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

TL;DR

This paper proves that solutions to certain strongly parabolic semi-linear PDEs with analytic coefficients and initial data are holomorphically extendable in space and time, using advanced functional analysis techniques.

Contribution

It establishes the space-time analyticity and holomorphic continuation of weak solutions in Besov spaces for a broad class of semilinear parabolic systems with variable coefficients.

Findings

Solutions have bounded holomorphic extensions in complex domains.

Holomorphic continuation relies on semigroup theory and maximal regularity in Besov spaces.

Results apply to models in mathematical finance, such as European options risk models.

Abstract

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.