# A comparison principle for a nonlinear parabolic PDE

We know the following comparison principle holds for the diffusion equation: Suppose that $u(x,t)$ and $v(x,t)$ satisfy $$\begin{cases} u_t\ge \Delta u+F(x,t,u), \ \ &x\in\Omega,\ \ t>0,\\ u(x,0)\ge u_0(x),\ \ &x\in\Omega,\\ \frac{\displaystyle\partial u}{\displaystyle\partial n}\ge\phi, \ \ &x\in\partial\Omega,\ \ t>0, \end{cases}$$ and $$\begin{cases} v_t\le \Delta u+G(x,t,v), \ \ &x\in\Omega,\ \ t>0,\\ v(x,0)\le v_0(x),\ \ &x\in\Omega,\\ \frac{\displaystyle\partial v}{\displaystyle\partial n}\le\varphi, \ \ &x\in\partial\Omega,\ \ t>0, \end{cases}$$ respectively. If we further assume that $$F(x,t,w)\ge G(x,t,w), \quad x\in\Omega,\quad t>0,\quad w\in\mathbb{R},$$ $$u_0(x)\ge v_0(x), \quad x\in\Omega,$$ and $$\phi(x)\ge \varphi(x), \quad x\in\partial\Omega,\quad t>0$$ hold, then we have $u(x,t)\ge v(x,t)$ for $x\in\overline{\Omega}$, $t>0$.

According to the discussion in

https://mathoverflow.net/questions/163574/a-comparison-principle-for-parabolic-equation

, the comparison principle holds for the porous medium equation $u_t= \Delta (u^2)$. Then is it easy to conclude that the comparison principle automatically holds for $u_t= \Delta (u^2)+f(x,t,u)$?

Any idea or comment or suggestion is welcome, thank you in advanced!