Consider this integro-differential heat equation taken from SciPy documentation page:
$ \nabla^2 P = \alpha \left(\iint_\Omega \cosh(P)dx dy \right)^2 $
which was found in this question.
In the considered python code the Newton-Krylov iterations are applied and perform quite nice for $\alpha=10$. However this method fails when $\alpha$ is changed from 10 to 15 or more. What might be the reason for this behaviour and is it possible to improve the situation?
Any references are appreciated, thank you.
EDIT: in the first part of the question I asked about possible physical background behind this problem. Several days after I decided to address to scipy-dev newsgroup and almost immediately received a reply from Pauli Virtanen:
This is just some random equation. There's no physical or otherwise background for it.
This information makes further discussion of the problem much less interesting, so I's rather close this question.