Studies on an inverse source problem for a space-time fractional diffusion equation by constructing a strong maximum principle

TL;DR

This paper develops a strong maximum principle for space-time fractional diffusion equations with nonlocal operators, enabling the unique determination of the temporal component of an inverse source problem.

Contribution

It introduces a new strong maximum principle for fractional diffusion equations with nonlocal operators, facilitating inverse problem solutions.

Findings

Established a weak Harnack's inequality for the equation.

Proved a strong maximum principle based on the inequality.

Achieved uniqueness in determining the temporal source component.

Abstract

In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result for an inverse source problem on the determination of the temporal component of the inhomogeneous term.