I have the following problem and I'm not sure how to get started, any help is appreciated!
Let u $\in C^2(U \times (0, \infty)) \cap C (\overline{U}\times[0,\infty)) $ be a solution of
\begin{cases} u_t-u_{xx}=\sin u &\text{in}\:\:U\times (0, \infty), \newline u=0 &\text{in} \:\: \partial U\times (0, \infty), \newline u=g &\text{in} \:\:U\times (t=0). \end{cases}
Show that if $g(x) \leq 1$, then $u(x,t)\leq e^t $ for all $x \in U, t> 0.$