Wyniki wyszukiwania

1

Scattering theory for a nonlinear system of wave equations with critical growth

100%

Miao C., Zhu Y.

Colloquium Mathematicum

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2006

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tom 106

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nr 1

69-81

EN

We consider scattering properties of the critical nonlinear system of wave equations with Hamilton structure ⎧uₜₜ - Δu = -F₁(|u|²,|v|²)u, ⎨ ⎩vₜₜ - Δv = -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). By using the energy-conservation law over the exterior of a truncated forward light cone and a dilation identity, we get a decay estimate for the potential energy. The resulting global-in-time estimates imply immediately the existence of the wave operators and the scattering operator.

2

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

100%

Miao C., Zhu Y.

Applicationes Mathematicae

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2006

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tom 33

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nr 3-4

323-344

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.

3

The Cauchy problem for the coupled Klein-Gordon-Schrödinger system

100%

Miao C., Zhu Y.

Annales Polonici Mathematici

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2006

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tom 89

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nr 2

163-195

EN

We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural restriction on the power of interactions. In the last part of this paper, we use special admissible pairs and Strichartz estimates to remove the restriction, thereby generalizing previous results and obtaining the well-posedness of the system.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Applicationes Mathematicae

1 Colloquium Mathematicum

3 Miao C.

3 Zhu Y.

3 2006

Scattering theory for a nonlinear system of wave equations with critical growth

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

The Cauchy problem for the coupled Klein-Gordon-Schrödinger system