Singularities generated by the triple interaction of semilinear conormal waves

Abstract

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 $N\ge 3$ in $u$ with $C^{\infty }({\mathbb{R}}_y^3)$ coefficients. We know from the work of Melrose & 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 $Q$ with vertex at $q$. We compute the principal symbol of $u$ at the cone and away from the hypersurfaces. We show that if ${\partial }_u^{3}P(q,u(q))\ne 0$, $u$ is an ellipitic conormal distribution.

To appear in Analysis&PDE