Reflection of transversal progressing waves in nonlinear strictly hyperbolic mixed problems.

*(English)*Zbl 0633.35051The authors consider the reflection of regularity for a semilinear hyperbolic boundary value problem as follows:
\[
P(y,D_ y)u=f(y,u,...,\nabla^{m-2}u)\quad in\quad \{y>0\},\quad B_ j(y,D_ y)u=0\quad on\quad \{y_ n=0\}\quad (j-1,2,...,\mu),
\]
where P is a linear strictly hyperbolic operator of order m with smooth coefficients and f is a smooth function of the variables (y,u,...). Assume that the boundary \(\{y_ n=0\}\) is not characteristic for \(P(y,D_ y)\), and that the boundary operators \(\{B_ j:\) \(j=1,2,...,\mu \}\) satisfy the uniform Lopatinski condition. Moreover, assume that there are N characteristic surfaces \(\Sigma_ 1,...,\Sigma_ N\) which intersect the boundary transversally along a manifold \(\Delta\). They consider the above problem in a small neighbourhood of the manifold \(\Delta\). The principal theorem of this paper is as follows: If the solution is conormal with respect to the characteristic surfaces in the past, then it is also conormal with respect to the union of these surfaces \(\{\Sigma_ n\}_{n=1,...,N}\). This is a continuation of their work [Duke Math. J. 53, 125-137 (1986; Zbl 0613.35050)] where they have treated the case \(N=2\).

Reviewer: M.Tsuji

##### MSC:

35L70 | Second-order nonlinear hyperbolic equations |

35L35 | Initial-boundary value problems for higher-order hyperbolic equations |

35L67 | Shocks and singularities for hyperbolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |