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

• Autorzy: Changxing Miao , Youbin Zhu
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 35L15: Initial value problems for second-order hyperbolic equations
• 35L05: Wave equation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 2006
• Tom: 33
• Numer: 3-4
• Strony: 323-344

Daty

wydano

2006

Twórcy

autor

Changxing Miao

• Institute of Applied Physics, and Computational Mathematics, P.O. Box 8009, Beijing 100088, China

autor

Youbin Zhu

• Department of Mathematics, Xidian University, P.O. Box 245, Xi'an, Shanxi 710071, China

Identyfikatory

DOI

10.4064/am33-3-7