发表机构
University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次研究二阶线性抛物型方程依赖于(x,t)的系数与初始条件的同时重建反问题，获稳定性估计与唯一性定理，对数稳定性估计为首次所得，可应用于舆论预测与流行病追踪。
AI 中文摘要
本研究首次针对二阶一般线性抛物型方程中依赖于(x,t)的未知系数与未知初始条件的同时重建反问题展开研究。输入数据依赖于(n+1)个变量，因此属于形式确定型数据。研究获得了该反问题的稳定性估计与唯一性定理，作为副产品，首次得到了带有时间反转数据的抛物型方程及不等式初始条件的对数稳定性估计。目前已知的关于带有形式确定型输入数据、未知系数的抛物型方程反问题的所有研究，均假设这些系数仅依赖于x或仅依赖于t，且初始条件被假定为已知，仅正文引用的两篇文献例外。本文所研究的反问题具有两项潜在应用：第一项是在平均场博弈（Mean Field Games）理论框架下的公众舆论预测；第二项是对流行病的时空暴发进行追踪。
英文摘要
This is the first publication, in which an inverse problem of the simultaneous reconstruction of an unknown (x,t) - dependent coefficient and an unknown initial condition in a general linear parabolic equation of the second order is considered. The input data depend on (n+1) variables and are, therefore, formally determined ones. Stability estimates and uniqueness theorem are obtained for this inverse problem. As a by-product, logarithmic stability estimates for initial conditions of parabolic equations and inequalities with time reversed data are obtained for the first time. All known publications about inverse problems for parabolic equations with formally determined input data and unknown coefficients assume that those coefficients depend either only on x or only on t. In addition, the initial condition is assumed to be known, except of two publications cited in the text. The inverse problem of this paper has two potential applications. The first one is in forecasting of public opinions in the framework of the Mean Field Games theory. The second one is in tracking spatiotemporal outbreaks of epidemics.
