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

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 $t\gg 1$ 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.