Unique determination of a time-dependent potential for wave equations from partial data

• Published in: Annales de l'Institut Henri Poincaré. Analyse non linéaire Vol. 34; no. 4; pp. 973 - 990
• Main Author: Kian, Yavar
• Format: Journal Article
• Language: English
• Published: Elsevier Masson SAS 01.07.2017; EMS
• Subjects: Analysis of PDEs; Carleman estimates; Inverse problems; Mathematics; Partial data; Time-dependent potential; Uniqueness; Wave equation; Wave equation; Inverse problems; 35R30; Uniqueness; 35L05; Time-dependent potential; Carleman estimates; Partial data; scalar time-dependent potential; wave equation; partial data Mathematics subject classification 2010 : 35R30
• ISSN: 0294-1449; 1873-1430
• DOI: 10.1016/j.anihpc.2016.07.003

We consider the inverse problem of determining a time-dependent potential q, appearing in the wave equation ∂t2u−Δxu+q(t,x)u=0 in Q=(0,T)×Ω with T>0 and Ω a C2 bounded domain of Rn, n⩾2, from partial observations of the solutions on ∂Q. More precisely, we look for observations on ∂Q that allows to recover uniquely a general time-dependent potential q without involving an important set of data. We prove global unique determination of q∈L∞(Q) from partial observations on ∂Q. Besides being nonlinear, this problem is related to the inverse problem of determining a semilinear term appearing in a nonlinear hyperbolic equation from boundary measurements.