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

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
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