Well-posedness of initial-boundary value problem for time-fractional diffusion/wave equation with time-dependent coefficients

We consider the well-posedness of the initial-boundary value problem for a time-fractional partial differential equation with the fractional order lying in (1,2]. For the case of time-dependent coefficients, it is difficult to give an explicit solution formula by the eigenfunction expansion method. In order to deal with the case of time-varying coefficients, we first show the unique existence and regularity of solution to a system of time-fractional ordinary differential equations. Then, the unique differential equation and improved regularity are derived by using the Galerkin method.