If $A$ generates the $C_0$-semigroup $\{T_t;t\ge0\}$, then $Au=f \Rightarrow u=-\int_0^\infty T_t f dt$?

Question

Let $X$ be a Banach space, $(A,D_A)$ be a linear operator on $X$ which can generate a (contractive, if needed) $C_0$-semigroup $\{T_t;t\ge0\}$, where $D_A\subset X$ is the domain of $A$. The question is the following:

If $u\in D_A$ and $f\in X$ satisfy the equation $$Au=f,$$ then $u=-\int_0^\infty T_t f dt$ ?

If the answer is yes, how to derive this formally? Or can anyone give some hints or reference? If no, can someone give some conditions such that the representation of $u$ hold true? Any comments will be appreciated.

Answer 1

$\int_{0}^{\infty}T_t fdt = \int_{0}^{\infty}T_t Au dt = \int_{0}^{\infty}\frac{d}{dt}(T_t u)dt= T_t u|_{t=0}^{\infty}=-u$ iff $T_tu\rightarrow 0$ as $t\rightarrow\infty$.