Wellposedness and regularity of a variable-order space-time fractional diffusion equation

We prove wellposedness of a variable-order linear space-time fractional diffusion equation in multiple space dimensions. In addition we prove that the regularity of its solutions depends on the behavior of the variable order (and its derivatives) at time t = 0, in addition to the usual smoothness assumptions. More precisely, we prove that its solutions have full regularity like its integer-order analogue if the variable order has an integer limit at t = 0 or have certain singularity at t = 0 like its constant-order fractional analogue if the variable order has a non-integer value at time t = 0.