# Periodic BBM — H4

Let \(\mathbb T=\mathbb R/(2\pi\mathbb Z)\). Suppose \(u\in C(\mathbb R;H^1(\mathbb T;\mathbb R))\) satisfies
\[
(1-\partial_x^2)u_t+\partial_x(u+u^2/2)=0
\]
in distributions. If there exist \(C\in\mathbb R\), \(T_0>0\), and a nonempty open set \(\omega\subset\mathbb T\) such that \(u=C\) on \(\omega\times(0,T_0)\), then
\[
u(x,t)=C\qquad\text{for every }(x,t)\in\mathbb T\times\mathbb R.
\]