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# Dynamics of Partial Differential Equations

## Volume 16 (2019)

### Number 2

### On global attractor of 3D Klein–Gordon equation with several concentrated nonlinearities

Pages: 105 – 124

DOI: https://dx.doi.org/10.4310/DPDE.2019.v16.n2.a1

#### Authors

#### Abstract

The global attraction is proved for solutions to 3D Klein–Gordon equation coupled to several nonlinear point oscillators. Our main result is a convergence of each finite energy solution to the set of all solitary waves as $t \to \pm \: \infty$. This attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation.

We justify this mechanism by the following strategy based on *inflation of spectrum by the nonlinearity*. We show that any *omega-limit trajectory* has the time-spectrum in the spectral gap $[-m, m]$ and satisfies the original equation. Then the application of the Titchmarsh convolution theorem reduces the time-spectrum to a single harmonic $\omega \in [-m, m]$.

#### 2010 Mathematics Subject Classification

35L70, 45J05, 47F05

E.K. is supported by Austrian Science Fund (FWF) under Grant No. P27492-N25 and RFBR grants 16-01-00100, 18-01-00524. A.K. is supported by Austrian Science Fund (FWF) under Grant No. P28152-N35 and RFBR grant 16-01-00100.

Received 25 September 2017

Published 14 March 2019