Suppose that $u(t,x)\in C_t^1C_x^2(\Omega_T)\cap C(\overline{\Omega_T})$ satisfying
$$
\begin{cases}
\partial _tu-\Delta u+c\left( x \right) u\le 0,\left( t,x \right) \in \Omega _T,\\
u\left( t,x \right) \le 0,\left( t,x \right) \in \Gamma _T.\\
\end{cases}
$$
where $c(x)\geq -c_0$ is a continuous lower-bounded function with $c_0>0$. Prove that $u(t,x)\leq 0,(t,x)\in \Omega_T$.
I guess that this question is about maximum principle, but this is not a heat equation because there is a term $c(x)u$ in the equation. I wonder how to adapt the original maximum principle to this question. Thanks a lot for your kind help!
Best Answer
Are you happy that $\partial_{t}v-\Delta v+(c+c_{0})v\le 0$ in $\Omega_{T}$ implies $\max_{\bar{\Omega}_{T}}v\le \max_{\Gamma_{T}}v^{+}$? If so your result follows by making $v=e^{-c_{0}t}u$.