Variational formulation of time-fractional parabolic equations

We consider initial/boundary value problems for parabolic PDE ∂tαu−Δu=f with fractional Caputo derivative ∂tα of order 1∕2<α<1 as time derivative and the usual Laplacian −Δ 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.