    u^\varepsilon(t)- h^0(v^0_\zeta(t)\\ v^\varepsilon(t)- v_\zeta^0(t)
    \end{pmatrix}\bigg \|_{X_1\times Y_1} &\leq C\textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0) t}\|u_0- h^0(v_0)\|_{X_1} (1 +{\textnormal{e}^{\textnormal{e}^{\omega_{\small{B}} t}+ct}})(M_A +  \varepsilon^{-\gamma}t^\gamma)\\
    &\quad+  \varepsilon C\|v_0\|_{Y_1}( \textnormal{e}^{\omega_B t}  +  {\textnormal{e}^{\textnormal{e}^{\omega_{\small{B}} t}+ct}} t^\delta )\\
     &\quad+C \textnormal{e}^{\omega_B t} \textnormal{e}^{\textnormal{e}^{\omega_B t}}\bigg(\|\text{pr}_{Y_F^\zeta} v_0\|_{Y_1}+ \frac{\|v_0\|_{Y_1}}{(\omega_B -\zeta^{-1}(\varepsilon\omega_A+\lambda_0) -N_F^\zeta)^\delta}\bigg).
\end{align*}
If in addition the initial values are on the slow manifold it holds that
\begin{align*}
    &\bigg\| \begin{pmatrix}
    u^\varepsilon(t)- h^0(v^0_\zeta(t)\\ v^\varepsilon(t)- v_\zeta^0(t)
    \end{pmatrix}\bigg \|_{X_1\times Y_1}\leq \\
    &\quad\leq C \textnormal{e}^{(\omega_B+c)t}  \|v_0\|_{Y_1}\bigg(\varepsilon (1+t^\delta) + \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)t}+\frac{1}{(\omega_B -\zeta^{-1}(\varepsilon\omega_A+\lambda_0) -N_F^\zeta)^\delta}+ \frac{1}{(N_S^\zeta-N_F^\zeta)^\delta}\bigg).
\end{align*} 

\end{prop}
\begin{proof}
The first estimate is a consequence of Theorem \ref{Thm 3.6} and Proposition \ref{Prop 4.9}. 
For the second estimate we apply the triangle inequality to the terms
\begin{align*}
    \|\text{pr}_{Y_F^\zeta} v_0\|_{Y_1}&\leq \|pr_{Y_F^\zeta} v_0-h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0)\|_{Y_1}+ \|h_{Y_F^\zeta}^{\varepsilon,\zeta}(v_0)\|_{Y_1},\\
    \|u_0-h^0(v_0)\|_{X_1}&\leq \|u_0-h^{\varepsilon,\zeta}_{X_1}(v_0)\|_{X_1}+ \|h^{\varepsilon,\zeta}_{X_1}(v_0)-h^0(v_0)\|_{X_1}.
\end{align*}
The first term in each of the two estimates equals to zero since the initial data is on the slow manifold and the second term can be estimated by Proposition \ref{Prop 4.6}.
\end{proof}

\begin{remark}
Again we want to put the final result into context. 
To this end we assume that the constant $N_F^\zeta$ is bounded independent of $\zeta$ and that in the asymptotic limit $N_S^\zeta-N_F^\zeta= C \zeta^{-\alpha}$.
Then, using the assumption that $\varepsilon \zeta^{-1}<1$ we obtain
\begin{align*}
   &\bigg\| \begin{pmatrix}
    u^\varepsilon(t)- h^0(v^0_\zeta(t)\\ v^\varepsilon(t)- v_\zeta^0(t)
    \end{pmatrix}\bigg \|_{X_1\times Y_1}\leq 
    C \textnormal{e}^{(\omega_B+c)t}  \|v_0\|_{Y_1}\big(\varepsilon  + \textnormal{e}^{\varepsilon^{-1}(\varepsilon\omega_A+\lambda_0)t}+\zeta^{-\alpha} +\zeta^{-\alpha\delta} \big).  
\end{align*}
Letting $\varepsilon,\,\zeta\to 0$ yields an explicit rate for the convergence of the semi-flow fast-reaction system to the semi-flow of the slow subsystem.

