## Asymptotic solutions of hyperbolic boundary value problems with diffraction.(English)Zbl 0601.35113

Advances in microlocal analysis, Proc, NATO Adv. Study Inst., Castelvecchio-Pascoli (Lucca)/Italy 1985, NATO ASI Ser., Ser. C 168, 165-202 (1986).
[For the entire collection see Zbl 0583.00014.]
This paper concerns the propagation of analytic singularities in boundary value problems. The author is mainly interested in the problem $(1)\quad (-\Delta +D^ 2_ t)u\in A({\mathbb{R}}\times \Omega);\quad u|_{{\mathbb{R}}\times \partial \Omega}\in A({\mathbb{R}}\times \partial \Omega),$ where $$\Omega$$ is an open subset of $${\mathbb{R}}^{n-1}$$ with analytic boundary. It is known that the problem (1) can be reduced locally to $(2)\quad Pu\in A(M),\quad u|_{\partial M}\in A(\partial M)$ where $$P(x,D)=D^ 2_{x_ n}+R(x,D_ x)$$, $$M=u'\times [0,a)$$, $$a>0$$ and u’ is an open neighbourhood of 0 in $${\mathbb{R}}^{n-1}$$. If $$p(x,\xi)=\xi^ 2_ n+r(x,\xi ')$$ denotes the principal symbol of P, $$r_ 0(x',\xi ')=r(x',0,\xi ')$$, the author considers (2) in the so called diffractive region $$G_+$$, where $$r_ 0(x',\xi ')=0$$. $$(\partial /\partial x_ n)r(x',0,\xi ')<0$$. The author constructs asymptotic solutions to the diffractive problem which are singular on the bicharacteristics of p but do not propagate singularities at the boundary.
Reviewer: M.Schneider

### MSC:

 35S05 Pseudodifferential operators as generalizations of partial differential operators 35C20 Asymptotic expansions of solutions to PDEs 58J45 Hyperbolic equations on manifolds 47Gxx Integral, integro-differential, and pseudodifferential operators

Zbl 0583.00014