The dependence on fractional orders of mild solutions to the fractional diffusion equation with memory

In this paper we investigate the Cauchy problem for a fractional diffusion equation and the time-fractional derivative is taken in the Caputo type sense. We give a representation of solutions under Fourier series and analyze initial value problems for the semi-linear fractional diffusion equation with a memory term. We also discuss the stability of the fractional derivative order for the time under some assumptions on the input data. Our key idea is to use Mittag-Leffler functions, the Banach fixed point theorem, and some Sobolev embeddings.

Automated summary

This paper investigates the Cauchy problem for a fractional diffusion equation in the Caputo type sense. It provides a representation of solutions using Fourier series and analyzes initial value problems for the semi-linear fractional diffusion equation with a memory term. The stability of the fractional derivative order for the time is also discussed under certain assumptions on the input data, using Mittag-Leffler functions, the Banach fixed point theorem, and Sobolev embeddings.