When can we apply sandwich or blow-up criterion?

Question

Suppose we have two PDE's like

$$\dfrac{\partial}{\partial t} f - F(\dfrac{\partial f}{\partial x_1}, \dfrac{\partial f}{\partial x_2}, \dfrac{\partial f}{\partial x_3}, f) = 0 $$

and

$$\dfrac{\partial }{\partial t}g - G( \dfrac{\partial g}{\partial x_1}, \dfrac{\partial g}{\partial x_2}, \dfrac{\partial g}{\partial x_3}, g) = 0 $$

if $F\le G$ for all $(x,t)$ we see that $\dfrac{\partial}{\partial t} f \le \dfrac{\partial}{\partial t} g$. If we use the same initial conditions $f(x,0) = g(x,0)$ can we conclude that $g$ will blow up before $f$ can? In particular is $f\le g$ for all time $t$?

Answer 1

Integrating the inequality $\partial_t f \leq \partial_t g$ and using the fundamental theorem of calculus shows that

$$ f(x,t)-f(x,0) \leq g(x,t)-g(x,0) $$

So if the initial conditions are equal, $f\leq g$. If they blow up, the singularity can still occur at the same time despite having the same initial conditions, like $f(x,t) = (1-t)^{-1}$ and $g(x,t) = (1+t)(1-t)^{-1}$.