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.