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

Delort, J.-M.

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