Let $(A, D(A))$ be the generator of a strongly continuous semigroup $(S(t))$ on $X$. Then the following Cauchy problem $$ \left\{\begin{array}{l} \dot{u}(t)=A u(t) \quad \text { on}\,\,\, (0,T) \\ u(0)=x \end{array}\right. $$ has a unique mild solution for all $x\in X$. This is a well-known result.
My confusion comes when they talk about the value of the semigroup $S(t)$ at $t=T$. I can't see why $S(T)$ has a meaning (because the Cauchy problem we solve is only on $(0,T)$).
If anyone can explain it to me please.