I'm study by myself parabolic PDEs by Avner Friedman's book. Initially, Friedman starts the first section of chapter $2$ with the following:
Consider the operator
$$(1.1) \ Lu \equiv \sum_{i,j=1}^n a_{ij}(x,t) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{i=1}^n b_i(x,t) \frac{\partial u}{\partial x_i} + c(x,t) u - \frac{\partial u}{\partial t}$$
in an $(n+1)$-dimensional domain $D$. We list some assumptions that will be needed in the future:
(A) $L$ is parabolic in $D$, i.e., for every $(x,t) \in D$ and for any real vector $\xi \neq 0$, $\sum_{i,j} a_{ij} \xi_i \xi_j > 0$;
(B) the coefficients of $L$ are continuous functions in $D$;
(C) $c(x,t) \leq 0$ in $D$.
The functions $u$ in $(1.1)$ are always assumed to have two continuous $x$-derivatives and one continuous $t$-derivative in $D$.
$\textbf{Notation.}$ For any point $P^0 = (x^0,t^0)$ in $D$, we denote by $S(P^0)$ the set of all points $Q$ in $D$ which can be connected to $P^0$ by a simple continuous curve in $D$ along which the $t$-coordinate is non-decreasing from $Q$ to $P^0$.
Further, he states the Weak Parabolic Maximum Principle as follows:
Let $(A), (B), (C)$ hold and let $D$ be bounded. Assume that $u$ is a continuous function in $\overline{D}$ and let $Lu \geq 0$ ($Lu \leq 0$) in $D$. Then for each $P \in D$ for which $u$ has a positive maximum (negative minimum) in $\overline{S(P)}$, that maximum (minimum) is obtained at some point lying in the complement of $S(P)$.
In contrast of this statement, I'm accustomed to the statement of the Weak Parabolic Maximum Principle in Evans' book and on slide $9$ of this presentation, which basically said that the maximum of $u$ is attained on the parabolic boundary.
I would like to know how I can see the Weak Parabolic Maximum Principle as stated by Friedman as the Weak Parabolic Maximum Principle as the statement on the presentation above.
Thanks in advance!