Vol. 14, No. 1, 2021

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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
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 3 in u with C(y3) 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 u3P(q,u(q))0, then u is an elliptic conormal distribution.

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