\end{remark}


\subsection{Example}\label{subsection example}

As a final step we apply our methods to a toy model.
The system reads
\begin{align}\label{eq 4.5.1}
\begin{split}
     \partial_t u^\varepsilon&= \Delta u^\varepsilon +\frac{1}{\varepsilon}\big[ (v^\varepsilon)^2- u^\varepsilon(u^\varepsilon-1)],\\
    \partial_t v^\varepsilon&= \Delta v^\varepsilon,\\
    u^\varepsilon(0)&= u_0,\quad v^\varepsilon(0)= v_0,
\end{split}
\end{align}
where the equations are posed on a periodic domain $\mathbb{T}^n\subset \mathbb{R}^n$ with $n\leq 3$.
The limit system satisfies
\begin{align}\label{eq 4.5.2}
\begin{split}
    0&=(v^0)^2- u^0(u^0-1),\\
     \partial_t v^0&= \Delta v^0,\\
     u^0(0)&=h^0(v_0) \quad v^0(0)= v_0.
\end{split}
\end{align}
In this example we work on the standard $L^2$ space, i.e. we have $X=Y= L^2(\mathbb{T}^n)$.

\begin{theorem}
Let $(u^\varepsilon,v^\varepsilon)$ be the solution of \eqref{eq 4.5.1} and denote $(u^0,v^0)$ the solution of the limit system \eqref{eq 4.5.2}.
Then there exist constants $C,\, c>0$ such that for all $\varepsilon\in (0,\varepsilon_0]$ and all $u_0,\, v_0 \in H^2(\mathbb{T}^n)$ with $\|u_0\|_{H^2(\mathbb{T}^n)}\leq \sigma$ and $\|v_0\|_{H^2(\mathbb{T}^n)}\leq \sigma$ it holds
\begin{align*}
    \sup_{0\leq t\leq T(\sigma)}\big(  \|u^\varepsilon(t)- u^0(t)\|_{H^2(\mathbb{T}^n)} +  \|v^\varepsilon(t)- v^0(t)\|_{H^2(\mathbb{T}^n)} \big) \leq C\big(\varepsilon +\textnormal{e}^{-\varepsilon^{-1} c t}\big),
\end{align*}
where the final time $T(\sigma)$ is given by
\begin{align*}
    T(\sigma):= \inf\big\{ t\in [0,T]:\max\{ \|u^\varepsilon(t)\|_{H^2(\mathbb{T}^n)}, \|v^\varepsilon(t)\|_{H^2(\mathbb{T}^n)}, \|u^0(t)\|_{H^2(\mathbb{T}^n)}, \|v^0(t)\|_{H^2(\mathbb{T}^n)}\}>\sigma\big\}.
\end{align*}
\end{theorem}
\begin{theorem}
Let the previous assumptions hold and set the dimension $n=1$. 
Moreover, there is a $\zeta_0$ such that for all $\zeta\in (0,\zeta_0]$ the following holds:
\begin{itemize}
    \item There exists a slow manifold $S_{\varepsilon,\zeta}$ that is $C^1$-regular and locally invariant under the semiflow generated by \eqref{eq 4.5.1}.
    \item Let $S_{0,\zeta}:=\{ (u,v)\in S_0\, :\, v\in B_{Y_S^\zeta}(0,\sigma)\}$ be the submanifold of the critical manifold which consists of all points whose slow components are in the bounded set $ B_{Y_S^\zeta}(0,\sigma)$.
    Then there exists a constant $C>0$ such that
    \begin{align*}
        \dist (S_{\varepsilon,\zeta},S_{0,\zeta})\leq C\big(\varepsilon+ \zeta^{1/2}\big).
    \end{align*}
    \item Suppose that $(h(v_0),v_0)\in S_{\varepsilon,\zeta}$ satisfying $\|v_0\|_{H^2(\mathbb{T}^n)}\leq \sigma$, $\|h^0(v_0)\|_{H^2(\mathbb{T}^n)}\leq \sigma$.
    Then there exists a constant $C>0$ such that 
    \begin{align*}
        \sup_{0\leq t\leq T(\sigma)}\big(  \|u^\varepsilon(t)- u_\zeta^0(t)\|_{H^2(\mathbb{T}^n)} +  \|v^\varepsilon(t)- v_\zeta^0(t)\|_{H^2(\mathbb{T}^n)} \big) \leq C\big(\varepsilon + \zeta^{1/2} \big).
    \end{align*}
\end{itemize}
\end{theorem}

