We begin a study of how singularities spread, due to interactions at the boundary, in mixed problems for semilinear strictly hyperbolic equations on domains \(\Omega \subset R^ n\). For second-order problems with Dirichlet conditions, solutions \(u\in H^ s_{loc}({\bar \Omega})\), \(s>(n+1)/2\), can have anomalous singularities of strength at most \(\sim 2s-n/2\). This was proved by F. David and the author [Am. J. Math. 109, 1087-1109 (1987; Zbl 0659.35068)] by an argument combining an \(H^ s\) propagation theorem for linear equations with an \(H^ s\) microlocal algebra lemma. When the high-order regime \((H^ r,r>2s-n/2)\) is considered, it becomes clear that to answer even the simplest questions about spreading at the boundary, such \(H^ s\) results are inadequate.
A theorem describing propagation of microlocal \(H^{s,s'}\) regularity along generalized bicharacteristics for linear equations is needed (so that one can “propagate in the second index”), together with appropriate \(H^{s,s'}\) microlocal algebra lemmas. We prove such \(H^{s,s'}\) results here, and give three applications to semilinear problems. For solutions whose strongest incoming singularities are confined to proper cones, we prove a high-order regularity propagation theorem. This enables one to identify regions into which singularities arising from interactions at the boundary cannot spread. We also prove a refinement of the 2s theorem of the paper cited above.
Finally, we discuss examples of second-order problems in which anomalous singularities of strength \(\sim 2s-n/2\) arise due to crossing at the boundary. Here the role of the \(H^{s,s'}\) results in detecting the singularities is emphasized. These examples show that for second-order mixed problems, the analogue of Beals’ 3s theorem does not hold for reflection at the boundary: although \(H^ r\) regularity for \(r<\sim 3s- n\) propagates in free space, for \(r>\sim 2s-n/2\) it does not in general reflect.