On the determination of nonlinear terms appearing in semilinear hyperbolic equations

We consider the inverse problem of determining 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 ${\mathrm{\partial }}_t^{2}u-{\mathrm{\Delta }}_{g}u+F(t,x,u)=0$ on $(0,T)\times M$ with $T>0$, from some partial knowledge of the solutions $u$ on the boundary of the time-space cylindrical manifold $(0,T)\times M$ or on the lateral boundary $(0,T)\times \mathrm{\partial }M$. We determine the expression $F(t,x,u)$ both on the boundary $x\in \mathrm{\partial }M$ and inside the manifold $x\in M$.