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.

We consider the inverse problem of determining how the physiological structure of a harvested population evolves in time, and of finding the time-dependent effort to be expended in harvesting, so that the weighted integral of the density, which may be, for example, the total number of individuals or the total biomass, has prescribed dynamics. We give conditions for the existence of a unique, global, weak solution to the problem. Our investigation is carried out using the method of characteristics...

Inertial manifold with delay (IMD) for dissipative systems of second order in time is constructed. This result is applied to the study of different asymptotic properties of solutions. Using IMD, we construct approximate inertial manifolds containing all the stationary solutions and give a new characterization of the K-invariant manifold.

A strictly hyperbolic quasi-linear 2×2 system in two independent variables with C2 coefficients is considered. The existence of a simple wave solution in the sense that the solution is a 2-dimensional vector-valued function of the so called Riemann invariant is discussed. It is shown, through a purely geometrical approach, that there always exists simple wave solution for the general system when the coefficients are arbitrary C^2 functions depending on both, dependent and independent variables.

These past few years, new types of computational architectures based on graphics processors have emerged. These technologies provide important computational resources at low cost and low energy consumption. Lots of developments have been done around GPU and many tools and libraries are now available to implement efficiently softwares on those architectures.This article contains the two contributions of the mini-symposium about GPU organized by Loïc Gouarin (Laboratoire de Mathématiques d’Orsay),...

We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations.

This work is concerned with the proof of $L_p-L_q$ decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation $u_{tt}-{\lambda }^2\left(t\right)b^2\left(t\right)(\Delta u-m^{2}u)=0$. The coefficient consists of an increasing smooth function $\lambda$ and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, $m^2$ is a positive constant. We study under which assumptions for λ and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n/2(1/p-1/q).

We establish an upper bound on the Kolmogorov’s entropy of the locally compact attractor for strongly damped wave equation posed in locally uniform spaces in subcritical case using the method of trajectories.