I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.

Definition: A non-local PDE is a PDE which has some terms which depend on the global behavior/value of the unknown function.

A prototypical non-local (or integro-differential equation) is the following(posed on some set $\Omega\in\mathbb{R}$:

$\partial_tu(x,t)=c\Delta u(x,t)+a(x)\int_{\Omega}K(x,y)u(y,t)dy$

where $K(x,y)=K(x-y)$ is a kernel function (could be $e^{-(x-y)^2}$ for example).

However, I am not sure if all the lessons from the local PDE spectral theory apply here. So I am looking for a good reference/monograph/review paper which describes the subtle points in nonlocal spectral theory and/or stability analysis, especially issues that do not arise in local PDE theory.

Some examples of stuff I am looking for:

1). One example is Evans function computation, whose theory is only complete for the local case.

2). Does there exist a Strum-Liouville type theory for non-local operators ?