Variational formulation of time-fractional parabolic equations

We consider initial/boundary value problems for parabolic PDE ${?}_t^{\alpha }u?\Delta u=f$ with fractional Caputo derivative ${?}_t^{\alpha }$ of order $1∕2<\alpha <1$ as time derivative and the usual Laplacian $?\Delta$ as space derivative (also called fractional diffusion equations in the literature). We prove well-posedness of corresponding variational formulations based entirely on fractional Sobolev–Bochner spaces, and extend existing results for possible choices of the initial value for $u$ at $t=0$.