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

TL;DR

This paper analyzes the long-term behavior of focusing damped cubic Klein-Gordon and Duffing equations, establishing conditions for blow-up, decay, or convergence to ground states on a Riemannian manifold.

Contribution

It provides a sharp classification of solution behaviors and a spectral criterion for nonconstant ground states in the context of damped focusing nonlinear equations.

Findings

Solutions either blow up, decay, or converge to ground states.

Complete classification of Duffing dynamics above ground state energy.

Spectral criterion for nonconstant ground states.

Abstract

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.