Global well-posedness for the radial defocusing cubic wave equation on $R^3$ and for rough data - Tristan Roy (Electronic Journal of Differential Equations, 2007)
We prove global well-posedness for the radial defocusing cubic wave equation $$displaylines{ partial_{tt} u - Delta u = -u^{3} cr u(0,x) = u_{0}(x) cr partial_{t} u(0,x) = u_{1}(x) }$$ with data $(u_0, u_1) in H^{s} imes H^{s-1}$, $1 > s >7/10$. The proof relies upon a Morawetz-Strauss-type inequality that allows us to control the growth of an almost conserved quantity.