Uniqueness in the inversion of distributed orders in ultraslow diffusion equations

Abstract

This paper deals with the distributed order time-fractional diffusion equations with non-homogeneous Dirichlet (Neumann) boundary condition. We first prove the wellposedness of the forward problem by means of eigenfunction expansion, which ensures 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. Finally we show a uniqueness result for the inverse problem in determining the weight function in the distributed order time derivative from one point observation.