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


Elena Kopylova (Faculty of Mathematics, University of Vienna, Austria)

Alexander Komech (Faculty of Mathematics, University of Vienna, Austria; and Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia)


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