Global well - posedness for the defocusing, cubic, nonlinear wave equation in three dimensions for radial initial in ${\dot{H}}^s\times {\dot{H}}^{s-1}$, $s>\frac{1}{2}$

In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$\mathrm{.}Weshowthatiftheinitialdataisradialandliesin$(\dot{H}^{s} \times \dot{H}^{s - 1}) \cap (\dot{H}^{1/2} \times \dot{H}^{-1/2})$forsome$s > \frac{1}{2}$, then the cubic initial value problem is globally well - posed. We use the I - method and the long time Strichartz estimates. This method is quite similar to the method used in [D2].