English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We discuss invariants and conservation laws for a nonlinear system of Klein–Gordon equations with Hamiltonian structure { utt −∆u+m 2u = −F1(|u| 2, |v|2)u, vtt −∆v +m 2v = −F2(|u| 2, |v|2)v for which there exists a function F (λ, μ) such that ∂F (λ, μ) ∂λ = F1(λ, μ), ∂F (λ, μ) ∂μ = F2(λ, μ). Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local L2-norm, and local energy also decays to zero if the initial energy satisfies E(u, v,R, 0) = 1 2 \ R (|∇u(0)| + |ut(0)| 2 +m|u(0)| + |∇v(0)| + |vt(0)| 2 +m|v(0)| + F (|u(0)|, |v(0)|)) dx <∞, and F1(|u| , |v|)|u| + F2(|u| , |v|)|v| − F (|u|, |v|) ≥ aF (|u|, |v|) ≥ 0, a > 0.

Invariants, conservation laws and time decay for a nonlinear system of Klein–Gordon equations with Hamiltonian structure