Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations

We study long-time dynamics of the damped focusing cubic Klein-Gordon equation on a compact three-dimensional Riemannian manifold, together with its space-independent reduction, the damped focusing Duffing equation. Under the geometric control condition on the damping and an assumption on the set of stationary solutions, we establish a sharp trichotomy for initial data with energy slightly above that of the ground state: every solution either blows up in finite time, decays exponentially to zero, or converges to a ground state. We provide a complete classification of the Duffing dynamics above the energy of the constant solution, use it to construct Klein-Gordon solutions realising each of the three behaviours in the case of a domain without boundary, and derive a simple spectral criterion ensuring that the ground states are nonconstant - and hence that different types of behaviour can indeed occur for solutions with initial energy above that of the ground state.