I was told the comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. I also know comparison principle can be regarded as the nonlinear version of maximum principle. I am vague about distinguishing these two theorem in a precise manner. Here is a version of weak MP from Evans p.368
Assume $u\in C^{(2,1)}(\Omega_T)\cap C(\bar{\Omega}_T)$ and \begin{equation} c\equiv 0\quad\text{in }\Omega_T \end{equation}
If \begin{equation}%\label{subconw1} u_t+Lu\le 0\quad\text{in }\Omega_T \end{equation} then \begin{equation*} \max_{\substack{\bar{\Omega}_T}}u=\max_{\substack{\bar{\Gamma}_T}}u \end{equation*} Likewise, if \begin{equation}%\label{supconw1} u_t+Lu\ge 0\quad\text{in }\Omega_T \end{equation} then \begin{equation*} \min_{\substack{\bar{\Omega}_T}}u=\min_{\substack{\bar{\Gamma}_T}}u \end{equation*}
I would like a comparison principle theorem by modifying above. Please help! (This is what I know https://www.ma.utexas.edu/mediawiki/index.php/Comparison_principle#Parabolic_case )