\begin{remark}
    For the second theorem we restrict the domain $\mathbb{T}$ to be one dimensional so that the spectral gap condition holds.
    Although in this special example, with no nonlinearity in the slow equation $v$, the result would also hold in higher dimensions.
\end{remark}

What remains is to check the assumptions from sections \ref{sect 3.1} and \ref{sect 4.1}.
\begin{itemize}
    \item[(i)] The Laplacian generates a holomorphic $C_0$-semigroup $(\textnormal{e}^{t\Delta})_{t\geq 0}$ on $X_1=H^2(\mathbb{T}^n)\subset L^2(\mathbb{T}^n)$ and from complex interpolation it follows that the spaces $X_\alpha= H^{2\alpha}(\mathbb{T}^n)$ are a suitable choice for the Banach scales.
    \item[(ii)] The Laplacian generates a bounded holomorphic $C_0$-semigroup $(\textnormal{e}^{t\Delta})_{t\geq 0}$ on $H^2(\mathbb{T}^n)$, where we can take $\omega_A=0$.
    
    \item[(iii)] To ensure global Lipschitz continuity for the nonlinearity $f(u,v)$ we apply a cutoff technique.
                 Let $\sigma>0$ and choose a $C^\infty$-cutoff function $\chi : H^2(\mathbb{T}^n)\to [0,1]$ such that $\chi(x)=1$ if $x\in B(0,\sigma)$ and $\chi(x)=0 $ if $\mathbb{T}^n\setminus B(0,2\sigma)$. 
                 Then, the nonlinearity 
                 \begin{align*}
                     f: H^2(\mathbb{T}^n)\times H^2(\mathbb{T}^n)\to H^{2\gamma}, (x,y)\mapsto \chi(y) y^2-\chi(x)x(x-1)
                 \end{align*}
                 satisfies
                 \begin{align*}
                     \|f(x_1,y_1)-f(x_2,y_2)\|_{H^{2\gamma}} &\leq L_f\big( \|x_1-x_2\|_{H^{2\gamma}}+\|y1-y_2\|_{H^{2\gamma}}\big),\\
                     \|f(u_1,v_1)-f(u_2,v_2)\|_{C^1([0,T];L^2)}&\leq L_f\big( \|u_1-u_2\|_{C^1([0,T];L^2)}+ \|v_1-v_2\|_{C^1([0,T];L^2)}\big)
                 \end{align*}
                 globally and if we choose $\sigma>0$ small enough, we then have  $L_f < 1/C_A$.
                 Moreover, the nonlinearity is differentiable with
                 \begin{align*}
                     \|\textnormal{D} f(x,y)\|_{\mathcal{B}(H^2\times H^2,H^{2\gamma})}\leq L_f.
                 \end{align*}
    \item[(iv)] The nonlinearity $f$ satisfies the assumptions for the implicit function theorem close to $u=0$.
    Moreover, choosing $\sigma>0$ such that in a neighborhood around $u=0$ we have that $\textnormal{D} f_u(u,v)=-2u-1\leq \lambda_0<0$.
   \item[(v)] The function $h^0$ given by the implicit function theorem can be computed explicitly, i.e. 
   \begin{align*}
       h^0(v^0)= \frac{1}{2} -\sqrt{\frac{1}{4} + (v^0)^2}.
   \end{align*}
\end{itemize}
Thus, all assumptions in section \ref{sect 3.1} are satisfied and it remains to show that the Laplacian satisfies the assumptions for the splitting.

