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


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English | ISBN: 1470409836 | 2015 | 92 Pages | PDF | 1 MB

The Hamiltonian $int_X(lvert{partial_t u} vert^2 + lvert{ abla u} vert^2 + mathbf{m}^2lvert{u} vert^2),dx$, defined on functions on $mathbb{R} imes X$, where $X$ is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation.

The author considers 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. The author shows that, when $X$ is the sphere, and when the mass parameter $mathbf{m}$ is outside an exceptional subset of zero measure, smooth Cauchy data of small size $epsilon$ give rise to almost global solutions, i.e. solutions defined on a time interval of length $c_Nepsilon^{-N}$ for any $N$. Previous results were limited either to the semi-linear case (when the perturbation of the Hamiltonian depends only on $u$) or to the one dimensional problem.

The proof is based on a quasi-linear version of the Birkhoff normal forms method, relying on convenient generalizations of para-differential calculus.
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