Let $(Y,\left\|\cdot\right\|_Y)$ be a Banach space and $A:D(A)\subset Y \to Y$ a closed operator. Studying dynamical systems of the form \begin{equation} u'=Au \end{equation} quickly leads to the notion of one-parameter semigroups:

$\textbf{Definition}$: A one-parameter semigroup $G$ is a collection $\left\{S(t):Y\to Y\right\}_{t \geq 0}$ of bounded operators which satisfy $S(0)=I$ and $\forall s,t > 0$: $S(s+t)=S(s)S(t)$. A one-parameter semigroup is called strongly continuous if the maps \begin{equation} t \mapsto S(t)u \end{equation} are continuous $\forall u \in Y$.

My problem is the following: what if the nice $C^1$-solutions $u(t)$ ($u(t) \subset D(A)$) of the $(u'=Au)$-system have not for all $t>0$ a uniform bound \begin{equation} \sup_{u(0) \in D(A),u(0)\text{ generates } C^1-solution}\frac{\left\|u(t)\right\|_Y}{\left\|u(0)\right\|_Y} \end{equation} but nontheless a subvectorspace $X \subset Y$ exists with $D(A) \subset X$, together with some norm $\left\|\cdot\right\|_X$ such that $(X,\left\|\cdot\right\|_X)$ is a Banach space of its own and \begin{equation} \sup_{u(0) \in D(A),u(0)\text{ generates } C^1-solution}\frac{\left\|u(t)\right\|_Y}{\left\|u(0)\right\|_X} \end{equation} does turn out to be finite for all $t>0$. Continuing this line of reasoning I end up with something like

$\textbf{Definition}$: Let $(Y,\left\|\right\|_Y)$ and $(X,\left\|\right\|_X)$ be Banach spaces and $X \subset Y$ such that $X$ is dense in $Y$ for the $Y$-norm. A bilateral one-parameter semigroup $G$ is a collection $\left\{S(t):X\to Y\right\}_{t \geq 0}$ of bounded operators which satisfy...

*$S(0)=I_{X \to Y}$

*$\forall u \in X$: $\varphi_u:[0,\infty) \to Y:t \mapsto S(t)u$ is a continuous map.

*$\forall u \in X$ and $f:[0,\infty) \to \mathbb{C}$ a $K_G$-function (see below for definition), we have $\int_0^{\infty} f(s)S(s)u\text{ d}s \in X$ and
\begin{equation}
S(t)\left(\int_0^{\infty} f(s)S(s)u\text{ d}s\right) = \int_0^{\infty} f(s)S(t+s)u\text{ d}t.
\end{equation}
$\textbf{Definition}$: The vector space $K_G$ consists of all complex functions with domain $[0,\infty)$ spanned by functions of the form $\chi_{[a,b]}$ or $f$ $C^1$ such that $\int_0^{\infty} \left\|f'(s)S(s)\right\|\text{d}s < \infty$, $\forall t \geq 0$ $\int_0^{\infty} \left\|f(t)S(s+t)\right\|\text{ d}s <\infty$, and $\left\|f(t)S(t)\right\| \to 0$ as $t\to \infty$.
*I think we could as well replace "spanned by functions of the form $\chi_{[a,b]}$ or $f$ $C^1$ such that..." in the definition of $K_G$ by "functions which are piecewise $C^1$ up to a set of 0 Lebesgue-measure"*

Are there any good reasons to reject this line of reasoning and this generalisation from the start? Are my concerns (which seem generic to me) already adressed somewhere in the literature?

EDIT: an example and elaboration is below in an answer.