Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit

Benoît Grébert; Tiphaine Jézéquel; Laurent Thomann

doi:10.3934/dcds.2014.34.3485

Article Contents

2014, Volume 34, Issue 9: 3485-3510. Doi: 10.3934/dcds.2014.34.3485

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Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit

1.
Laboratoire de Mathématiques J. Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, 44322 Nantes Cedex 03
2.
INRIA & ENS Cachan Bretagne, Avenue Robert Schuman, 35170 Bruz

Received: February 28, 2013

Revised: November 30, 2013

Published: August 2014

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Abstract

We consider the Klein-Gordon equation (KG) on a Riemannian surface $M$ $$ \partial^{2}_t u-\Delta u-m^{2}u+u^{2p+1} =0,\quad p\in \mathbb{N}^{*},\quad (t,x)\in \mathbb{R}\times M,$$ which is globally well-posed in the energy space. This equation has a homoclinic orbit to the origin, and in this paper we study the dynamics close to it. Using a strategy from Groves-Schneider, we get the existence of a large family of heteroclinic connections to the center manifold that are close to the homoclinic orbit during all times. We point out that the solutions we construct are not small.

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Mathematics Subject Classification: 37K45, 35Q55, 35Bxx.

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References

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