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**Striated solutions of semilinear, two speed wave equations.**
*(English)*
Zbl 0537.35057

Indiana Univ. Math. J. (to appear).

In Ann. Math., II. Ser. 111, 531-552 (1980; Zbl 0407.35057), the authors showed that if P is a 2nd order, strictly hyperbolic, scalar operator in one space dimension, then the propagation of singularities for the semilinear equation \(Pu=f(u,Du)\) is the same as the case when f is linear. That is, singularities do not interact to produce anomalous singularities and the regularity of u at any point \(p_ 0=(x_ 0,t_ 0)\) is determined completely by the regularity of the initial data at the two points where the backward characteristic curves from \(p_ 0\) intersect the initial line. The purpose of this paper is to prove that this ”linear” behavior persists in higher dimensions for a special class of hyperbolic operators, called two speed operators, and special initial data.

For scalar operators we require the order of P to be less than or equal to two. More generally, we require that through any initial hypersurface \(\mathcal B\) there pass exactly two characteristic surfaces for P. An additional hypothesis, valid for second order equations and symmetric hyperbolic systems is also imposed. The class of two-speed operators includes, in addition to the wave operator, many Lorentz invariant systems which arise in mathematical physics: Maxwell’s equations in a vacuum (wave operator form), the Pauli electron equation, the Dirac equation, the coupled Maxwell-Dirac and Klein Gordon-Dirac equations. We assume that the initial data is infinitely differentiable parallel to a foliation of \({\mathbb{R}}^ n\) by n-1 dimensional hypersurfaces and call such initial data striated. Striated initial data arise naturally in a variety of physical situations where the initial data is smooth in all but one variable. In addition to the obvious case of cartesian coordinates we mention data which is smooth in the angular variables in spherical coordinates.

For scalar operators we require the order of P to be less than or equal to two. More generally, we require that through any initial hypersurface \(\mathcal B\) there pass exactly two characteristic surfaces for P. An additional hypothesis, valid for second order equations and symmetric hyperbolic systems is also imposed. The class of two-speed operators includes, in addition to the wave operator, many Lorentz invariant systems which arise in mathematical physics: Maxwell’s equations in a vacuum (wave operator form), the Pauli electron equation, the Dirac equation, the coupled Maxwell-Dirac and Klein Gordon-Dirac equations. We assume that the initial data is infinitely differentiable parallel to a foliation of \({\mathbb{R}}^ n\) by n-1 dimensional hypersurfaces and call such initial data striated. Striated initial data arise naturally in a variety of physical situations where the initial data is smooth in all but one variable. In addition to the obvious case of cartesian coordinates we mention data which is smooth in the angular variables in spherical coordinates.

### MSC:

35L70 | Second-order nonlinear hyperbolic equations |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

35A20 | Analyticity in context of PDEs |

35L67 | Shocks and singularities for hyperbolic equations |