I have a question about exercise 5.9. of the book A Basic Course in Partial Differential Equations, by Qing Han.
Let assume $\Omega \subset \mathbb{R}^n$ is a bounded domain and $f$ and $u_0$ are continuous in $\overline{\Omega}$ and $\phi$ continuous in $\Omega \times [0,T]$. Suppose $u \in C^{2,1} ( \Omega \times (0,T]) \cap C(\overline{\Omega} \times [0,T])$ solves the following equation.
\begin{equation*} \begin{cases} u_t-\Delta u= e^{-u} - f(x) & \quad (x,t) \in \Omega \times (0,T] \\ u(x,0)=u_0 & \quad x \in \Omega \\ u(x,t)= \phi & \quad (x,t) \in \partial \Omega \times (0,T] \end{cases} \end{equation*}
Prove that $$ -M < u <T e^{M} +M, \qquad \qquad \Omega \times (0,T] $$ where $$ M=T \sup_{\Omega} |f| + \max \Big\{ \sup_{\Omega} |u_0|, \sup_{\partial \Omega \times (0,T)} |\phi| \Big\}. $$
I know that I must apply the strong maximum principle for some candidate $v$ which is defined by $u$. But I do not know how to handle it.
