## Construction de solutions singulières pour des équations aux dérivées partielles non linéaires. (Construction of singular solutions for nonlinear partial differential equations).(French)Zbl 0618.35074

Sémin., Équations Dériv. Partielles 1985-1986, Exposé No. 9, 15 p. (1986).
Let P be an operation of order $$m:$$ $u\to Pu=\sum_{| \alpha | =m}P_{\alpha}(x,u,...,\partial_ x^{\beta}u,...)_{| \beta | \leq m-1}\partial_ x^{\alpha}u+R(x,\partial^{\beta}u)$ where the $$P_{\alpha}$$ and R are holomorphic in their arguments for $$\kappa$$ near $$0\in {\mathbb{C}}^{n+1}$$, $$\partial^{\beta}_{\kappa}u$$ near $$u_ 0^{(\beta)}\in {\mathbb{C}}$$, $$\beta \in {\mathbb{N}}^{n+1}$$; $$| \beta | \leq m-1$$. In this paper local solutions of $$Pu=0$$ are constructed that are holomorphically ramified near a complex, smooth hypersurface S passing through the solution $$0\in {\mathbb{C}}^{n+1}$$ of $$s(x)=0$$ and that are of the form $u(x)=a(x)+\sum^{+\infty}_{k=0}b_ k(x)s^{\gamma_ k}(x),$ where $$\{\gamma_ k\}$$ is a sequence in $$R^*_+$$, strictly increasing to $$+\infty$$, the functions a and $$b_ k$$ are holomorphic in a neighborhood $$\Omega$$ of 0, and the series for u converge on $$\Omega\setminus S$$, and S is simply characteristic for $$P_{u_ n}$$ of P in a, i.e. the principal symbol $$p_ m$$ satisfies $p_ m(x,ds(x))|_ S\equiv 0,\quad \partial_{\xi}p_ m(0,ds(0))\neq 0.$
Reviewer: N.Kazarinoff

### MSC:

 35L75 Higher-order nonlinear hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A20 Analyticity in context of PDEs
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