Wellposedness and regularity of the variable-order time-fractional diffusion equations

Abstract

It was proved that the solution to a time-fractional partial differential equation lacks regularity at the initial time t=0, which is often inconsistent with the physical problem being modeled even though the nature of the physical problem away from t=0 is well captured by a time-fractional partial differential equation. We argue that the fundamental reason why this phenomenon occurs lies in the incompatibility between the nonlocality of the time-fractional partial differential equations and the locality of the classical initial condition at t=0. We show that the variable-order time-fractional partial differential equations, in which the order varies in time as t goes to 0 to accommodate the impact of the local initial condition at time t=0, turn out to be a natural candidate to eliminate the nonphysical singularity of the solutions to the (constant-order) time-fractional partial differential equations and open up opportunities for modeling multiphysics phenomena from nonlocal to local dynamics and vice versa. We prove the wellposedness of a variable-order linear time-fractional diffusion partial differential equation in multiple space dimensions. We also prove that the regularity of its solution 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 solution has 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 time-fractional partial differential equations if the variable order has a non-integer value at time t=0.