The unique identification of variable-order fractional wave equations

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.