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Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-021-04241-5
Commun. Math. Phys. 388, 1557–1601 (2021) Communications in
Mathematical
Physics
Description and Classification of 2-Solitary Waves for
Nonlinear Damped Klein–Gordon Equations
Raphaël Côte1, Yvan Martel2, Xu Yuan2,3, Lifeng Zhao4
1IRMA UMR 7501, Université de Strasbourg, CNRS, 67000 Strasbourg, France.
E-mail: cote@math.unistra.fr
2CMLS, École polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France.
E-mail: yvan.martel@polytechnique.edu
3Present address: Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong
Kong.
E-mail: xu.yuan@cuhk.edu.hk
4School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui,
China. E-mail: zhaolf@ustc.edu.cn
Received: 24 March 2021 / Accepted: 30 September 2021
Published online: 22 October 2021 – © The Author(s), under exclusive licence to Springer-Verlag GmbH
Germany, part of Springer Nature 2021
Abstract: We describe completely 2-solitary waves related to the ground state of the
nonlinear damped Klein–Gordon equation
∂ttu+2α∂tu−u+u−|u|p−1u=0
on RN,for1N5 and energy subcritical exponents p>2. The description is
twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we
construct and classify the full family of 2-solitary waves in the case of opposite signs.
Close to the sum of two remote solitary waves, it turns out that only the components
of the initial data in the unstable direction of each ground state are relevant in the large
time asymptotic behavior of the solution. In particular, we show that 2-solitary waves
have a universal behavior: the distance between the solitary waves is asymptotic to logt
as t→∞. This behavior is due to damping of the initial data combined with strong
interactions between the solitary waves.
1. Introduction
1.1. Setting of the problem. We consider the nonlinear focusing damped Klein–Gordon
equation
∂ttu+2α∂tu−u+u−f(u)=0(t,x)∈R×RN,(1.1)
where f(u)=|u|p−1u,α>0, 1 N5, and the exponent pcorresponds to the
energy sub-critical case, i.e.
2<p<∞for N=1,2 and 2 <p<N+2
N−2for N=3,4,5.(1.2)
R. C. was partially supported by the French ANR contract MAToS ANR-14-CE25-0009-01. Y. M. and
X. Y. thank IRMA, Université de Strasbourg, for its hospitality. L. Z. thanks CMLS, École Polytechnique, for
its hospitality. L. Z. was partially supported by the NSFC Grant of China (11771415).
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