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

Changxing Miao, and Youbin Zhu. "Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure." Applicationes Mathematicae 33.3-4 (2006): 323-344. <http://eudml.org/doc/279483>.

@article{ChangxingMiao2006,
abstract = {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.},
author = {Changxing Miao, Youbin Zhu},
journal = {Applicationes Mathematicae},
keywords = {Klein-Gordon equations; variational principle; local energy; Lagrange density function; Morawetz-type identity},
language = {eng},
number = {3-4},
pages = {323-344},
title = {Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure},
url = {http://eudml.org/doc/279483},
volume = {33},
year = {2006},
}

TY - JOUR
AU - Changxing Miao
AU - Youbin Zhu
TI - Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure
JO - Applicationes Mathematicae
PY - 2006
VL - 33
IS - 3-4
SP - 323
EP - 344
AB - 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.
LA - eng
KW - Klein-Gordon equations; variational principle; local energy; Lagrange density function; Morawetz-type identity
UR - http://eudml.org/doc/279483
ER -