Interaction of progressing waves through a nonlinear potential. (English) Zbl 0554.35085

Sémin. Goulaouic-Meyer-Schwartz 1983-1984, Équat. dériv. part., Exposé No. 12, 13 p. (1984).
In this paper, a wave equation with nonlinear potential \(Pu=f(t,x,u)\) in \(\Omega \subset {\mathbb{R}}_ t\times {\mathbb{R}}^ n_ x\) was considered, where f is a \(C^{\infty}\)-function in all variables and P is a strictly linear t-hyperbolic differential operator of second order. The local existence of progressing waves was given in Section 1. In Section 2 the simplest case of a single wave was considered where no interaction occurs. The regularity properties of the solution were discussed and more specific results were obtained. The conormal rings were presented in Section 3, and the P-completeness was discussed in Section 4. The interaction of two waves was considered in Section 5, while that of three waves was considered in Section 6. In both cases new singularities are produced in the solutions, but for two waves these singularities do not propagate.
Reviewer: L.Y.Shih


35L70 Second-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
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