Singularities generated by the triple interaction of semilinear conormal waves

TL;DR

This paper investigates how conormal singularities propagate in solutions to semilinear wave equations with polynomial nonlinearities, especially focusing on the effects of triple wave interactions and the resulting distribution properties.

Contribution

It provides a detailed analysis of the propagation of conormal singularities in semilinear wave equations, including computation of principal symbols and conditions for ellipticity after triple interactions.

Findings

Conormal singularities persist after triple wave interactions.

The principal symbol of the solution can be explicitly computed.

Ellipticity of the solution depends on the third derivative of the polynomial nonlinear term.

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 $N \geq 3$ in $u$ with $C^\infty(\mathbb{R}^3_y)$ 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 $\partial_u^3 P (q, u(q)) \neq 0$, $u$ is an ellipitic conormal distribution.