Solutions to semilinear wave equations of very low regularity

Abstract

This paper finds solutions to semilinear wave equations with strongly anomalous propagation of singularities. For very low Sobolev regularity we obtain solutions whose singular support propagates along any ray inside or outside the light cone. In one dimension these solutions exist for any Sobolev exponent $s<\frac{1}{2}$ in space, while classical results show that the singular support of solutions with higher regularity is contained in the light cone. The spatial Fourier transform of these anomalous solutions is supported in a half-line. We obtain wellposedness results in such function spaces when the problem is ill-posed for Sobolev data without the support condition and, in some cases, obtain wellposedness below $L^2(\mathbb{R})$. The results are based on new multiplier theorems for Sobolev spaces satisfying the support condition. Extensions to higher space dimensions are given.