  Analytic smooth solutions of a general, strongly parabolic semi-linear Cauchy
problem of $2m$-th order in $\mathbb{R}^N\times (0,T)$ with analytic
coefficients (in space and time variables) and analytic initial data (in space
variables) are investigated. They are expressed in terms of holomorphic
continuation of global (weak) solutions to the system valued in a suitable
Besov interpolation space of $B^{s;p,p}$-type at every time moment $t\in
[0,T]$. Given $0 < T'< T\leq \infty$, it is proved that any $B^{s;p,p}$-type
solution $u\colon \mathbb{R}^N\times (0,T)\to \mathbb{C}^M$ with analytic
initial data possesses a bounded holomorphic continuation $u(x + \mathrm{i}y,
\sigma + \mathrm{i}\tau)$ into a complex domain in $\mathbb{C}^N\times
\mathbb{C}$ defined by $(x,\sigma)\in \mathbb{R}^N\times (T',T)$, $|y| < A'$
and $|\tau | < B'$, where $A', B'> 0$ are constants depending upon~$T'$. The
proof uses the extension of a weak solution to a $B^{s;p,p}$-type solution in a
domain in $\mathbb{C}^N\times \mathbb{C}$, such that this extension satisfies
the Cauchy-Riemann equations. The holomorphic extension is obtained with a help
from holomorphic semigroups and maximal regularity theory for parabolic
problems in Besov interpolation spaces of $B^{s;p,p}$-type imbedded (densely
and continuously) into an $L^p$-type Lebesgue space. Applications include risk
models for European options in Mathematical Finance.
