= Classical solution of an abstract Cauchy problem
{title2=$u'=Lu+f$}
= Classical solutions of an abstract Cauchy problem
{synonym}
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 $\|LS(r)\|\leq C/r$ into the integrable bound $Cr^{\theta-1}$.
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