We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations...

We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier–Lebesgue space $ℱL^{s,r}\left(T\right)$ with $s\ge \frac{1}{2},2<r<4,(s-1)r<-1$ and scaling like $H^{\frac{1}{2}-ϵ}\left(𝕋\right)$, for small $ϵ>0$. We also show the invariance of this measure.

Dans cet article, on introduit pour certaines classes d’opérateurs pseudo-différentiels à caractéristiques multiples des invariants, définis en chaque point caractéristique, permettant d’exprimer des conditions nécessaires ou suffisantes d’hypoellipticité ; une formule explicite est donnée permettant de les calculer dans un système de coordonnées.

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.

On définit des invariants entiers mesurant l’irrégularité d’un point singulier d’un système différentiel. Les propriétés de ces invariants, l’étude de la variation de l’ordre de la singularité par perturbation linéaire ainsi qu’une généralisation d’un théorème de W. Jurkat et D.A. Lutz permettent de donner une méthode de calcul de cet ordre.

We consider the identification of a distributed parameter in an elliptic variational inequality. On the basis of an optimal control problem formulation, the application of a primal-dual penalization technique enables us to prove the existence of multipliers giving a first order characterization of the optimal solution. Concerning the parameter we consider different regularity requirements. For the numerical realization we utilize a complementarity function, which allows us to rewrite the optimality...

We consider the identification of a distributed parameter in an elliptic variational inequality. On the basis of an optimal control problem formulation, the application of a primal-dual penalization technique enables us to prove the existence of multipliers giving a first order characterization of the optimal solution. Concerning the parameter we consider different regularity requirements. For the numerical realization we utilize a complementarity function, which allows us to rewrite the optimality...