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Zbl 0665.35051
Alinhac, S.
Interaction d'ondes simples pour des équations complètement non- linéaires. (Interaction of simple waves for completely nonlinear equations). (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 21, No. 1, 91-132 (1988). ISSN 0012-9593

The author studies a class of solutions of completely nonlinear hyperbolic equations of the form $$ (1)\quad F(x,u(x),...,\partial\sp{\alpha} u(x),...)=0,\quad \vert \alpha \vert \le m, $$ in an open subset $\Omega$ of ${\bbfR}\sp n$. Put $\Omega\sb{\pm}=\Omega \cap \{\pm t>0\}$, where t is a time coordinate. If $\Omega\sb+$ is a domain of influence of $\Omega\sb-$ for the linearization at u of the operator in (1), the author describes the $C\sp{\infty}$ singularities of u in $\Omega\sb+$ in cases where $u\vert\sb{\Omega\sb-}$ is conormal with respect to a finite number of characteristic hypersurfaces. His results generalize earlier semilinear theorems of Bony. Cauchy problems with conormal data for strictly hyperbolic operators are also studied.
[P.Godin]

MSC 2000:: *35L75 Nonlinear hyperbolic PDE of higher $(>2)$ order
35L67 Shocks, etc.
35A20 Analytic methods (PDE)
35B40 Asymptotic behavior of solutions of PDE
35A30 Geometric theory for PDE, transformations

Keywords: interaction of simple waves; completely nonlinear; domain of influence; linearization; $C\sp{\infty }$ singularities; characteristic hypersurfaces; semilinear; Cauchy problems; conormal data

Cited in: Zbl 0791.35079

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