Long-time asymptotic estimate and a related inverse source problem for time-fractional wave equations

TL;DR

This paper derives long-time asymptotic estimates for time-fractional wave equations, revealing solution sign behavior and establishing uniqueness in an inverse source problem involving the temporal component.

Contribution

It provides the first long-time asymptotic analysis for time-fractional wave equations and applies it to prove uniqueness in an inverse source problem.

Findings

Solutions exhibit strict positivity or negativity for large time under certain conditions.

Asymptotic estimates help determine solution behavior over long time periods.

Uniqueness of the inverse source problem is established based on the asymptotic analysis.

Abstract

Lying between traditional parabolic and hyperbolic equations, time-fractional wave equations of order $\alpha\in(1,2)$ in time inherit both decaying and oscillating properties. In this article, we establish a long-time asymptotic estimate for homogeneous time-fractional wave equations, which readily implies the strict positivity/negativity of the solution for $t\gg1$ under some sign conditions on initial values. As a direct application, we prove the uniqueness for a related inverse source problem on determining the temporal component.