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


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