从边界测量识别反应-扩散方程中的未知时滞和空间变化系数
Identifying unknown time delay and spatially varying coefficients in a reaction-diffusion equation from boundary measurements
- School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）
- Department of Mathematics, City University of Hong Kong（香港城市大学数学系）
- School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对含多个延迟项的线性反应-扩散系统，利用边界通量测量和奇异时间行为，同时恢复未知时滞和空间变化系数，并建立Lipschitz稳定性估计，为首个此类结果。
AI中文摘要：
本文研究了一类包含多个延迟贡献的线性反应-扩散系统的逆问题，即延迟扩散、延迟时间导数和延迟源项。目标是从边界通量测量中同时恢复未知时间滞后$\ au>0$和空间异质系数。恢复策略利用了由预设初始历史与边界数据之间的不兼容性所产生的奇异时间行为。与先前针对延迟方程的逆问题（假设$\ au$已知且限于ODE或抽象设置）相比，我们的方法在具有空间变化系数的抛物型PDE框架中运作。一旦识别出$\ au$，在时间无关系数$p=p(x)$，$q=q(x)$的情况下，我们通过Carleman不等式建立了Lipschitz稳定性估计。这是首个从边界数据同时恢复延迟抛物型PDE中未知延迟和空间依赖系数的结果。
英文摘要：
This work investigates an inverse problem for a general class of linear reaction-diffusion systems incorporating multiple delayed contributions, namely retarded diffusion, retarded time derivatives, and retarded source terms. The objective is to simultaneously recover the unknown time lag $τ>0$ and spatially heterogeneous coefficients from boundary flux measurements alone. The recovery strategy exploits the singular temporal behavior generated by an incompatibility between the prescribed initial history and the boundary data. In contrast to prior inverse problems for delay equations, which assume $τ$ known and are confined to ODE or abstract settings, our approach operates in a parabolic PDE framework with spatially varying coefficients. Once $τ$ is identified, we establish a Lipschitz stability estimate via a Carleman inequality, in the case of time-independent coefficients $p=p(x)$, $q=q(x)$. This is the first result to simultaneously recover an unknown delay and spatially dependent coefficients in a delayed parabolic PDE from boundary data.
