Sub-exponentially long timescale stability for nonlinear Klein–Gordon equation with potential

We prove a Nekhoroshev type theorem for the nonlinear Klein–Gordon equation under Dirichlet boundary conditions and c ∈ [1, + ∞), which is uniform with respect to c ≥ 1. More precisely, we prove the sub-exponential long time stability result in Gevrey spaces around the origin for the above equation, where using Birkhoff normal form technique for infinite dimensional Hamiltonian systems and the so-called tame property of the non-linearity.