I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing 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).

The paper says in order to have an existence theory for the system, one would have to show that the theory works when the quantity defined by the integral expression takes a 'wide range of possible values'.

I find it unclear what is meant by a 'wide range': is there some sense in which this would be understood in PDE analysis? What would it mean for the range to be 'sufficiently wide'? Does the fact that the equation does not satisfy a PDE locally mean it cannot have an existence theory or useful regularity result?