Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities

The main purpose of this article is to establish the Runge-type approximation in $L^2(0,T;\widetilde{H}^s(\Omega))$ for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calder\'on problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calder\'on problem of nonlocal wave equations with polyhomogeneous nonlinearities.