## On the global attraction to solitary waves for the Klein-Gordon equation coupled to a nonlinear oscillator.(English)Zbl 1096.35020

Summary: The long-time asymptotics are analyzed for all finite energy solutions to a model $$U(1)$$-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as $$t\rightarrow \pm \infty$$ to the set of ‘nonlinear eigenfunctions’ $$\psi (x)e^{-\mathrm i\omega t}$$.

### MSC:

 35B41 Attractors 35L70 Second-order nonlinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations 35Q51 Soliton equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems

### Keywords:

one space dimension; finite energy solutions
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### References:

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