Front d’onde des fonctions non linéaires et polynômes. (Wave front of nonlinear functions and polynomials). (French) Zbl 0698.35101

Lebeau, G.

Sémin. Équations Dériv. Partielles 1988-1989, Exp. No. 10, 5 p. (1989).

Let \(\Omega\) be an open domain on \(R\times R^ d\) and denote \(\omega =\Omega \cap \{t=0\}\), \(\square =\partial^ 2/\partial t^ 2- \sum^{d}_{j=1}\partial^ 2/\partial x^ 2_ j.\) The following problem is considered \[ \square u=F(t,x,u,\nabla u),\quad \nabla u=(\partial_ tu,\nabla_ xu),\quad u|_{t=0}=u_ 0\in H^{s+1}_{loc}(\omega),\quad \partial u/\partial t|_{t=0}=u_ 1\in H^ s_{loc}(\omega), \] where F is a \(C^{\infty}\) function, \(s>d/2\) (or \(s>d/2-1\) if F is not dependent on \(\nabla u)\). A spectral decomposition method is exposed to prove the following result (Theorem 2): Let \(u\in H^ s(R^ d)\) be \((s=d/2+\rho\), \(\rho >0)\) with values in R and \(F\in C^{\infty}(R,R)\). Then, for all \(N\geq 1\), and for all \(\mu \in [d/2+N\rho\), \(d/2+(N+1)\rho]\), \[ WF^{\mu}(F(\mu))\subset \cup^{N}_{j=1}WF^{\mu}(u^ j), \] where \(WF^{\mu}\) is the usual Sobolev wave front.

Reviewer: I.Onciulescu

MSC:

35L70	Second-order nonlinear hyperbolic equations
35L67	Shocks and singularities for hyperbolic equations

Keywords:

nonlinear wave equation; wave front set