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 ∂2 t u − ∆xu + 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 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 ∈ W 1,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.