Problème de Cauchy semi-linéaire en 3 dimensions d'espace. Un résultat de finitude. (A semilinear Cauchy problem in three dimensional spaces. A finiteness result) (Q1105124)

We prove that if u is a solution of the semilinear wave equation \[ (\partial^ 2_ t-\Delta)u=\sum_{0\leq j\leq j_ 0}p_ j(t,x)u^ j,\quad (t,x)\in {\mathbb{R}}^ 4,\quad u\in H^ s,\quad s>2, \] such that the Cauchy data of u, \(u|_{t=0}\), \((\partial u/\partial t)|_{t=0}\) are Hörmander Fourier integral distributions on some analytic Lagrangian, then for every real \(\sigma\), there exist a subanalytic homogeneous isotropic subset \(L_{\sigma}\) of \(T^*{\mathbb{R}}^ 4\) such that \(WF^{\sigma}(u)\subset L_{\sigma}\). In particular, for every integer k, u is \(C^ k\) on a dense open subset.