Spectral properties of Non-local Differential operators on real line

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 ?

• You won't get answers, unless you explain what you means by non-local. Notions of non-something are vague. You can have a precise idea of what you mean, while nobody shares your understanding. – Denis Serre Jan 25 '18 at 16:30
• ok, I add a definition. – mystupid_acct Jan 25 '18 at 16:37
• A more common name for these kind of operators/equations is integro-differential. – Igor Khavkine Jan 25 '18 at 23:05
• It seems to me that a sane framework for such issues is that of pseudo-differential operators, or possibly Fourier integral operators. In both cases, there is a substantial body of existing work. But if by "Sturm-Liouville theory" you mean an implicit assertion that the spectrum (in whatever useful sense) is discrete, you are surely doomed to be frustrated in any general situation, because even the simplest examples can have continuous spectrum... – paul garrett Jan 26 '18 at 1:22
• To push this more into the microlocal (ie pseudodifferential) direction: If $K(x,y) = \int e^{i(x-y)\xi} |\xi|^2 d\xi$ you get again (modulo constants) the Laplacian, so for arbitrary kernels you cannot treat $K$ as a small perturbation. So you need to add some condition on the regularity of $K$. – mcd Jan 26 '18 at 9:10