Interaction des singularités pour les équations de Klein-Gordon non linéaires. (Interaction of singularities for nonlinear Klein-Gordon equations). (French) Zbl 0555.35118

Bony, J. M.

Sémin. Goulaouic-Meyer-Schwartz 1983-1984, Équat. dériv. part., Exposé No. 10, 27 p. (1984).

The author considers the nonlinear Klein-Gordon equation \(u_{tt}- u_{xx}-u_{yy}=f(t,x,y,u),\) the problem is to determine the regularity of solutions u for \(t>0\) when the singularities of u for \(t<0\) are given. The cases when the singularities are located in a finite number of hypersurfaces \(\Sigma_ j\) (i.e., u belongs to \(H^ s\) near \(\Sigma_ j\), and to \(H^{s+k}\) otherwise) are studied in some details. To study the propagation of singularities of solutions the author uses a kind of symbolic calculus, called the operator 2-micro-differentials which is a vector field singular at the origin and tangent to the surfaces bearing the initial singularities.

Reviewer: G.-Z.Tu

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35G25	Initial value problems for nonlinear higher-order PDEs
35A20	Analyticity in context of PDEs

Keywords:

nonlinear Klein-Gordon equation; regularity; singularities; propagation of singularities; symbolic calculus; operator 2-micro-differentials