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 $\square u=P\left(y,u\right)$, where $P\left(y,u\right)$ is a polynomial of degree $N\ge 3$ in $u$ with $C^{\infty }\left({ℝ}_y^3\right)$ 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 $u\left(y\right)$ 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 ${\partial }_u^{3}P\left(q,u\left(q\right)\right)\ne 0$, then $u$ is an elliptic conormal distribution.