Propagation, interaction and reflection of discontinuous progressing waves for semilinear hyperbolic systems.

*(English)*Zbl 0687.35021The author considers the interaction and reflection of discontinuities of solutions for semilinear strictly hyperbolic systems of first order as follows:
\[
L(x,D_ x)u=\sum^{n}_{i=0}A_ i(x)\partial u/\partial x_ i=f(x,u).
\]
The discontinuities of solutions are always treated here in “conormal” sense. Therefore the results are expressed in a term of the “propagation of conormal smoothness”. For the interaction of discontinuities, the author assumes that the characteristic surfaces intersect transversally along a spacelike manifold. For the reflection of discontinuities, he adds the assumptions that the boundary is not characteristic for \(L(x,D_ x)\) and that the boundary conditions satisfy the uniform Lopatinski condition. Roughly speaking, the principal result of this paper can be written as follows: even if the equations may contain the nonlinear term f(x,u) as the above, the phenomena of the interaction and reflection of discontinuities are locally similar to the ones of typical “linear” hyperbolic problems. Next the author shows that the local results can be extended to a suitable domain of determinacy.

Reviewer: M.Tsuji

##### MSC:

35B99 | Qualitative properties of solutions to partial differential equations |

35L60 | First-order nonlinear hyperbolic equations |

58J47 | Propagation of singularities; initial value problems on manifolds |

35L50 | Initial-boundary value problems for first-order hyperbolic systems |

35L67 | Shocks and singularities for hyperbolic equations |