Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data

Yavar Kian (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E8 Dynamique quantique et analyse spectrale

We consider the inverse problem of determining a time-dependent damping coefficient a and a time-dependent potential q, appearing in the wave equation ∂ 2 t u − ∆xu + a(t, x)∂tu + q(t, x)u = 0 in Q = (0, T) × Ω, with T > 0 and Ω a C 2 bounded domain of R n , n 2, from partial observations of the solutions on ∂Q. More precisely, we look for observations on ∂Q that allow to determine uniquely a large class of time-dependent damping coefficients a and time-dependent potentials q without involving an important set of data. We prove global unique determination of a ∈ W 1,p (Q), with p > n + 1, and q ∈ L ∞ (Q) from partial observations on ∂Q.

Mots clés: time-dependent potential; time-dependent damping coefficient; Inverse problems; wave equation; uniqueness; Carleman estimates; partial data

Domaines: Equations aux dérivées partielles [math.AP]

Yavar Kian. Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data. SIAM Journal on Mathematical Analysis, 2016, 48 (6), pp.4021-4046. ⟨hal-01295818v2⟩