We prove the wellposedness of a nonlinear variable-order fractional ordinary differential equation and the regularity of its solutions, which is determined by the values of the variable order and its high-order derivatives at time t=0. 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 exhibits singular behaviors at t=0 like in the case of the constant-order fractional differential equations if the variable order has a non-integer value at time t=0.

We then extend the developed techniques to prove the wellposedness of a variable-order linear time-fractional diffusion equation in multiple space dimensions and the regularity of its solutions, which depends on the behavior of the variable order at t=0 in the similar manner to that of the fractional ordinary differential equations. 

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