Consider the following parabolic equation $$ \begin{cases} \partial_t u = \partial_{xx}u + \sin u,\ (x,t)\in\mathbb{R}\times(0,\infty), \\ u(x,0) = u_0(x), \end{cases} $$ in which $u_0(x):\mathbb{R}\to\mathbb{R}$ is $2\pi$-periodic and odd.
I want to know how to derive the solution $u$ itself is also $2\pi$-periodic and odd for each $t>0$. Any hint?