Long time dynamics for damped Klein-Gordon equations

TL;DR

This paper studies the long-term behavior of solutions to damped nonlinear Klein-Gordon equations, showing solutions either blow up or stabilize to stationary states, with implications for energy subcritical nonlinearities.

Contribution

It establishes a comprehensive long-time dynamics framework for damped Klein-Gordon equations, including blow-up and convergence to stationary solutions, using advanced PDE and dynamical systems techniques.

Findings

Finite energy radial solutions either blow up or converge to stationary states.

Global solutions are proven to be bounded in energy.

Results apply to a broad class of energy subcritical nonlinearities.

Abstract

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, $1\textless{}p\textless{}(d+2)/(d-2)$ as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).