I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing an existence theory for a system of equations. One of the equations in the system is not a PDE and so the system is not local: it's not written in the paper, but it should be an integral of $t$ taken from $0$ to $\infty$ (hence it does not satisfy a local PDE at every point). To clarify, the other two equations in the system are elliptic PDEs.

In order to have an existence theory for the system, one would have to show that the theory works when $\phi$ takes a 'wide range of possible values', but I find it unclear what it would mean for $\phi$ to take a sufficiently wide range of values.

Edit: System of equations added (first equation is the Dirac equation).

$D \psi_t = 0,$

$\phi = 2 \int_0^{\infty} \chi_t \bigg((w_t^{-})^2|\psi_t^{-}|^2 + (w_t^{+})^2|\psi_t^{+}|^2 \bigg) u_t^2 \: \text{d} t ,$

$H_{\Sigma} - Tr_{\Sigma} K = 0 .$

I will not define all the symbols but $\chi$ is an indicator function which is $1$ on the manifold being studied and $0$ outside of it.