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Reflection of transversal progressing waves in nonlinear strictly hyperbolic mixed problems. (English) Zbl 0633.35051
The 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
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