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The Hamiltonian \\int_X(\\lvert{\\partial_t u}\\rvert^2 + \\lvert{\\nabla u}\\rvert^2 + \\mathbf{m}^2\\lvert{u}\\rvert^2)\\,dx, defined on functions on \\mathbb{R}\\times 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_N\\epsilon^{-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.

We prove that small smooth solutions of weakly semi-linear Klein-Gordon equations on the torus ( d ≥ 2) exist over a larger time interval than the one given by local existence theory, for almost every value of the mass. We use a normal form method for the Sobolev energy of the solution. The difficulty, in comparison with previous results obtained on the sphere, comes from the fact that the set of differences of eigenvalues of on ( d ≥ 2) is dense in ℝ.

We prove that higher Sobolev norms of solutions of quasi-linear Klein-Gordon equations with small Cauchy data on S1\\mathbb S^1 remain small over intervals of time longer than the ones given by local existence theory. This result extends previous ones obtained by several authors in the semi-linear case. The main new difficulty one has to cope with is the loss of one derivative coming from the quasi-linear character of the problem. The main tool used to overcome it is a global paradifferential calculus adapted to the Sturm-Liouville operator with periodic boundary conditions.

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 an earlier paper by the authors. The proof relies on almost orthogonality properties of products of eigenfunctions of positive elliptic selfadjoint operators on a compact manifold and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.

The three bilinearities uv,uv¯,u¯vu v, \\overline {uv},\\overline {u}v for functions u,v:R2×[0,T]⟼Cu, v : \\mathbb {R}^2 \\times [0,T] \\longmapsto \\mathbb {C} are sharply estimated in function spaces Xs,bX_{s,b} associated to the Schrödinger operator i∂t+Δi \\partial _t + \\Delta. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.

Consider (M,g) a Riemannian manifold and denote by Δ g (resp. ∇ g ) the Laplacian (resp. the gradient) associated tog. Letm> 0 be given and consider a local solution u to the Cauchy problem\\[\\begin{align*} (\\partial_t^2 - \\Delta_g + m^2)u &= f(x,u,\\partial_t u,\\nabla_g u) \\\ u|_{t = 0} &= \\epsilon u_0 \\\ \\partial_t u|_{t = 0} & = \\epsilon u_1,\\\ \\end{align*} \\caption{(8.1)}\\]p wherefis a real polynomial in (u, ∂ t u, ∇ g u), vanishing at some orderp≥ 2 at 0, withC ∞coefficients inx, where the data (u₀,u₁) are in$C_0^\\infty (M)$, real valued, and where ϵ > 0 goes to zero. Denote by ]T *(ϵ),T *(ϵ)[ (withT *(ϵ) < 0

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