The backward problem for a multi-term time-fractional diffusion equation

TL;DR

This paper studies the backward problem for a multi-term time-fractional diffusion equation, demonstrating conditions for existence, uniqueness, stability, and the solution's smoothing properties despite inherent ill-posedness.

Contribution

It provides a rigorous analysis of the backward problem's well-posedness under smooth data and characterizes the asymptotic behavior of the multinomial Mittag-Leffler function involved.

Findings

Solutions exist, are unique, and stable with sufficiently smooth data.

The solution belongs to the domain of the operator A for any positive time.

Conditional stability can be achieved with an appropriate a priori bound.

Abstract

This paper is devoted to the investigation of the backward problem for a multi-term time-fractional diffusion equation. Backward problems for fractional diffusion equations are typically studied using regularization methods due to their ill-posedness in the sense of Hadamard; that is, a small change in u(T) may lead to large changes in the initial data. Nevertheless, we show that if sufficiently smooth current data are considered, then the solution exists, is unique, and is stable. A principal difficulty in the analysis of the backward problem stems from the structure of the solution, in which the multinomial Mittag-Leffler function appears in the denominator. Accordingly, a precise characterization of the asymptotic behavior of this function is required. Such asymptotic properties are nontrivial and have been rigorously established in the authors' recent work, which serves as a fundamental basis for the present study. In addition, we investigate the conditional stability of the backward problem. It is shown that, although the problem is ill-posed in general, stability can be restored under an appropriate a priori bound imposed on the initial data. The main novelty of the paper lies in proving the best smoothing property of the solution, showing that it belongs to the domain of the operator A for any positive time.