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Singularities generated by the triple interaction of semilinear conormal waves. (English) Zbl 1467.35010

Summary: We study the local propagation of conormal singularities for solutions of semilinear wave equations \(\square u=P(y,u)\), where \(P(y,u)\) is a polynomial of degree \(N\geq 3\) in \(u\) with \(C^\infty(\mathbb{R}^3_y)\) coefficients. We know from the work of Melrose and Ritter and Bony that if \(u\) is conormal to three waves which intersect transversally at point \(q\), then after the triple interaction \(u(y)\) is a conormal distribution with respect to the three waves and the characteristic cone \(\mathcal{Q}\) with vertex at \(q\). We compute the principal symbol of \(u\) at the cone and away from the hypersurfaces. We show that if \(\partial_u^3 P(q,u(q))\neq 0\), then \(u\) is an elliptic conormal distribution.

MSC:

35A18 Wave front sets in context of PDEs
35A21 Singularity in context of PDEs
35L15 Initial value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
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