A property of exponential stability in nonlinear wave equations near the fundamental linear mode (Q1808325)

This paper considers the nonlinear second-order hyperbolic PDE \(u_{tt}-c^2u_{xx}=\varepsilon\psi(u)\), with the boundary conditions \(u(t,0)= u(t,\pi)=0\), where \(\psi\) is an analytic function such that \(\psi(0)=0\), \(\psi' (0)\neq 0\). A stability property, valid up to times exponentially large in \(1/ \varepsilon\), is proven for the neighboorhood of the trajectory of the linear mode with lowest frequency.