AI 中文总结
该研究针对共形横截各向异性流形上的抛物方程，利用部分输入输出数据，结合多种分析工具证明了时变势的唯一可恢复性。
AI 中文摘要
我们研究共形横截各向异性流形上抛物方程中时变势的恢复问题，输入为初始状态和支撑于前端面附近的侧向Dirichlet数据，观测量为终端状态和后端面附近的Neumann迹。假设横截流形上衰减测地线射线变换对所有足够小的常数衰减系数满足单射性，我们证明这类部分输入输出数据可唯一确定该势，且未对横截流形施加任何单性假设。证明结合了共形约化、横截高斯束拟模、边界Carleman估计以及具有指定侧向支撑的几何光学解。
英文摘要
We study the recovery of a time-dependent potential in a parabolic equation on a conformally transversally anisotropic manifold. The initial state and a lateral Dirichlet datum supported near the front face are used as inputs, while the terminal state and the Neumann trace near the back face are observed. Assuming injectivity of the attenuated geodesic ray transform on the transversal manifold for all sufficiently small constant attenuations, we prove that these partial input-output data uniquely determine the potential. No simplicity assumption is imposed on the transversal manifold. The proof combines a conformal reduction, transversal Gaussian beam quasimodes, boundary Carleman estimates, and geometric optics solutions with prescribed lateral supports.
Comments30 pages, no figures
