Let \(U\) a bounded domain in \(\mathbb {R}^n\). Consider a smooth solution \(u\) of the equation \[ \begin {cases} u_t=\Delta u+c(x,t)u & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ),\\ u(x,0)=f(x). \end {cases} \]
Show that, if \(c(x, t) \le -\gamma \) for all \(x\) and \(t\), then
\[ \max _{x\in U}|u(x,t)|\le e^{-\gamma t}\max _U |f|. \]