We analyze the well-posedness and smoothing properties of a variable-order time-fractional wave partial differential equation in multiple space dimensions. Accordingly, we prove the unique determination of the variable order in this model with the observations of the unknown solutions on an arbitrarily small spatial domain over a sufficiently small time interval. The proved theorem provides a guidance where the measurements should be taken and ensure the unique identification of the variable order.

中文翻译：

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分数阶分数阶波动方程的唯一辨识

我们分析了在多个空间维度上的变阶时间分数波偏微分方程的适定性和平滑特性。因此，通过在足够小的时间间隔内在任意小的空间域上观察未知解，我们证明了该模型中变量阶数的唯一确定。证明定理为应该进行测量的方向提供了指导，并确保对可变阶次进行唯一标识。