Simultaneous determination of initial value and source term for time-fractional wave-diffusion equations

We consider initial boundary value problems for time fractional diffusion-wave equations: $$d_t^{\alpha} u = -Au + \mu(t)f(x)$$ in a bounded domain where $\mu(t)f(x)$ describes a source and $\alpha \in (0,1) \cup (1,2)$, and $-A$ is a symmetric ellitpic operator with repect to the spatial variable $x$. We assume that $\mu(t) = 0$ for $t > T$:some time and choose $T_2>T_1>T$. We prove the uniqueness in simultaneously determining $f$ in $\Omega$, $\mu$ in $(0,T)$, and initial values of $u$ by data $u\vert_{\omega\times (T_1,T_2)}$, provided that the order $\alpha$ does not belong to a countably infinite set in $(0,1) \cup (1,2)$ which is characterized by $\mu$. The proof is based on the asymptotic behavior of the Mittag-Leffler functions.