# Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres

@inproceedings{Delort2015QuasiLinearPO,
title={Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres},
author={Jean-Marc Delort},
year={2015}
}
The Hamiltonian $\int_X(\abs{\partial_t u}^2 + \abs{\nabla u}^2 + \m^2\abs{u}^2)\,dx$, defined on functions on $\R\times X$, where $X$ is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. We consider perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of $u$. The associated PDE is then a quasi-linear Klein-Gordon equation. We show that, when $X$ is the sphere, and when the mass parameter $\m$ is…

### Long time existence results for Hamiltonian non-linear Klein-Gordon equations on some compact manifolds

Consider a nonlinear Klein-Gordon equation on X, a compact Riemannian manifold without boundary, $(∂_t^2 − ∆ + m^2)w = N (w)$, where N is a smooth non-linearity. If the non-linearity vanishes at

### Almost global solutions to two classes of 1-d Hamiltonian Derivative Nonlinear Schr\"odinger equations

Consider two kinds of 1-d Hamiltonian Derivative Nonlinear Schrodinger (DNLS) equations with respect to different symplectic forms under periodic boundary conditions. The nonlinearities of these

### Almost Global Solutions to Hamiltonian Derivative Nonlinear Schrödinger Equations on the Circle

• Jing Zhang
• Mathematics
Journal of Dynamics and Differential Equations
• 2019

### Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres

• Mathematics
• 2004
Consider a nonlinear Klein-Gordon equation on a compact manifold M with smooth Cauchy data of size e → 0. Denote by T e the maximal time of existence of a smooth solution. One always has T e ≥ c e

### Birkhoff Normal Form for Some Nonlinear PDEs

Abstract: We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave

### Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds

• Mathematics
• 2005
This paper is devoted to the proof of almost global existence results for Klein‐Gordon equations on Zoll manifolds (e.g., spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the

### Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds

• Mathematics
• 2006
We prove a long time existence result for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds. This generalizes a preceding result concerning the case of spheres, obtained in

### Long-Time Existence for Semi-Linear Klein–Gordon Equations with Quadratic Potential

We prove that small smooth solutions of semi-linear Klein–Gordon equations with quadratic potential exist over a longer interval than the one given by local existence theory, for almost every value

### Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces

• Mathematics
• 2006
Cet article est consacre a la preuve de resultats d'existence presque globale pour des equations de Klein-Gordon sur des hypersurfaces compactes de revolution avec des non-linearites non