Stabilization for the Klein–Gordon–Zakharov system

Article type: Research Article

Authors: Li, Weijia | Shangguan, Yuqi^{; *} | Yan, Weiping

Affiliations: College of Mathematics and Information Science, Guangxi University, Nanning 530004, P.R. China

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: This paper deals with global stability dynamics for the Klein–Gordon–Zakharov system in R2. We first establish that this system admits a family of linear mode unstable explicit quasi-periodic wave solutions. Next, we prove that the Kelvin–Voigt damping can help to stabilize those linear mode unstable explicit quasi-periodic wave solutions for the Klein–Gordon–Zakharov system in the Sobolev space Hs+1(R2)×Hs+1(R2)×Hs+1(R2) for any s⩾1. Moreover, the Kelvin–Voigt damped Klein–Gordon–Zakharov system admits a unique Sobolev regular solution exponentially convergent to some special solutions (including quasi-periodic wave solutions) of it. Our result can be extended to the n-dimension dissipative Klein–Gordon–Zakharov system for any n⩾1.

Keywords: Stabilizability, quasi-periodic wave solutions, Klein–Gordon–Zakharov system, Kelvin–Voigt damping

DOI: 10.3233/ASY-231856

Journal: Asymptotic Analysis, vol. 135, no. 3-4, pp. 305-348, 2023