I derived the following differential equation from a problem in signal analysis.
$$ 0 = f - u - \frac{1}{2\lambda}\nabla^T\cdot\left(e^{-\lVert \nabla u \rVert/\lambda} \frac{\nabla u}{\lVert \nabla u\rVert}\right) $$
here $u = u(x,y)$, $f$ is infinitely times differentiable, the domain is a rectangle $(x,y) \in [0,X]\times[0,Y]$.
The question is... if $\nabla u$ has very large magnitude compared to $\lambda > 0$ then the solution is roughly $u = f$, however I do struggle to understand what happens when $\nabla u$ is very small. Is there something I can say even roughly? It seems to me it would resemble a diffusion equation, however there's the gradient normalized (so it's a unitary vector field) so I can't really say much about it.
Any help would be appreciated.