Bounding the solution to an inhomogeneous heat equation.

Question

I have a PDE question that I am struggling to get started on. Suppose we have a function $u\in \mathcal{C}^2\Big(U\times(0,\infty)\Big)\cap \mathcal{C}\Big(\overline{U}\times[0,\infty)\Big)$ which solves $$\begin{cases} u_t-\Delta u=\sin(u)& U\times(0,\infty)\\ u=0& \partial U\times (0,\infty)\\ u=g& \partial U\times\{t=0\} \end{cases}.$$ If $g(x)\le 1$, I want to show that $u(x,t)\le e^t$ for all $x\in U$ and $t>0$. What is a good approach I can employ to solve this question?