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

Changxing Miao ; Youbin Zhu

Applicationes Mathematicae, Tome 33 (2006), p. 323-344 / Harvested from The Polish Digital Mathematics Library

Résumé

We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧$u_{tt} - Δu + m²u = -F₁(|u|²,|v|²)u$, ⎨ ⎩$v_{tt} - Δv + m²v = -F₂(|u|²,|v|²)v$ 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 $E(u,v,ℝⁿ,0) = 1/2 ∫_{ℝⁿ}(|∇u(0)|² + |u_t(0)|² + m²|u(0)|² + |∇v(0)|² + |v_t(0)|² + m²|v(0)|² + F(|u(0)|²,|v(0)|²))dx < ∞$, and F₁(|u|²,|v|²)|u|² + F₂(|u|²,|v|²)|v|² - F(|u|²,|v|²) ≥ aF(|u|²,|v|²) ≥ 0, a > 0.

Détail

Publié le : 2006-01-01

Zbl 1108.35111

EUDML-ID : urn:eudml:doc:279483