  We study the well-posedness for initial boundary value problems associated
with time fractional diffusion equations with non-homogenous boundary and
initial values. We consider both weak and strong solutions for the problems.
For weak solutions, we introduce a new definition of solutions which allows to
prove the existence of solution to the initial boundary value problems with
non-zero initial and boundary values and non-homogeneous terms lying in some
arbitrary negative-order Sobolev spaces. For strong solutions, we introduce an
optimal compatibility condition and prove the existence of the solutions. We
introduce also some sharp conditions guaranteeing the existence of solutions
with more regularity in time and space.
