On the reachable space for parabolic equations

TL;DR

This paper characterizes the reachable states for heat equations with lower order terms in a Euclidean ball, showing they correspond to functions extendable as holomorphic functions in specific complex domains, with results for both linear and semilinear cases.

Contribution

It provides a precise description of the reachable space for heat equations with lower order terms, extending previous results to semilinear cases and complex domain characterizations.

Findings

Reachable states are holomorphically extendable functions in certain complex domains.

The results apply to both linear and small-data semilinear heat equations.

The proofs rely on well-posedness in spaces of holomorphic functions.

Abstract

In this article, we provide a description of the reachable space for the heat equation with various lower order terms, set in the euclidean ball of $\mathbb{R}^d$ centered at $0$ and of radius one and controlled from the whole external boundary. Namely, we consider the case of linear heat equations with lower order terms of order $0$ and $1$, and the case of a semilinear heat equations. In the linear case, we prove that any function which can be extended as an holomorphic function in a set of the form $\Omega_\alpha = \{ z\in\mathbb{C}^d \big| |\Re(z)| + \alpha |\Im(z)| < 1\}$ for some $\alpha \in (0,1)$ and which admits a continuous extension up to $\overline\Omega_\alpha$ belongs to the reachable space. In the semilinear case, we prove a similar result for sufficiently small data. Our proofs are based on well-posedness results for the heat equation in a suitable space of holomorphic functions over $\Omega_\alpha$ for $\alpha > 1$.