We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not...

We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not...

We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy...

We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧$u_{tt}-\Delta u+m²u=-F₁\left(\right|u|²,|v\left|²\right)u$, ⎨ ⎩$v_{tt}-\Delta v+m²v=-F₂\left(\right|u|²,|v\left|²\right)v$ 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 $E(u,v,ℝⁿ,0)=1/2{\int }_{ℝⁿ}\left(\right|\nabla u\left(0\right)|²+|u_t\left(0\right)|²+m²|u\left(0\right)|²+|\nabla v\left(0\right)|²+|v_t\left(0\right)|²+m²|v\left(0\right)|²+F(\left|u\left(0\right)\right|²,\left|v\left(0\right)\right|²\left)\right)dx<\infty$, 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.