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Singular solutions for semilinear hyperbolic equations. I. (English) Zbl 0629.35087

For \(x=(x_ 1,...,x_ n)\in R^ n\) the authors consider a semi-linear equation \[ (1)\quad P(u):=L(x,D)u+\sum_{\mu \in F}b_{\mu}(x)(X^ Bu)^{\mu}=0. \] Here L is a strictly hyperbolic linear partial differential operator of order m; B is a set of multi-indices \(\{\beta \in Z^ n_+:| \beta | \leq m-1\}\); \((D^ Bu)^{\mu}:=\prod_{\beta \in B}(D^{\beta}u)^{\mu_{\beta}}\) for a multi-index \(\mu =(...,\mu_{\beta},...)_{\beta \in B}\); F is a finite set of \(\mu\) with \(| \mu | \geq 2\); and the coefficients of L and \(b_{\nu}\) belong to \(C^{\infty}(\Omega)\), where \(\Omega\) is some neighborhood of \(x=0.\)
The main result asserts that for a generic noncharacteristic surface S given by the equation \(\phi (x)=0\) it is possible to construct solutions u(x)\(\in {\mathcal D}'(\Omega)\) of (1) and \(\bar u{}_ N(x)\in {\mathcal D}'(\Omega)\), \(N\in Z_+\) of the form \[ \bar u_ N(x)=(\phi (x)+i0)^{\sigma_ 0}\sum^{N}_{k=0}\nu_ k(x)(\phi (x)+i0)^{k/r} \] with \(\nu_ k\in C^{\infty}(\Omega)\) such that \(u-u_ N\in C^{\ell}(\Omega)\), \(\ell \in Z_+\). The exponents \(\sigma_ 0\) and r are determined by the equation (1) and the singular support of u is contained in S.
Reviewer: V.Durikovic

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
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