  This paper deals with the distributed order time-fractional diffusion
equations with non-homogeneous Dirichlet (Nuemann) boundary condition. We first
prove the wellposedness of the weak solution to the initial boundary value
problem for the distributed order time-fractional diffusion equation by means
of eigenfunction expansion, which ensure that the weak solution has the
classical derivatives. We next give a Harnack type inequality of the solution
in the frequency domain under the Laplace transform, from which we further show
a uniqueness result for an inverse problem in determining the weight function
in the distributed order time derivative from point observation.
