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

Abstract

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.