Determination of singular time-dependent coefficients for wave equations from full and partial data

2017/06/22 by Guanghui Hu, Yavar Kian, Hu, Guanghui +1
Earth and Planetary Sciences · Engineering · Mathematics · #35L05 #35R30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Microwave Imaging and Scattering Analysis #Numerical methods in inverse problems #Seismic Imaging and Inversion Techniques

paper · pdf · doi:10.48550/arxiv.1706.07212

openalex publication_date 2017/06/22 · openalex created_date 2017/06/30 · openalex updated_date 2026/07/28

Abstract

We study the problem of determining uniquely a time-dependent singular potential q, appearing in the wave equation ∂_t²u-Δ_x u+q(t,x)u=0 in Q=(0,T)×Ω with T>0 and Ω a \mathcal C² bounded domain of \mathbb Rⁿ, n≥2. We start by considering the unique determination of some singular time-dependent coefficients from observations on ∂ Q. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.

Citations

• Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials
• Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations
• The L^p Carleman estimate and a partial data inverse problem
• Stability result for elliptic inverse periodic coefficient problem by\n partial Dirichlet-to-Neumann map

Related

• Unique determination of a time-dependent potential for wave equations\n from partial data
• Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data
• Recovering time-dependent singular coefficients of the wave-equation-One Dimensional Case
• Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations
• Inhomogeneous wave equation with t-dependent singular coefficients