Propagation of Singularities for Semilinear Hyperbolic Equations in One Space Variable

Abstract

In this paper we investigate the propagation of singularities for general semi-linear hyperbolic equations in two variables. In order to make clear the motivation for this work, we give a little history. In [91, Reed showed that in one space dimension solutions ju = f (u, Du) are C~o except at points Kx, t> where the backward characteristics through the point intersect singular points of the initial data at t = 0; thus, as in the linear case, the singularities lie on rays issuing from singularities at t = 0. The counterexample of Lascar [61 and the theorems of Rauch [71 show that when the number of space dimensions is greater than one, the solutions u of Elu = f (u) may have other singularities too. Roughly what is true is the following. Suppose that the solution u is in Hk and that the backward characteristics through Kx, t> do not intersect the singular support of the initial data. Then u will be in HP at Kx, t> where p > k depends on k and the number of space dimensions. Such results are valid for higher order equations and more strongly nonlinear equations. In this paper we reexamine the result of Reed to understand what is special about one space dimension. First, we prove that the C~o result holds in one space dimension for all semilinear hyperbolic equations of second order. Second, we construct a third order semi-linear counterexample in which the propagation of singularities is not that predicted by the highest order linear part. A new singularity is created when two singularities cross and the new singularity propagates in the direction of the third characteristic. Our example makes it clear that the same phenomenon will also occur for higher order equations. Thus, we have the following general picture: the only semi-linear hyperbolic equations of order >2 for which the propagation of singularities is like the linear case are second order