Making use of the Hilbert structure of $L^2(\mathbb{T}^n)$ we introduce the splitting as 
\begin{align*}
    L^2=Y= Y_S^\zeta\oplus Y_F^\zeta
\end{align*} 
by truncating at a certain Fourier mode.

Assuming that $-(k_0+1)^2< \zeta^{-1}(\varepsilon \omega_A +\lambda_0)\leq - k_0^2$ we set 
\begin{align*}
   Y_S^\zeta= \text{span}\{x\mapsto \textnormal{e}^{ikx}: |k|\leq k_0\},\qquad Y_F^\zeta= \text{cl}_{L^2}(\text{span} \{x\mapsto \textnormal{e}^{ikx}: |k|> k_0\}). 
\end{align*}

\begin{itemize}
    \item[(vi)] Since $Y_S^\zeta$ is a finite-dimensional space the Laplacian generates a $C_0$-group on $Y_S^\zeta$, where $(\textnormal{e}^{t\Delta_{Y_S^\zeta}})_{t\in \mathbb{R}}$ coincides with $(\textnormal{e}^{t\Delta})|_{Y_S^\zeta}$ for $t\geq 0$.
    \item[(vii)] For the realization of the Laplacian in $Y_F^\zeta$ we have that $0$ is in its resolvent.
    This can be seen by the representation of the inverse, which is given by
    \begin{align*}
        \Delta_{Y_F^\zeta}^{-1}: L^2(\mathbb{T}^n) \to  H^2(\mathbb{T}^n),\quad \sum_{|k|\geq k_0} \hat f(k) \textnormal{e}^{ikx} \mapsto \sum_{|k|\geq k_0} \frac{\hat f(k) \textnormal{e}^{ikx}}{|k|^2}
    \end{align*}
    and is well-defined as $k=0$ does not appear in the sum.
    \item[(viii)] We note that the semigroup $(\textnormal{e}^{t\Delta})_{t\geq 0}$ can be expressed as
    \begin{align*}
         \textnormal{e}^{t\Delta} f= \bigg[ x\mapsto \sum_{k\in \mathbb{Z}^n} \textnormal{e}^{-|k|^2 t} \hat f (k) \textnormal{e}^{ikx}\bigg].
    \end{align*}
    Applying Plancherel's theorem for $y_S\in Y_S^\zeta$ and $t\geq 0$ yields
    \begin{align*}
        \|\textnormal{e}^{-t\Delta} y_S\|_{H^2(\mathbb{T}^n)}\leq \textnormal{e}^{(k_0-1)^2t}\|y_S\|_{H^2(\mathbb{T}^n)}.
    \end{align*}
    Thus, we can take $N_S^\zeta= -\zeta^{-1}\lambda_0 -(k_0-1)^2$ and similarly we may take $N_F^\zeta= -\zeta^{-1} \lambda_0- k_0^2$.
    Then,
    \begin{align*}
        N_S^\zeta-N_F^\zeta= 2 k_0-1 \geq 2 \sqrt{-\zeta^{-1}\lambda_0}-C.
    \end{align*}
    Therefore, $N_S^\zeta-N_F^\zeta\approx \mathcal{O}(\zeta^{-1/2})$ for $\zeta$ small enough.
    \item[(ix)] We take $\zeta>0$ such that $\varepsilon \zeta^{-1}<1$.
\end{itemize}
This completes checking the assumptions for section \ref{sect 4.1}.
We conclude this example with some observations.
\begin{remark}
    We note some differences between the results obtained for fast-reaction systems in this work and the results in \cite{hummel2022slow} for fast-slow systems.
    One key issue are the restrictions on the operator $A$. 
    In this work the growth bound on the semigroup generated by the operator $A$ need not satisfy $\omega_A<0$ since the fast-reaction dominates the terms on the right-hand side.
    Therefore, we only require that $\| \textnormal{e}^{\varepsilon^{-1}(\varepsilon A -\textnormal{D}_x f(h^0(v^0),v^0))t} \|_{\mathcal{B}(X_1)}\leq M_A \textnormal{e}^{\varepsilon^{-1}(\varepsilon \omega_A- \lambda_0)t} $ where $\varepsilon \omega_A- \lambda_0< 0$.
    Consequently, we can apply our results to the case where $A=\Delta$ in $\Omega\subset \mathbb{R}^n$ for $n\geq 2$, as shown in the above example.
    On the other hand, we need more requirements on the nonlinear function $f$, one of which is, that $f$ has to satisfy the implicit function theorem in a neighborhood around the equilibrium $f(u,v)=0$. 
\end{remark}







