Long time dynamics for damped Klein-Gordon equations

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 su...

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Bibliographic Details
Published in:arXiv.org 2015-05
Main Authors: Burq, N, Raugel, G, Schlag, W
Format: Article
Language:English
Subjects:
Banach spaces
Energy dissipation
Mathematical analysis
Nonlinear equations
Nonlinearity
Online Access:Get full text
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Summary: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).
ISSN:2331-8422