ON THE WAVE EQUATION WITH QUADRATIC NONLINEARITIES IN THREE SPACE DIMENSIONS

The Cauchy problem for the nonlinear wave equation square u = (partial derivative u)(2), u(0) = u(0), u(t)(0) = u(1) in three space dimensions is considered. The data (u(0), u(1)) are assumed to belong to (H) over cap (r)(s)(R-3) x (H) over cap (r)(s) (1)(R-3), where (H) over cap (r)(s) is defined by the norm parallel to f parallel to((H) over cap sr) := parallel to <xi >(s) (f) over cap parallel to L-xi(r)', <xi > = (1 + vertical bar xi vertical bar(2))(1/2), 1/r + 1/r' = 1. Local well-posedness is shown in the parameter range 2 >= r > 1, s > 1 + 2/r. For r = 2 this coincides with the result of Ponce and Sideris, which is optimal on the H-s-scale by Lindblad's counterexamples, but nonetheless leaves a gap of 1/2 derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for square u = partial derivative u(2) are obtained, too.