Cauchy problem for general time fractional diffusion equation

Abstract

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.