Unique continuation for the periodic Benjamin-Bona-Mahony equation

Abstract

We prove unique continuation for the real periodic Benjamin--Bona--Mahony equation in the energy space: a solution that equals a constant on a nonempty space--time strip is that constant everywhere. In particular, Rosier's unique continuation conjecture [23] holds without any smallness or mean assumption. It even suffices to observe, at one point $x_0$ and for $t$ in an interval, the solution and the potential $ψ=(1-\partial_x^2)^{-1}(u+u^2/2)$ of its flux: $u(x_0,t)=C$ and $ψ(x_0,t)=C+C^2/2$ imply $u\equiv C$, and the second condition cannot be dropped when $C=-1$. The proof uses an exponentially weighted Hamiltonian of indefinite sign with a coercive dissipation identity, and it applies to every nonconstant real polynomial flux. Consequently, Rosier's stabilization results for two localized feedback laws and for a boundary feedback hold unconditionally, for arbitrary data. Stationary step profiles show that the theorem fails in $H^s$ for $0\leq s<1/2$. We also give a generating-series proof of single-point unique continuation for the linearized equation and discuss related control results.