  We consider the local well-posedness for 3-D quadratic semi-linear wave
equations with radial data: $\Box u = a |\partial_t u|^2+b|\nabla_x u|^2$,
$u(0,x)=u_0(x)\in H^{s}_{\mathrm{rad}}$, $\partial_t u(0,x)=u_1(x)\in
H^{s-1}_{\mathrm{rad}}$. It has been known that the problem is well-posed for
$s\ge 2$ and ill-posed for $s<3/2$. In this paper, we prove unconditional
well-posedness up to the scaling invariant regularity, that is to say, for
$s>3/2$ and thus fill the gap which was left open for many years. For the
purpose, we also obtain a weighted fractional chain rule, which is of
independent interest. Our method here also works for a class of nonlinear wave
equations with general power type nonlinearities which contain the space-time
derivatives of the unknown functions. In particular, we prove the Glassey
conjecture in the radial case, with minimal regularity assumption.
