ON EXISTENCE AND SCATTERING WITH MINIMAL REGULARITY FOR SEMILINEAR WAVE-EQUATIONS

We prove existence and scattering results for semilinear wave equation s with low regularity data. We also determine the minimal regularity t hat is needed to ensure local existence and well-posedness, and we giv e counterexamples to well-posedness. More specifically, we show that e quations of the type square u=\ u \(p), with initial data (u, u(t)) in H-7(R(n))x H-y-1(R(n)), have a local solution if y greater than or eq ual to y(p, n), and we construct counterexamples if y < y(p, n). The e xistence results rely on mixed-norm space-time estimates of Strichartz -type. (C) 1995 Academic Press, Inc.