Interaction of Singularities and Propagation into Shadow Regions in Semilinear Boundary Problems

Abstract

We consider solutions, u ∈ H ^s_loc (Ω̄), s > (dim Ω + 1)/2, satisfying

$$ Pu = f(t,z,u) $$

((1.1))

on Ω = R _t × ω, where ω ⊂ R ⁿ _z is an open set with smooth boundary; P is a second-order differential operator with C ^∞ coefficients on R ⁿ⁺¹_{(t, z)} , noncharacteristic with respect to bΩ and strictly hyperbolic with respect to the planes t = c; and f ∈ C ^∞.Interactions between singularity-bearing bicharacteristics taking place in the interior of Ω in t > t̄ can produce anomalous singularities in u, that is, singularities not present in the function u satisfying Pu = 0, u│ _{b Ω} = u│ _{b Ω} in t < t̄ These singularities can have strength at most ~ 3s - n (Beals [1]) and are generated by processes, crossing and self-spreading, that have been well-understood for some time (Beals [2], [3]). In this paper we shall describe propagation and interaction at the boundary, where generalized bicharacteristics [6], which typically contain segments of reflecting, grazing, or gliding rays, carry singularities.

Supported by NSF Grant DMS-8701654 and an Alfred P. Sloan Research Fellowship.