Classical solution of an abstract Cauchy problem ($u^{\prime }=Lu+f$)

For a closed operator $L$ on a Banach space $H$, a positive-time classical solution belongs to $C([0,T],H)\cap C^1((0,T],H)$, takes values in $D(L)$ for positive time, and has $Lu\in C((0,T],H)$. It satisfies the differential equation and the initial condition in $H$. Classical regularity also at time zero requires the corresponding regularity of the initial datum, in particular $u_0\in D(L)$. For an analytic semigroup and Hölder continuous forcing, the variation-of-constants formula gives such a solution: subtracting the forcing at the current time turns the singular estimate $\Vert LS(r)\Vert \le C/r$ into the integrable bound $C{r}^{\theta -1}$.