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?