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

We consider initial boundary value problems for time fractional diffusion-wave equations: d αt u = − A u + μ ( t ) f ( x ) in a bounded domain where μ(t)f (x) describes a source and α ∈ (0, 1) ∪ (1, 2), and −A is a symmetric ellitpic operator with repect to the spatial variable x. We assume that μ(t) = 0 for t > T :some time and choose T2 > T1 > T . We prove the uniqueness in simultaneously determining f in Ω, μ in (0,T), and initial values of u by data u|ω×(T1,T2), provided that the order α does not belong to a countably infinite set in (0, 1) ∪ (1, 2) which is characterized by μ. The proof is based on the asymptotic behavior of the Mittag-Leffler functions.