What I have in mind is that on $(0,T)\times\Omega$, we have a parabolic pde operator $L$, we have unique solution to 


$Lu = f$ when f is Hoelder for some coefficient strictly between 0 and 1.


This is a theorem from Friedman's book on pdes. While i can plough through the proofs which were full of estimates, i don't get the intuitions why they work or why wouldn't they work (Or would they?) for an operator such as


$\bigtriangleup u - \partial\_{t\_1} - \partial\_{t\_2} = f$


Where f is a suitable function and the domain is something like $(0,T)\times(0,S)\times\Omega$?


The only answer i found regarding intuition was from User's guide to viscosity solution by crandall-ishii-lions page 52. It says


$u\_t+ F(x,u,Du,D^2u)=0$ more or less correspond to $\lambda t+ F = 0$ for large $\lambda$.


I don't quite understand what this means? Does it work if we do $u\_t+u\_s+F = 0$?
What is so nice about having a time evolution that does not (or maybe it does, but i am ignorant) work in higher dimensions?



What fails when we try to extend existence and unique for parabolic PDEs for 'PDEs which are 'parabolic in two components''?
Change the variables: define
$$
u(t,s,x)=v(\underbrace{\frac{t+s}{2}}\_{\tau},\underbrace{{t-s}}\_{\sigma},x).
$$
You get
$
\partial\_t u+\partial\_s u=\partial\_{\tau} v
$
and the equation becomes
$$
\partial\_{\tau} v+F(x,v,D\_x
v,D\_x^2 v)=0,$$
where the function $v$ depends on the real parameter $\sigma$.


