I'm reading his PDE in chapter 7.1 the last theorem is about the strong maximum principle for parabolic equation when $c\geq 0$ at page 398. I have some problem at the second step:
Since $u_t +Ku=-cu\leq 0$ on $\{ u\geq 0 \}$, we deduce from the weak maximum principle that $u\leq v$.
Who can give some detail to show how the weak maximum principle applies? Thanks in advance!
On the one hand, $$ u_t + K u \leq 0 $$ on $\{ u \geq 0\}$. On the other hand, $$ v_t + K v = 0, $$ with $v = u^+$ on $\Delta_T$.
If we subtract second equation from the first one, we get $$ (u-v)_t + K(u-v) \leq 0 \quad \implies \quad w_t + K w \leq 0, $$ where $w := u-v$. (We can do that, since $\frac{d}{dt}$ and $K$ are linear operators). Moreover, $w = 0$ on $\Delta_T$.
Hence, applying Weak Maximum Principle for $w$ (Theorem 8), we obtain $$ \max_{W_T} w = \max_{\Delta_T} w = 0, $$ and therefore, $w \leq 0$. Thus, $u \leq v$.
