Suppose $U=\Omega \times (0,\infty),~\Omega$ is a bounded domain in $\mathbb{R}^n$ and $u \in C^{2,1}(\overline{U})$ satisfies $$u_t \leq \Delta u+cu~~\text{in}~U,$$ where $c \leq 0$ is a constant.
Question: If $u \geq 0,$ show that the weak maximum principle holds for $u.$
I'm still stuck in proving this. Any help is much appreciated.
If you examine Evans' proofs in this section, then you'll notice that it's not particularly important that the coefficients $a^{ij}$, $b^i$, and $c$ are continuous. What is important to his proofs is that the coefficients are bounded in $U_T$, $T<\infty$. Evans obtains this simply from continuity of the coefficients and compactness of $U_T$, but if you assume the coefficients are a priori bounded then this isn't necessary. You should be able to derive your desired result by noting this and proving the weak maximum principle (via exactly Evans' proof) for all $U_T$, $0<T<\infty$.
As for $c\leq 0$ vs $c\geq 0$, note the sign on the second-order coefficients of $L$, versus the coefficients of the Laplacian. Everything should work fine after properly accounting for the change in sign between the hypotheses of each theorem.
Edit: In fact, as Jeff has kindly pointed out to me, the equation in question is constant-coefficient, so Evans' theorem can almost be used without modification. The main modification needed is to deal with the infinite time interval; again, this can be done essentially by proving the weak maximum principle on every $U_T$ for $T$ finite, then letting $T\to\infty$.
