I have a question concerning strongly continuous semigroups and applications to PDE. A strongly continuous semigroup on a Banach space $X$ is a map $T: \mathbb{R}_+ \rightarrow L(X)$ such that:
(1) $T(0)=id$
(2) $\forall t,s \geq 0$: $T(t+s)=T(t)T(s)$
(3) $\forall x_0\in X: \| T(t)x_0 - x_0 \| \rightarrow 0, \text{as } t\downarrow 0$.
(cf. Wikipedia)
I am interested in the applications of semigroups to PDEs of the type \begin{equation} u_t-Au=f \end{equation}
My question is: Why is the convergence in the 3rd condition the "right" convergence to claim? Stronger convergence limits us to bounded operators as generators, but why do we claim this convergence?
Thank you in advance!
Best, Luke