Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $u_{0}$ and $f$ be continuous in $\overline{\Omega}$, and $\varphi$ be continuous on $\partial \Omega \times [0,T]$. Suposse $u \in C^{2,1}(\Omega \times (0,T])\cap C(\overline{\Omega}\times [0,T])$ is a solution of
$u_{t}-\Delta u = e^{-u} - f(x)$ in $\Omega\times (0,T]$
$u(.,0)= u_{0}$ on $\Omega$
$u=\varphi$ on $\partial \Omega \times (0,T)$
Prove that
$-M\leq u \leq Te^{M} + M $ in $ (0,T]$
Where $M=T\sup\limits_{\Omega} |f|+\sup\Big\{ \sup\limits_{\Omega} |u_{0}|, \sup\limits_{\partial \Omega \times (0,T)}|\varphi| \Big\}$
Well, my strategy was to check that equation $v=e^{u}$ satisfies a linear heat equation as
$v_{t}-\Delta v +cv$, where $c$ is a continuous function on $\Omega \times (0,T]$. En effet,
$v_{t}-\Delta v +(|\nabla u|^2 + f)v=1$ in $\Omega\times (0,T]$. But , it must verified that there is a nonnegative constant $c_{0}$ s.t $(|\nabla u|^2+f)\geq -c_{0}$, to apply weak principle ,my candidate went $c_{0} := \inf\limits_{\overline{\Omega}} (|\nabla u|^{2}+f)$, but, I imagine which there is no exist signal control $c_{0}$, because there is no information about the $f$ signal. Could anyone give me a tip?