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Propagation et interaction des singularités pour les solutions des équations aux dérivées partielles non-linéaires. (French) Zbl 0573.35014
Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 1133-1147 (1984).
[For the entire collection see Zbl 0553.00001.]
The article gives a survey on results on microlocal analysis for general partial differential operators of the type \[ (1)\quad F(x,c(x),u(x),...,\partial^{\beta}u(x),...)_{| \beta | \leq m}=0, \] \(x\in {\mathbb{R}}^ n\), c(x): given control function, c not necessary an element of \(C^{\infty}\), \(F\in {\mathbb{C}}^{\infty}\); in particular the known results on the propagation of singularities from the starting point in 1978 until 1983 are presented.
In the first part general microlocal regularity theorems for a solution \(u\in H^ s\) of (1) are given, investigating the question if u belongs to \(H^{s'}\) for some \(s'>s\) at least in noncharacteristic points \((x_ 0,\xi_ 0)\in {\mathbb{R}}^ n\times {\mathbb{R}}^ n\setminus \{0\}\) and depending on the regularity of c.
In the second part the calculus of paramultiplication, developed by the author, is discussed and applied to linearize (1), implying that functions belonging to \(H^ s\) and microlocally to \(H^{s'}\) build an algebra.
In the third part the interaction of singularities is studied, here distinguishing the cases \(n=1\) respectively \(n>1\) (space variable dimension); also the Cauchy problem is discussed. Some conjectures are formulated. No proofs are given. The relevant papers of M. Beals, the author, B. Lascar, M. Reed and J. Rauch are quoted in the references.
Reviewer: R.Racke

MSC:
35G20 Nonlinear higher-order PDEs
35A20 Analyticity in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators