I am considering the following section of Peter Grindrods „Pattern and Waves“.
We begin by considering a scalar equation $$ u_t=\Delta u+f(u,x,t),\quad x\in\Omega\subseteq\mathbb{R}^n, t>0. $$ Here $f$ is smooth (say, continously differentiable).
Suppose $\overline{u}\colon\Omega\times [0,T]\to B$, some bounded subset of $\mathbb{R}$, and $$ \overline{u}_t\geq\Delta\overline{u}+f(\overline{u},x,t); $$ then we say that $\overline{u}$ is a super-solution. If $\underline{u}\colon\Omega\times [0,T]\to B$, and $$ \underline{u}_t\leq\Delta\underline{u}+f(\underline{u},x,t), $$ then we say that $\underline{u}$ is a sub-solution.
Now suppose that there exist constants $\alpha,\beta (\alpha^2+\beta^2\neq 0)$ such that $$ \alpha\overline{u}-\beta\nabla\overline{u}.n\geq \alpha\underline{u}-\beta\nabla\underline{u}.n,\quad x\in\partial\Omega, t>0, $$ (n is the outer normal to $\partial\Omega$); and that $$ \overline{u}(x,0)\geq\underline{u}(x,0), x\in\Omega. $$ Then we claim that $$ \overline{u}(x,t)\geq \underline{u}(x,t),\quad x\in\Omega. $$
The author also gives a short "proof".
To see this, set $$ w=\overline{u}-\underline{u}. $$ Then the mean value theorem implies $$ w_t-\Delta w\geq f_u(\underline{u}+\theta (\overline{u}-\underline{u}))w $$ for some mapping $\theta\colon\Omega\times [0,T]\to [0,1]$. The result follows from the strong maximum principle for linear parabolic equations [15] (The point is that $w$ is initially nonnegative and the boundary conditions ensure that if it becomes negative it must do so at an interior point of $\Omega$. We then can construct a contradiction.)
Unfortunately, I have no opportunity to have a look at the mentioned maximum principle cited from book [15] (which is a book by P.C. Fife).
So we set $w=\overline{u}-\underline{u}$ and note that $$ w_t-\Delta w\geq f(\overline{u},x,t)-f(\underline{u},x,t)=f_u(\underline{u}+\theta(\overline{u}-\underline{u})w,x,t)w $$ where the last identity comes from applying the mean value theorem.
Up to here, I can follow the proof.
Now, i write this inequality by using a parabolic linear operator in order to bring it in the usual textbook form for maximum principles:
$$ Lw = \sum_{i=1}^n\frac{\partial^2 w}{\partial^2 x_i}-f_u(\underline{u}+\theta(\overline{u}-\underline{u},x,t)w-(-w_t)\geq 0. $$ Now, if $$ f_u(\underline{u}+\theta(\overline{u}-\underline{u},x,t)\leq 0, $$ I could use the strong maximum principle for the function $-w$, i.e. making some statement about non-negative maxima of the function $-w$ (resp. about negative minima of the function $w$).
However, I can neither find the assumption $$ f_u(\underline{u}+\theta(\overline{u}-\underline{u},x,t)\leq 0 $$ nor deduce it from the text.
But let's suppose for a moment that $$ f_u(\underline{u}+\theta(\overline{u}-\underline{u},x,t)\leq 0. $$
The proof by contradiction which the author mentions maybe works like this:
Assume we have at least one point $(x_0,t_0), t_0>0$ such that $$ \overline{u}(x,t)<\underline{u}(x,t). $$ Then $-w(x_0,t_0)>0$.
By initial- and boundary conditions, the point $(x_0,t_0)$ had to be an inner point of $\Omega\times [0,T]$ since $w<0$ can only happen at inner points.
From this, can we deduce that in the interior of $\Omega\times [0,T]$ we have a negative minimum $M$ (say at some point $(x',t')$ which equals $(x_0,t_0)$ in case this is the only point such that $\overline{u}<\underline{u}$) of $w$ and that - by the strong maximum principle - $$ w(x,t)=M $$ for all $(x,t)$ which are on the horizontal line on which $(x',t')$ lies? (in particular for boundary points on this line, which gives a contradiction since there $w$ cannot be negative)?
Thats how I did understand the proof.