\begin{appendix}
\section{Tools from Functional Analysis}\label{appendix A}
In this section we present useful definitions and theorems from the theory of semigroups that are used throughout the paper.
The details to the brief summary we present here can be found in \cite{amann1995linear} and \cite{ lunardi2012analytic} and the references therein.

\begin{definition}
Let $X$ be a Banach space. 
Then a family of linear, bounded operators $\big( S(t)\big)_{t\geq 0}\subset \mathcal{L}(X)$, where $S(t): X\to X$ for all $t\geq 0$, is called a semigroup if
\begin{align*}
    S(0)= I,\qquad S(s+t)=S(s)S(t) \quad \forall t\geq 0.
\end{align*}
It is called a strongly continuous or $C_0$-semigroup if in addition
\begin{align*}
    \|S(t)x-x\|_X\to 0,\quad t\to 0^+ \quad \forall x\in X.
\end{align*}
\end{definition}

\begin{definition}
Define the set $D(A):= \{ x\in X :\, \exists \, \lim_{t\to 0^+} \frac{S(t)x-x}{t}\}$ and $Ax:= \lim_{t\to 0^+} \frac{S(t)x-x}{t}$ for $x\in D(A)$.
Then, $A$ is called the infinitesimal generator of $\big( S(t)\big)_{t\geq 0}$.
\end{definition}

\begin{theorem}
Let $x\in D(A)$ and assume that $\sup_{t\geq 0} \|S(t)\|<\infty$.
Then
\begin{itemize}
    \item[i)] $S(t)x \in D(A)$ for all $t\geq 0$.
    \item[ii)] $AS(t)x = S(t)A x$ for all $t\geq 0$.
    \item[iii)] The mapping $t\mapsto S(t)x$ is differentiable for each $t\geq 0$ and $\frac{\textnormal{d}}{\textnormal{d}t} S(t)x= AS(t)x$ for all $t\geq 0$.
\end{itemize}
\end{theorem}

\begin{theorem}
Let $\big( S(t)\big)_{t\geq 0}$ be a $C_0$-semigroup on $X$. 
Then there exists an $\omega\in \mathbb{R}$ and $M\geq 1$ such that
\begin{align*}
    \|S(t)\|_{\mathcal{B}(X)}\leq M \textnormal{e}^{\omega t}\quad \forall t\geq 0.
\end{align*}
\end{theorem}

\begin{theorem}
Let $A$ be the generator of a $C_0$-semigroup $\big( S(t)\big)_{t\geq 0}$ on $X$.
If $B\in \mathcal{B(X)}$, then $A+B$ is the infinitesimal generator a $C_0$-semigroup $\big( T(t)\big)_{t\geq 0}$ on $X$ satisfying
\begin{align*}
   \|T(t)\|_{\mathcal{B}(X)}\leq M \textnormal{e}^{(\omega +M\|B\|_{\mathcal{B}(X)})t}\quad \forall t\geq 0.
\end{align*}
\end{theorem}

\begin{theorem}
Let $A$ be the generator of a $C_0$-semigroup $\big( S(t)\big)_{t\geq 0}$ on $X$ and let $x\in D(A)$.
Then the mapping $t\mapsto S(t)x $ solves the abstract Cauchy problem on X for $t\geq 0$
\begin{align*}
    \frac{\textnormal{d}}{\textnormal{d}t} y=Ay, \quad y(0)=x.
\end{align*}
\end{theorem}

