  In the present work, we consider the Cauchy problem for the time fractional
diffusion equation involving the general Caputo-type differential operator
proposed by Kochubei. First, the existence, the positivity and the long time
behavior of solutions of the equation without source term are established by
using the Fourier analysis technique. Then, based on the representation of the
solution of the inhomogenous linear ordinary differential equation with the
general operator, the similar problems for the diffusion equation with source
term are studied.

Analysis of PDEs