Singularities generated by the triple interaction of semilinear conormal waves

Sá Barreto, Antônio; Wang, Yiran

doi:10.2140/apde.2021.14.135

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Singularities generated by the triple interaction of semilinear conormal waves

Antônio Sá Barreto and Yiran Wang

Vol. 14 (2021), No. 1, 135–170

DOI: 10.2140/apde.2021.14.135

Abstract

We study the local propagation of conormal singularities for solutions of semilinear wave equations $□ u = P (y, u)$ , where $P (y, u)$ is a polynomial of degree $N \geq 3$ in $u$ with $C^{\infty} (ℝ_{y}^{3})$ 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 $𝒬$ 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.

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Keywords

nonlinear wave equations, propagation of singularities, wave front sets, conformal distributions

Mathematical Subject Classification

Primary: 35A18, 35A21, 35L70

Milestones

Received: 26 October 2018

Revised: 31 July 2019

Accepted: 7 October 2019

Published: 19 February 2021

Authors

Antônio Sá Barreto
Department of Mathematics Purdue University West Lafayette, IN United States

Yiran Wang
Department of Mathematics Emory University Atlanta, GA United States

Vol. 14, No. 1, 2021