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

Abstract

In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is ${\dot{H}}^{1/2}\times {\dot{H}}^{-1/2}$. We show that if the initial data is radial and lies in $({\dot{H}}^s\times {\dot{H}}^{s-1})\cap ({\dot{H}}^{1/2}\times {\dot{H}}^{-1/2})$ for some $s>\frac{1}{2}$, 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].

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