Recovery of Time-Dependent Damping Coefficients and Potentials Appearing in Wave Equations from Partial Data

We consider the inverse problem of determining a time-dependent damping coefficient a and a time-dependent potential q, appearing in the wave equation ∂ₜ²u-Δₓ u+a(t,x)∂ₜu+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 allow us to determine uniquely a large class of time-dependent damping coefficients a and time-dependent potentials q without involving an important set of data. Assuming that a is known on ∂ Q, we prove global unique determination of a∈ W1,p(Q) with $p>n+1$, and q∈ L^∞(Q) from partial observations on ∂ Q. Our problem is related to the determination of nonlinear terms appearing in nonlinear wave equations.