There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the original interval as its diagonal. In this short note we provide a direct argument that the analogue of this result holds in any dimension. For the heat equation on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ at positive time all solutions are analytically extendable to a geometrically determined subdomain $\mathcal{E}(Ω)$ of $\mathbb{C}^d$ containing $Ω$. This domain is sharp in the sense that there is no larger domain for which this is true. If $Ω$ is a ball we prove an almost converse of this theorem. Any function that is analytic in an open neighborhood of $\mathcal{E}(Ω)$ is reachable in the sense that it can be obtained from a solution of the heat equation at positive time. This is based on an analysis of the convergence of heat equation solutions in the complex domain using the boundary layer potential method for the heat equation. The converse theorem is obtained using a Wick rotation into the complex domain that is justified by our results. This gives a simple explanation for the shapes appearing in the one-dimensional analysis of the problem in the literature. It also provides a new short and conceptual proof in that case.
Exact Controllability of Nonlinear Heat Equations in Spaces of Analytic Functions
利用Carleman估计可以建立非线性抛物型方程轨迹的可控性。然而，通过这种方法，并不清楚如何确定给定的函数是否确实是可达的。在Gevrey空间中，通过考虑一维非线性热方程，基于控制输入的直接方法研究了抛物型方程的可达状态.假设非线性部分是空间变量x、未知量y及其导数$\partial$ x y的解析函数。通过研究空间变量的非线性Cauchy问题和空间导数喷流与时间导数喷流之间的关系，得到了小初值和小终值下的精确能控性结果，该结果可推广为复平面上某个球上的解析函数. 2010年数学学科分类：35K40、93B05
