## 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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