On almost global existence and local well-posedness for some 3-D quasi-linear wave equations

We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time $L^2$ estimate for the variable coefficient wave equation plays a central role for this purpose. Assuming radial symmetry, we establish the almost global existence of a strong solution for every small initial data in $H^2 \times H^1$. We also show that the initial value problem is locally well-posed.