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

2023/07/31 by Loreti, Paola, Sforza, Daniela, Yamamoto, Masahiro
#35R11 #35R25 #35R30 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.16665

Abstract

We consider initial boundary value problems for time fractional diffusion-wave equations: d_t^α u = -Au + 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\vert_{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.

Related

• Decay rates and initial values for time-fractional diffusion-wave equations
• Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation
• Uniqueness of solution to boundary value problems for time-fractional wave equations
• Identification of time-varying source term in time-fractional diffusion equations
• Uniqueness of the inverse source problem for fractional diffusion-wave equations