  In this paper, we prove a sharp local well-posedness result for spherically
symmetric solutions to quasilinear wave equations with rough initial data, when
the spatial dimension is three or higher. Our approach is based on Morawetz
type local energy estimates with fractional regularity for linear wave
equations with variable $C^1$ coefficients, which rely on multiplier method,
weighted Littlewood-Paley theory, duality and interpolation. Together with
weighted linear and nonlinear estimates (including weighted trace estimates,
Hardy's inequality, fractional chain rule and fractional Leibniz rule) which
are adapted for the problem, the well-posed result is proved by iteration. In
addition, our argument yields almost global existence for $n=3$ and global
existence for $n\ge 4$, when the initial data are small, spherically symmetric
with almost critical Sobolev regularity.
