For a function $f(x,t)$ I'm aware of a fairly strong result about the PDE:
$\sigma^2 f_{xx} + f_t = 0,$
$f(x,0) = h(x)$,
which guarantees that any local maxima of $f$ or any of its derivatives in $x$ can only exist along the boundary $t = 0$. I'm wondering if a similar result holds for a more complicated version of the PDE:
$\sigma^2(g_{xx},g_x,g,x,t) g_{xx} + g_t = 0,$
$g(x,0) = h(x)$.
I've been able to show that any local maxima of $g$ and $g_x$ exist only along the boundary $t = 0$ but I am having trouble with $g_{xx}$.
I'm wondering whether or not the maximum principle can be extended this far, and if so what conditions I would need on the function $\sigma$ to make it work.