Wellposedness and smoothing properties of history-state-based variable-order time-fractional diffusion equations

We prove the wellposedness of history-state-based variable-order linear time-fractional diffusion equations in multiple space dimensions. We also prove that the regularity of their solutions depends on the behavior of the variable order at the initial time t = 0, in addition to the usual smoothness assumptions. More precisely, we prove that their solutions have full regularity (i.e., the solutions can achieve high-order smoothness under high-order regularity assumptions of the data) as their integer-order analogs if the variable order has an integer limit at t = 0 or exhibits singular behaviors at t = 0 like in the case of the constant-order time-fractional diffusion equations if the variable order has a non-integer value at t = 0.