  This article is devoted to forward and inverse problems associated with
time-independent semilinear nonlocal wave equations. We first establish
comprehensive well-posedness results for some semilinear nonlocal wave
equations. The main challenge is due to the low regularity of the solutions of
linear nonlocal wave equations. We then turn to an inverse problem of
recovering the nonlinearity of the equation. More precisely, we show that the
exterior Dirichlet-to-Neumann map uniquely determines homogeneous
nonlinearities of the form $f(x,u)$ under certain growth conditions. On the
other hand, we also prove that initial data can be determined by using passive
measurements under certain nonlinearity conditions. The main tools used for the
inverse problem are the unique continuation principle of the fractional
Laplacian and a Runge approximation property. The results hold for any spatial
dimension $n\in \N$.
