Spreading of singularities at the boundary in semilinear hyperbolic mixed problems. II. Crossing and self-spreading

Abstract:

The creation of anomalous singularities in solutions to nonlinear hyperbolic equations due to crossing or self-spreading in free space is by now rather well understood. In this paper we study how anomalous singularities are produced in mixed problems for semilinear wave equations $\square u = f(u)$ on the half-space ${\mathbf {R}}_ + ^{n + 1},u \in H_{{\operatorname {loc}}}^s,s > (n + 1)/2$, due to crossing and self-spreading at boundary points. Several phenomena appear in the problems considered here which distinguish spreading at the boundary from spreading in free space: (1) Anomalous singularities of strength $\sim 2s - n/2$ can arise when incoming singularity-bearing rays cross or self-spread at a point on the boundary. A consequence of this, announced in [14], is that the analogue of Beals’s $3s$ theorem fails for reflection in second-order mixed problems. Although ${H^r}$ regularity for $r < \sim 3s -n$ propagates along null bicharacteristics in free space, for $r < \sim 2s -n/2$ it does not in general reflect. (2) For nonlinear wave equations in free space, anomalous singular support is never produced by the interaction of fewer than three bicharacteristics, unless self-spreading occurs. However, anomalous singularities can arise when a pair of rays cross at a boundary point. (3) Suppose $\square u = {u^2}$ and $u \in {C^\infty }$ on the boundary. For certain choices of initial data, anomalous singularities of strength $\sim 2s - n/2$ arise at the boundary from three sources: interactions of incoming rays with incoming rays, incoming rays with reflected rays, and reflected rays with reflected rays. Singularities produced by the incoming-reflected interactions differ in sign from and are strictly weaker than the other two types, so some cancellations occur. As the incoming rays approach being gliding rays, the difference in strength decreases and hence the cancellations become increasingly significant.