Cauchy problem for general time fractional diffusion equation

TL;DR

This paper investigates the existence, positivity, and long-term behavior of solutions to a general time fractional diffusion equation involving a Caputo-type operator, using Fourier analysis and solution representation methods.

Contribution

It introduces new analysis techniques for the Cauchy problem with a general Caputo-type operator, extending previous results to broader fractional diffusion models.

Findings

Solutions exist and are positive under certain conditions

Long-time behavior of solutions is characterized

Methods are developed for equations with source terms

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.