Let $U$ be a bounded open set with smooth boundary $\partial U$. Show that $C^2$ solutions of the initial-boundary value problem:
$u_t-\Delta u+ \cos (u)=0$ in $U\times\mathbb R^+$
$u=0 $ on $\partial U\times\mathbb R^+$
$u(x,0)=u_0(x)$ in $U$
I believe I should use the maximum principle to solve the problem. But the nonlinear term $\cos(u)$ can have different sign. It make it hard to apply the theorem. Anyone know how to deal with that term? Please help, thanks!!