On the determination of nonlinear terms appearing in semilinear hyperbolic equations

We consider the inverse problem of determining the shape of a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary ( M , g ) of dimension n = 2 , 3 . We prove results of unique recovery of the nonlinear term F ( t , x , u ) , appearing in the equation∂ t 2 u − Δ g u + F ( t , x , u ) = 0 on ( 0 , T ) × M with T > 0 , from partial knowledge of the solutions u on the lateral boundary ( 0 , T ) × ∂ M . We obtain, what seems to be, the first result of determination of the expression F ( t , x , u ) on the boundary x ∈ ∂ M for such a general class of nonlinear terms. With additional assumptions on the manifold and some extended measurements at t = 0 and t = T , we prove also the recovery of F inside the manifold x ∈ M .