AI 中文总结
针对$n \geq 2$的$\mathbb{R}^n$区域中的热算子，研究从初值到终值映射确定有界超指数衰减热生成系数$V$的反问题，通过构造指数增长解和高斯矩生成函数推导加权$L^2$估计，将薛定谔方程的初值到终态反问题扩展至抛物场景。
AI 中文摘要
我们研究介质中热方程的一个反问题，该介质产生的内部热生成速率与热密度成正比。目标是通过初值到终值映射来确定热生成系数$V$，该映射将每个初始热密度与其对应的最终热密度关联起来。我们针对$n \geq 2$、以$\mathbb{R}^n$为模型的区域中的热算子，陈述并证明了一个唯一性结果。我们假设$V$是有界的且超指数衰减。我们的方法依赖于为对应的热算子构造指数增长的解。我们方法的一个关键贡献是提供了一个加权$L^2$估计，该估计源自高斯随机变量的矩生成函数。这将先前针对薛定谔方程研究过的初值到终态反问题，扩展到了抛物型场景中。
英文摘要
We study an inverse problem for the heat equation in a medium that generates internal heat at a rate proportional to the heat density. The goal consists of determining the heat-generation coefficient $V$ from the knowledge of the initial-to-final map, which assigns to every initial heat density its final heat density. We state and prove a uniqueness result for the heat operator in a region modelled by $\mathbb{R}^n$ with $n \geq 2$. We assume that $V$ is bounded and decays super-exponentially. Our approach relies on constructing exponentially-growing solutions for the corresponding heat operator. A key contribution of our approach is to provide a weighted $L^2$-estimate, which follows from the moment generating function of a Gaussian random variable. This extends the initial-to-final-state inverse problem, previously studied for the Schrödinger equation, to the parabolic setting.
Comments21 pages
