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Analytic smooth solutions of a general, strongly parabolic semi-linear Cauchyproblem of $2m$-th order in $\mathbb{R}^N\times (0,T)$ with analyticcoefficients (in space and time variables) and analytic initial data (in spacevariables) are investigated. They are expressed in terms of holomorphiccontinuation of global (weak) solutions to the system valued in a suitableBesov 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 $B^{s;p,p}$-typesolution $u\colon \mathbb{R}^N\times (0,T)\to \mathbb{C}^M$ with analyticinitial 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~$T'$. Theproof uses the extension of a weak solution to a $B^{s;p,p}$-type solution in adomain in $\mathbb{C}^N\times \mathbb{C}$, such that this extension satisfiesthe Cauchy-Riemann equations. The holomorphic extension is obtained with a helpfrom holomorphic semigroups and maximal regularity theory for parabolicproblems in Besov interpolation spaces of $B^{s;p,p}$-type imbedded (denselyand continuously) into an $L^p$-type Lebesgue space. Applications include riskmodels for European options in Mathematical Finance.

Space-Time Analyticity of Weak Solutions to Semilinear Parabolic Systems with Variable Coefficients