I see. If we suppose $\nabla u$ to exist so we don't need to worry about the Laplacian. I think under this condition it may be hopeful. I want to work on a different equation where the standard method is not amenable so I want to try this method.

One last thing: is it obvious that the Lipschitz constant is independent of $x \in \Omega$ and $t$? When $f:[a,b] \to \mathbb{R}$ then (I believe) the Lipschitz constant depends on $b$.

Thanks. How did you get your first displayed equation? I could only get $f(\frac u2) -\frac 12f(\frac v2) \leq \frac 12f(\frac{2u-v}{2})$? Well I guess we don't need that second line anyway.