\begin{definition}
Let $A$ be a densely defined closed linear operator on $X$ with $0\in \varrho(A)$.
Let $\theta \in (0,1)$.
Then we call $(\cdot,\cdot)_\theta$ an exact admissible interpolation functor, i.e. it is an exact interpolation functor such that $X_1$ is dense in $(X_0,X_1)_\theta$ whenever $X_1 \overset{d}{\hookrightarrow}X_0$.
\end{definition}

\begin{definition}
We define a family of Banach spaces $(X_\alpha)_{\alpha\in [-1,\infty)}$ and a family of operators $(A^\alpha)_{\alpha\in [-1,\infty)}\in \mathcal{B}(X_\alpha,X_{\alpha+1})$ as follows
\begin{itemize}
    \item for $k\in \mathbb{N}_0$ we set $X_k:= D(A^k)$ endowed with the norm $\|x\|_{X_k}:= \|A^k x\|_{X}$.
    In particular, $X_0=D(A^0)=X$;
    \item $X_{-1}$ is defined as the completion of $X$ with respect to the norm $\|x\|_{X_{-1}}:= \|A^{-1}x\|_{X}$;
    \item For $k\in \mathbb{N}_0$, $\theta\in (0,1)$ and $\alpha=k+\theta$ we define
    $X_\alpha:= (X_k,X_{k+1})_\theta$ and $A^\alpha= A^k\big|_{D(A^\alpha)}$, where
    $D(A^\alpha)=\{ x\in X_{k+1}:\, A^k x \in X_\alpha\}$.
\end{itemize}
The family $(X_\alpha,A^\alpha)_{\alpha\in [-1,\infty)}$ is a densely injected Banach scale in the sense that $X_\alpha\overset{d}{\hookrightarrow}X_\beta$ whenever $\alpha\geq \beta$ and $A^\alpha: X_{\alpha+1} \to X_\alpha$ is an isomorphism for all $\alpha\in \mathbb{R}$.
Moreover, $A^\alpha: X_{\alpha+1}\to X_\alpha$ is a densely defined closed operator with $0\in \varrho(A^\alpha)$ for all $\alpha\in \mathbb{R}$.
The family $(X_\alpha,A^\alpha)$ is called an interpolation-extrapolation scale.
\end{definition}

\begin{theorem}
Let $A$ be the generator of a $C_0$-semigroup $\big( S(t)\big)_{t\geq 0}$ on $X$ and let $\omega_S\in \mathbb{R}$ be the growth bound of the semigroup $S(t)$.
Then $A^\alpha:X_{\alpha+1}\to X_\alpha$ also generates a $C_0$-semigroup $\big( S_\alpha(t)\big)_{t\geq 0}$ with the same growth bound. 
For all $\alpha,\, \beta \in [0,\infty)$ with $\alpha\geq \beta$ the following diagram commutes
\[\begin{tikzcd}
	{X_\alpha} & {X_\alpha} \\
	{X_\beta} & {X_\beta}
	\arrow[hook, from=1-1, to=2-1]
	\arrow[hook, from=1-2, to=2-2]
	\arrow["{S_\alpha(t)}", from=1-1, to=1-2]
	\arrow["{S_\beta(t)}", from=2-1, to=2-2]
\end{tikzcd}\]
and for all $\omega>\omega_S$ there is a constant $C$ depending on $\alpha$ and $\beta$ such that
\begin{align*}
   \|S_\beta(t)\|_{\mathcal{B}(X_\beta,X_\alpha)} \leq C t^{\beta-\alpha} \textnormal{e}^{\omega t}\quad \forall t>0.
\end{align*}

\end{theorem}





\end{appendix}

\section*{Acknowledgements}
The authors would like to thank F. Hummel and B.Q. Tang for the fruitful discussions. 
Furthermore, the authors acknowledge the support of the DFG under grant No.$456754695$.
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