## Conormalité des ondes semi-linéaires le long des caustiques. (Conormality of semilinear waves along caustics).(French)Zbl 0703.35113

Sémin. Équations Dériv. Partielles 1988-1989, Exp. No. 15, 12 p. (1989).
Let $$\Omega$$ be a domain of $${\mathbb{R}}\times {\mathbb{R}}^ d$$. Denote $$\omega =\Omega \cap \{t=0\}\subset {\mathbb{R}}^ d$$, $$\square =\partial^ 2/\partial t^ 2-\sum^{d}_{j=1}\partial^ 2/\partial x^ 2_ j.$$ The following problems $\square u=P((t,x),u),\quad u|_{t=0}=u_ 0,\quad \partial u/\partial t)|_{t=0}=u_ 1,\quad \square u=P((t,x),u),\quad u|_{t<0}=v,$ are considered, where P(z,Z) is a polynomial in Z with coefficients $$C^{\infty}$$ in z, $$u_ 0\in H^{\gamma}_{loc}(\omega)$$, $$u_ 1\in H^{\gamma - 1}_{loc}(\omega)$$ (respectively $$v\in H^{\gamma +}_{loc}(\Omega)).$$
The author presents a study of the propagation of conormal regularity of a solution $$u\in H^{\gamma +}_{loc}(\Omega)$$ which consists in supposing $$u_ 0$$, $$u_ 1$$ (resp. v) conormals in one or more subvarieties of $$\omega$$ (resp. $$\Omega \cap \{t<0\})$$ and then to study the singularities of the solution in the future.
Reviewer: I.Onciulescu

### MSC:

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

### Keywords:

propagation of conormal regularity
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