This study addresses the inverse source problem for the fractional diffusion-wave equation, characterized by a source comprising spatial and temporal components. The investigation is primarily concerned with practical scenarios where data is collected subsequent to an incident. We establish the uniqueness of either the spatial or the temporal component of the source, provided that the temporal component exhibits an asymptotic expansion at infinity. Taking anomalous diffusion as a typical example, we gather the asymptotic behavior of one of the following quantities: the concentration on partial interior region or at a point inside the region, or the flux on partial boundary or at a point on the boundary. The proof is based on the asymptotic expansion of the solution to the fractional diffusion-wave equation. Notably, our approach does not rely on the conventional vanishing conditions for the source components. We also observe that the extent of uniqueness is dependent on the fractional order.

翻译：本研究探讨分数阶扩散波方程的反源问题，该方程源项由空间与时间分量构成。研究主要关注数据在事件发生后采集的实际场景。我们证明，在时间分量于无穷远处具有渐近展开的条件下，源项的空间或时间分量具有唯一性。以反常扩散为例，我们获取以下任一量的渐近行为：部分内部区域或区域内某点的浓度，或部分边界或边界上某点的通量。证明基于分数阶扩散波方程解的渐近展开。值得注意的是，本方法不依赖于源项分量需满足的传统零化条件。我们还观察到，唯一性的程度取决于分数阶阶数。