# Maximum principle for the heat equation involving an aditional linear term

Let $\Omega$ be a bounded open region in $\mathbb{R}^n$, $\alpha >0$, $T>0$ and $f \in C^0 (\overline{\Omega})$ with $f \geq 0$ on $\Omega$. Also let $u \in C_1^2 (\Omega _T) \cap C^0 (\overline{\Omega}_T)$ satisfy $$\begin{cases} u_t -\Delta u+ \alpha u = f(x) & \text{ in } \Omega _T \\ u=0 & \text{ on } \Gamma _T, \end{cases}$$ where $\Omega_T=\Omega \times (0,T]$ is the parabolic cylender, and $\Gamma_t$ is the parabolic boundary of $\Omega_T$. Prove that $u \geq 0$ and $u_t \geq 0$ in $\Omega \times \left[ 0,T \right]$.