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Abstract

This paper deals with global stability dynamics for the Klein–Gordon–Zakharov system in $R^{2}$ . 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 $H^{s + 1} (R^{2}) \times H^{s + 1} (R^{2}) \times H^{s + 1} (R^{2})$ 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

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Stabilization for the Klein–Gordon–Zakharov system

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