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

According to the discussion in


, 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!


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