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Interactions totalement non linéaires. (Totally nonlinear interactions). (French) Zbl 0654.35064
Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1987, Exp. No. 12, 8 p. (1987).
The author describes the singularities of a sufficiently smooth real solution of the equation \(F(x,u,\partial^{\alpha})=0\), \(| \alpha | \leq m\), on an open set \(\Omega\) of R n, where F is a \(C^{\infty}\) function of its arguments. It is assumed that \(p_ m=\sum_{| \alpha | =m}\partial F/\partial u_{\alpha}\cdot \xi^{\alpha}\) is strictly hyperbolic with respect to \(\xi_ 1\) and that every bicharacteristic starting from \((x,\xi)\in T\quad *\Omega \cap (x_ 1>0)\) such that \(p_ m(x,\xi)=0\), meets \(T\quad *\Omega \cap (x_ 1<0).\) One tries to obtain singularities of linear type or to control the birth of singularities which are the offsprings of nonlinear interactions. The four parts of the paper are: result in dimension 2 or to first order, paradifferential calculus made precise, iterate action of not very regular vector fields, and Cauchy problem with singular data in one point.
Reviewer: R.Vaillancourt
MSC:
35L67 Shocks and singularities for hyperbolic equations
58J40 Pseudodifferential and Fourier integral operators on manifolds
35S05 Pseudodifferential operators as generalizations of partial differential operators
35Q99 Partial differential equations of mathematical physics and other areas of application
58J47 Propagation of singularities; initial value problems on manifolds
58J45 Hyperbolic equations on manifolds
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